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1,370	# Solve by completing the square Solve by completing the square could take a little bit more time to do than solving by factoring. However, the steps are straightforward. Before showing examples, you need to understand what a perfect square trinomial is Binomial × same Binomial = Perfect square trinomial (x + 4) × (x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 (x + a) × (x + a) = x2 + ax + ax + a2 = x2 + 2ax + a2 Important observation: What is the relationship between the coefficient of the second term and the last term? For x2 + 8x + 16, coefficient of the second term is 8 and the last term is 16 (8/2)2 = 42 = 16 For, x2 + 2ax + a2, the coefficient of the second term is 2a and the last term is a2 (2a/2)2 = a2 So, what is the relationship? The last term is obtained by dividing the coefficient of the second term by 2 and squaring the result Now, let's say you have x2 + 20x and you want to find the last term to make the whole thing a perfect square trinomial Just do (20/2)2 = 102. The perfect square trinomial is x2 + 20x + 102= (x + 10) × (x + 10) Example #1: Solve by completing the square x2 + 6x + 8 = 0 x2 + 6x + 8 = 0 Subtract 8 from both sides of the equation. x2 + 6x + 8 - 8 = 0 - 8 x2 + 6x = - 8 To complete the square, always do the following 24 hours a day 365 days a year. It is never going to change when you solve by completing the square! You are basically looking for a term to add to x2 + 6x that will make it a perfect square trinomial. To this end, get the coefficient of the second term, divide it by 2 and raise it to the second power. The second term is 6x and the coefficient is 6. 6/2 = 3 and after squaring 3, we get 32 x2 + 6x = - 8 Add 32 to both sides of the equation above x2 + 6x + 32 = - 8 + 32 x2 + 6x + 32 = - 8 + 32 (x + 3)2 = -8 + 9 (x + 3)2 = 1 Take the square root of both sides √((x + 3)2) = √(1) x + 3 = ±1 When x + 3 = 1, x = -2 When x + 3 = -1, x = -4 Example #2: Solve by completing the square x2 + -6x + 8 = 0 instead of x2 + 6x + 8 = 0 The second term this time is -6x and the coefficient is -6. -6/2 = -3 and after squaring -3, we get (-3)2 = 9 x2 + -6x = - 8 Add (-3)2 to both sides of the equation above x2 + -6x + (-3)2 = - 8 + (-3)2 (x + -3)2 = -8 + 9 (x + -3)2 = 1 Take the square root of both sides √((x + -3)2) = √(1) x + -3 = ±1 When x + -3 = 1, x = 4 When x + -3 = -1, x = 2 Example #3: Solve by completing the square 3x2 + 8x + -3 = 0 3x2 + 8x + -3 = 0 Divide everything by 3. Always do that when the coefficient of the first term is not 1 (3/3)x2+ (8/3)x + -3/3 = 0/3 x2+ (8/3)x + -1 = 0 Add 1 to both sides of the equation. x2 + (8/3)x + -1 + 1 = 0 + 1 x2 + (8/3)x = 1 The second term is (8/3)x and the coefficient is 8/3. 8/3 ÷ 2 = 8/3 × 1/2 = 8/6 and after squaring 8/6, we get (8/6)2 x2 + (8/3)x = 1 Add (8/6)2 to both sides of the equation above x2 + (8/3)x + (8/6)2 = 1 + (8/6)2 (x + 8/6)2 = 1 + 64/36 (x + 8/6)2 = 36/36 + 64/36 = (36 + 64)/36 = 100/36 Take the square root of both sides √((x + 8/6)2) = √(100/36) x + 8/6 = ±10/6 x + 8/6 = 10/6 x = 10/6 - 8/6 = 2/6 = 1/3 x + 8/6 = - 10/6 x = -10/6 - 8/6 = -18/6 = -3 To solve by completing the square can become quickly hard as shwon in example #3 ## Recent Articles 1. ### Average age word problem Aug 09, 16 01:40 PM Average age of Dipu and Apu is 22 years. Average age of Dipu and Tipu is 24 years. Age of Dipu is 21 years. What are the ages of Apu, and tipu ? Let<\|endoftext\|>	4.71875
1,459	## Motion in One Dimension- Before you go through this article, make sure that you have gone through the previous article on Motion in One Dimension. We have discussed- • In one-dimensional motion, particle moves along a straight line. • It is also known as rectilinear motion. Various parameters related to motion are- ## Speed- Rate of change of distance of a particle is called as its speed. OR Distance covered by a particle per unit time is called as its speed. Mathematically, ### Example- Consider- • A particle moves in a circular path of radius 7 m. • It starts its journey from point A and ends at the same point in time 11 seconds. Then, Speed of the particle = Distance covered by the particle during its motion / Time taken = Circumference of the circle / Time taken = (2 x 22/7 x 7) / 11 = 4 m/s ### Characteristics- The characteristics of speed are- • It is a scalar quantity. • Speed is always positive. It can never be negative. • Speed can increase or decrease with time. • Speed is never zero for a moving particle. • The SI unit of speed is meter per second (m/s). • The dimensional formula of speed is [M0L1T-1]. ## Types of Speeds- There are mainly following four types of speeds- 1. Uniform Speed 2. Variable Speed 3. Average Speed 4. Instantaneous Speed ### 1. Uniform Speed- A particle is said to be moving with uniform speed if it covers equal distances in equal intervals of time. OR A particle is said to be moving with uniform speed if it moves continuously with constant speed. It is worth remembering- The direction of motion of the particle may change while moving with constant speed. ### 2. Variable Speed- A particle is said to be moving with variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time. ### 3. Average Speed- The average speed of a particle is that constant speed with which the particle covers the same distance in a given time as it does while moving with variable speed during the given time. Mathematically, It is the ratio of total distance travelled by the particle to the total time taken in which the distance is travelled. If Δx is the distance travelled by particle in time Δt, then average speed is given by- ### 4. Instantaneous Speed- Instantaneous speed is the speed of particle at a particular instant of time. Mathematically, Instantaneous speed is the limiting value of average speed as the time interval becomes infinitesimally small. If Δx is the distance covered by a particle in the time interval Δt, then- ## Velocity- Rate of change of displacement of a particle is called as its velocity. OR Rate of change of position vector of a particle is called as its velocity. Mathematically, Its direction is same as that of displacement. ### Example- Consider- • A particle moves in a circular path of radius 7 m. • It starts its journey from point A and ends at the same point in time 11 seconds. Velocity of the particle = Displacement of the particle / Time taken = 0 / 11 = 0 m/s ### Characteristics- The characteristics of velocity are- • It is a vector quantity. • Velocity can be positive, negative or zero. • Velocity can increase or decrease with time. • Velocity may be zero for a moving particle. • The SI unit of velocity is meter per second (m/s). • The dimensional formula of velocity is [M0L1T-1]. ## Types of Velocities- There are mainly following four types of velocities- 1. Uniform Velocity 2. Variable Velocity 3. Average Velocity 4. Instantaneous Velocity ### 1. Uniform Velocity- A particle is said to be moving with uniform velocity if it covers equal displacements in equal intervals of time. OR A particle is said to be moving with uniform velocity if it moves continuously in the same direction with constant speed. It is worth remembering- ### 2. Variable Velocity- A particle is said to be moving with variable velocity if it covers equal displacements in unequal intervals of time or unequal displacements in equal intervals of time. OR A particle is said to be moving with variable velocity if either magnitude of velocity or direction or both changes during its motion. ### 3. Average Velocity- The average velocity of a particle is that constant velocity with which the particle undergoes same displacement in a given time as it undergoes while moving with variable velocity during the given time. Mathematically, It is the ratio of total displacement of the particle to the total time interval in which the displacement occurs. Consider- • At any time t1, the position vector of the particle is $\overrightarrow{r1}$. • At time t2, the position vector is $\overrightarrow{r2}$. Then for this interval, average velocity is given by- ### 4. Instantaneous Velocity- Instantaneous speed is the velocity of particle at a particular instant of time. Mathematically, Instantaneous velocity is the limiting value of average velocity as the time interval becomes infinitesimally small. If Δr is the displacement of particle in the time interval Δt, then- ## Difference Between Speed And Velocity- Some important differences between speed and velocity are- Speed Velocity Rate of change of distance of a particle is called as its speed. Rate of change of displacement of a particle is called as its velocity. It is a scalar quantity. It is a vector quantity. Speed is either positive or zero. It can never be negative. Velocity can be positive, negative or zero. Speed is never zero for a moving particle. Velocity can be zero for a moving particle. Speed tells nothing about the direction of motion of the particle. Velocity tell the direction of motion of the body. ## Important Points- It is important to note the following points- ### Point-01: According to convention, • If the particle moves upwards or towards right, its velocity is taken positive. • If the particle moves downwards or towards left, its velocity is taken negative. ### Point-02: Velocity tell the direction of motion of the particle as- • If the velocity is positive, particle must be moving towards positive direction. • If the velocity is negative, particle must be moving towards negative direction. ### Point-03: • If motion takes place in the same direction, then average speed and average velocity are same. • This is because distance and displacement are then same. ### Point-04: • The magnitude of instantaneous velocity is always instantaneous speed. • Instantaneous speed and instantaneous velocity differs only by direction. • However, magnitude of average velocity is not always average speed. • Speedometer of an automobile measures instantaneous speed of the automobile. To gain better understanding about Speed and Velocity, Watch this Video Lecture Next Article- Acceleration and Retardation Get more notes and other study material of Class 11 Physics. Get more notes and other study material of Class 12 Physics.<\|endoftext\|>	4.46875
793	What is Down syndrome? Down syndrome is a chromosomal condition caused by extra genetic material. Typically, our cells contain 23 pairs of chromosomes. In the case of Down syndrome, a person has an extra full or partial copy of chromosome 21. This extra genetic material changes how the body and brain develop. People with Down syndrome have a few common physical traits, but each individual is unique and can lead a healthy active life. We don’t know for sure why Down syndrome happens. Even though it’s a genetic condition, the majority of the cases are not passed on from the parents or family to the baby. Most cases happen because in the early stages of the baby’s development, there is a problem when the cells are dividing. However, there are some factors that may increase the risk of having a baby with Down syndrome, such as: - Mother’s age. The risk of Down syndrome increases with the mom’s age. Even though the risk is greater as your age increases, about 80 percent of babies with Down syndrome are born to women age 35 or less. This is because younger women have more babies than older women. - Having had a baby with Down syndrome. Up to age 40, for each pregnancy your chances of having another baby with Down syndrome is about 1 in 100 (1 percent). After age 40, the risk is based on your age. Talk to a genetic counselor to understand your risk of having another baby with Down syndrome. - Being a carrier of a genetic translocation. Both, men and women, can pass a genetic translocation to their baby. These cases are not very common. If you had a baby with Down syndrome before or if you or your partner have a family history of Down syndrome, it’s best to talk to a genetic counselor. During pregnancy your health care provider will offer screening tests to see if your baby is more likely to have Down syndrome. These tests are offered to all pregnant women as part of regular prenatal care. However, a screening test won’t tell you for sure if your baby has Down syndrome. It only tells you if there is a higher risk. To know for sure you will need a diagnostic test. How do you know if your baby has Down syndrome? If you get an abnormal screening test result, your provider will recommend a diagnostic test. A diagnostic test will confirm if a baby has Down syndrome. There are few diagnostic tests: - Amniocentesis (also called amnio). This test checks the amniotic fluid surrounding your baby in the uterus to check for Down syndrome. You can get an amnio at 15 to 20 weeks of pregnancy. - Chorionic villus sampling (also called CVS). This test checks the tissue from the placenta to see if a baby has Down syndrome. You can get a CVS at 10 to 13 weeks of pregnancy. - Cordocentesis (also called percutaneous umbilical cord sampling or PUBS). For this test your provider inserts a thin needle into an umbilical cord vein to take a small sample of your baby’s blood to check for chromosome defects. You can get this test between 18 and 22 weeks of pregnancy. There’s a much greater risk of miscarriage with cordocentesis than with an amnio or a CVS. So you only get this test if other tests are unclear and your provider can’t confirm if your baby has Down syndrome any other way. Down syndrome is also identified at birth by physical traits like: almond-shaped eyes that slant up, low muscle tone, a single line across the center of the palm of the hand, and a flattened face. But these traits won’t tell you for sure if your baby has Down syndrome, a chromosomal test call karyotype is needed to confirm this diagnosis.<\|endoftext\|>	3.828125
201	Fulufjället is made up of sandstone with components of partially weathered and fertile diabase. The sandstone formed when the area lay at the bottom of a warm sea south of the equator 1,200 million years ago. Here and there you can see marks made by waves that have been preserved on the boulders. For millions of years sun, wind, water, cold and glaciers have weathered and sculpted Fulufjället into its current shape. The fields of boulders have been caused by frost erosion; the rock precipices and ravines were formed by brooks and rivers from the melting ice sheet. The sculpting is still going on, especially by Njupeskär, where the canyon is becoming deeper and deeper. Where brooks and rivers have weathered the bedrock, you can find stones made of greenish Särnatin guanite, a type of rock found only in north-western Dalarna.<\|endoftext\|>	3.671875
920	What is the Soil Food Web? The soil food web refers to the complex relationships between the diverse groups of fauna and flora found in soil. These groups include bacteria, fungi, protozoa, nematodes, microarthropods, and the larger plants and animals found in and around soil. The composition of each specific web is greatly influenced by biological, chemical and physical forces in the environment. The practice of soil food web health management (SFWHM) was developed by Dr. Elaine Ingham, a respected soil microbiologist. Her research over the past three decades has greatly broadened our understanding of how life in the soil affects the health of plants. The most significant finding is that plants thrive best under the specific soil food web that it had naturally evolved under over millenia. Hence, SFWHM emphasizes the cultivation of soil food webs that are specific to the plants of interest for maximum plant health. Why is it important? A healthy soil food web offers great ecological importance to all landscapes, ranging from small backyard gardens to old growth forests. Soil contains countless microorganisms continuously performing varied functions that help maintain healthy ecosystems. When we improve the biology in the soil, we are promoting vital ecosystem services that are used by society at large. Such services include: - Decomposition of organic matter - Bacteria and fungi are the major decomposers in soil. Soil animals such as mites, millipedes, earthworms and termites aid in the decomposition process by shredding the organic material and dispersing microbes throughout the soil. This community of decomposers play a crucial role in waste management and pollution control. - Nutrient cycling - When predators such as protozoa and nematodes exist in soil, their faecal excretions from consuming bacteria and fungi release nutrients that plants then absorb. These nutrients are often necessary for plants to flourish. The presence of these predators result in the elimination of any need for chemical fertilizers. - Retention of nutrients - Nutrient leaching refers to the loss of nutrients in the soil due to movement of water down through the soil. Bacteria and fungi prevent nutrient leaching by storing the nutrients in the soil in their bodies. Thus, soil that is rich in these organisms are more resistant to nutrient leaching. Conventional industrial farming relies heavily on salt-based fertilizers that not only kill bacteria and fungi, but also pollute the surrounding watershed. - The movement of gases and water into and through the soil is important for life within the soil. This is achieved primarily through bioturbation, the process of creating channels within the soil through the activities of organisms such as ants, earthworms, termites and plant roots. Increased bioturbation has been linked to increased decomposition of organic pollutants in soil. - Disease suppression - Soil that is abundant with a diversity of microorganisms reduces the viability of plant and human pathogens in the soil. - Toxin decomposition - In aerobic conditions, bacteria decomposers can break down pesticides and pollutants in soil - Fungi degrade and sequester contaminants in the environment. A practice called “Mycoremediation” is a form of bioremediation which utilizes fungi as the major decomposer of toxins. The importance of soil microorganisms cannot be overstated. Virtually every facet of our daily lives requires, directly or indirectly, the work of microorganisms; from the food we eat, the water we drink, the materials we consume, the clothes we wear, the air we breathe. Current mainstream practices typically destroy the life in the soil and cannot be maintained indefinitely. If we are to have any hope of enduring as a species well into the future, we would be wise to bring back and maintain biological diversity in our soils, and acknowledge the complex processes that occur beneath our feet. RootShoot's role in Soil Health RootShoot provides microscopic soil analysis that allows one to better understand their soil. Soil food web health can be determined for any scale of plant production, from planter pots to massive farms. Through our analysis, we can help you determine if your soil is compatible in terms of biological diversity to support your desired plants. [Soil FoodWeb Incorporated]; Dr. Elaine Ingham 1 Biodiversity and Ecosystem Functioning in Soil; Brussard, Lijbert Teaming with Microbes; Jeff Lowenfels & Wayne Lewis<\|endoftext\|>	4
825	Building walls on the seafloor may become the next frontier of climate science, as engineers seek novel ways to hold back the sea level rises predicted to result from global warming. By erecting barriers of rock and sand, researchers believe they could halt the slide of undersea glaciers as they disintegrate into the deep. It would be a drastic endeavour but could buy some time if climate change takes hold, according to a new paper published on Thursday in the Cryosphere journal, from the European Geosciences Union. Though the notion may sound far-fetched, the design would be relatively straightforward. “We are imagining very simple structures, simply piles of gravel or sand on the ocean floor,” said Michael Wolovick, a researcher at the department of geosciences at Princeton University in the US who described the plans as “within the order of magnitude of plausible human achievements”. The structures would not just be aimed at holding back the melting glaciers, but at preventing warmer water from reaching the bases of the glaciers under the sea. New research is now being undertaken by scientists showing how the effects of the warmer water around the world, as the oceans warm, may be the leading cause of underwater melting of the glaciers. Wolovick and his fellow researchers ran computer models to check on the likely impacts of the structures they believe would be needed, taking as their starting point the Thwaites glacier in Antarctica, which at 80-100km is one of the widest glaciers in the world. They found that creating a structure of isolated columns or mounds on the sea floor, each about 300 metres high, would require between 0.1 and 1.5 cubic km of aggregate material. This would make such a project similar to the amount of material excavated to form Dubai’s Palm Islands, which took 0.3 cubic km of sand and rock, or the Suez canal, which required the excavation of roughly one cubic km. Building a structure of this kind would have about a 30% probability of preventing a runaway collapse of the west Antarctic ice sheet, according to the models. Using more complex designs that would be harder to accomplish in the harsh conditions on the sea bottom in the south polar regions, a small underwater wall could be built, which they calculate would have a 70% chance of succeeding in blocking half the warm water from reaching the ice shelf. Glaciers melting under rising temperatures at the poles have the potential to discharge vast amounts of fresh water into the oceans, sending sea levels rising faster than they have for millennia. The Thwaites glacier alone, an ice stream the size of Britain and likely to be the biggest single source of future sea level rises, could trigger the melting of enough water to raise global sea levels by three metres. Many of these glaciers extend far under the sea, and scientists have begun to explore their subsea melting as well as the easier-to-measure reductions in the or visible parts. The undersea research vessel now known as Boaty McBoatface after its much larger relative was named the Sir David Attenborough is to be deployed at the Thwaites glacier for just this purpose. Building undersea walls could be accomplished by similar vessels but would have to be precisely positioned and strong enough to withstand the immense pressure of the ice. The authors hope that by creating their experimental models they can foster future research into the engineering needed to bring about such projects, which would take many years or decades to be worked out and implemented. Geo-engineering solutions such as this one should not deter the world from reducing greenhouse gas emissions, Wolovick said. “The more carbon we emit, the less likely it becomes that the ice sheets will survive in the long term at anything close to their present volume,” he said. The forthcoming report from the Intergovernmental Panel on Climate Change is expected to warn afresh of the potential for sea level rises to inundate low-lying areas if warming is not held to 1.5C above pre-industrial levels.<\|endoftext\|>	4.21875
919	Patterned Ground (Polygonal Terrain) When NASA's Phoenix Mars lander set down in 2008, it was at 68° north latitude, analogous to northern Alaska and Canada on Earth. Much of the mission focused on analyzing the soil and water ice found directly underneath the non-roving lander. But the photos Phoenix took of its surroundings looked remarkably similar to ones taken at equivalent latitudes on Earth. In particular, the flat plain extending to the horizon showed a network of polygons (straight-edged geometric shapes) in the ground a few meters (yards) wide separated by shallow troughs. Similar polygons occur commonly on Earth in regions where the ground is deeply frozen but a shallow surface layer warms on a daily or annual basis. When temperatures drop, ice forms and expands. Repeated freezing and thawing of wet soil pushes larger rocks upward toward the surface. Small rocks and soil particles settle under the large rocks in a process called frost heaving. Over many repeated cycles, soil with larger stones contains less water than areas of fine-grained sediments. The water-saturated areas expand and contract more, producing a horizontal force that shoves larger stones into clusters and lines. Eventually such freeze-thaw cycles smoothen out the rough edges to produce polygons, circles, and stripes of patterned ground. A glacier moving across a landscape is like a giant bulldozer: it knocks down nearly everything in its path. Even big outcrops of bedrock don't escape being scarred — bedrock is scratched, scraped, rounded, and has pieces of outcrop torn away. Where does all the debris go? When a glacier reaches its limit — where melting overtakes the flow of ice — the piled-up debris is left in what geologists call a moraine. (The name originated in French.) Glaciers can leave several kinds of moraine, but a terminal moraine is one of the most significant. Terminal moraines mark the farthest advance of a glacier or ice sheet. Long Island, New York, is built from two linked ones: the Harbor Hill and Ronkonkoma moraines. Every moraine represents a line on the ground that marks the outer edge of a glacier or ice sheet. Or it marks where a glacier paused as its front edge retreated. On Mars, scientists have identified probable moraines that were left by glaciers that formed on the northwest sides of the four giant Tharsis volcanoes. But these moraines lie on top of delicate landscape features that a moving sheet of ice would totally destroy. What happened? The moraines, scientists think, were left by cold-based glaciers. These mountain-born ice sheets covered the ground, but the base of the glacier was frozen firmly to the surface and did not move, thus preserving the underlying landscape. The ice sheet flowed internally and dropped debris carried from upstream at points where the ice front paused or stopped. When a glacier is warm-based (or "wet-based"), water underneath the ice seeks to escape. If the water can't sink into the ground under the glacier, it carves or melts a tunnel-like channel into the glacier ice. And like all streams, it carries along sand, gravel, and small rocks as it flows. Now take away the stream and the ice, and what's left? A ridge of small-scale debris meandering in a stream-like way across an ice-free landscape. Such features are common in glacial landscapes on Earth; geologists call them by their Irish name, esker. Scientists have identified what may be possible eskers on Mars in several areas. Or they may be ridges that formed in some other way. In many cases, resolving the question will probably require sending a lander or rover to investigate at ground level. Lobate Debris Aprons Lobate debris aprons are piles of rock debris lying below cliffs or escarpments. They have a gentle slope and many show lineations (lines) on their surfaces that resemble glaciers on Earth that are covered by rocks. Radar studies show that pure water ice makes up the debris aprons under a thin rocky surface coating. Lineated Valley Fill Lineated valley fill appears to be what happens when lobate debris aprons flow enough to fill a valley. The top surface of lineated valley fill consists of ridges and grooves a few meters (yards) high that appear to flow around obstacles. They resemble the flow features seen on terrestrial valley glaciers.<\|endoftext\|>	4.4375
1,901	This web page was produced as an assignment for an undergraduate course at Davidson College "Language Gene" Discovered Popular Press v. Scientific Literature What if a gene involved in the development of language was present in other animals? Would humans then be able to directly communicate with animals using the same language? Image from IMDB.com. Permission pending Popular Press Article: Click here to view article. On August 14, 2002, BBC News Online published an article entitled, " First language gene discovered", in which it indicated that a team of British and German researchers had identified the first gene that was definitively related with human language. They determined that when this gene, FOXP2, is mutated, people suffer from a disorder that is characterized by serious language and grammar-learning abilities. The article also indicates that the scientists compared the versions of the FOXP2 gene in mice and several types of monkeys to the human version of the gene. In this study they found slight differences in the monkey and mice versions of the gene compared to humans. This mutation in two nucleotides of human DNA apparently happened nearly 200,000 years ago, and have allowed speech. The researchers predict that the product of the gene is related to an ability to control facial movements. The article also mentions that one of the authors of the report says, " We don't think this is THE speech gene." and that they also expect from 10 to 1,000 other genes related to speech and language to be discovered. Background on the FOXP2 Gene The FOXP2 gene was actually isolated by Lai et al.(2001). It is located on chromosome 7q31 in humans. It's protein product is a transcription factor that contains a polyglutamine tract and a forkhead DNA binding domain (OMIM, 2003). Figure 1: Graphic of chromosome 7 showing the location of the FOXP2 gene. Image from NCBI's human map viewer. Scientific Journal Article Enard et al., 2002 did further genetic analysis of FOXP2, a gene that had actually been discovered by Lai et al., 2001. Their study included the sequencing of cDNAs that encode FOXP2 protein in different species of monkeys and mice, and investigating the variation of the FOXP2 gene among humans. After sequencing complementary DNA of FOXP2 protein in the chimpanzee, gorilla, rhesus macaque, orangutan and the mouse they compared them to human cDNA of FOXP2. They found two amino-acid positions in the mouse orthologue of the human FOXP2 protein. They found that out of 1,880 human-rodent gene pairs, FOXP2 is one of the 5% most conserved proteins. Among the chimpanzee, gorilla, and rhesus macaque's FOXP2 proteins, there are no differences in amino acid sequence with each other, one difference from the mouse, and two differences from the human protein. The FOXP2 protein orangutan differed from the mouse in two amino acids, and from the human in three amino acids. Enard et al. concluded that though the FOXP2 protein is highly conserved, two of the three amino-acid differences between humans and mice must have occurred on the human lineage after speciation between humans and the chimpanzee took place. After analysis of the FOXP2 protein structures for humans, chimpanzees, orangutans, and mice, they indicated that of these amino-acid changes on the human lineage, at least one is related to a functional change in the protein since the amino acid in humans can potentially be phosphorylated by protein kinase C which would create a minor change in the secondary structure of the protein. Figure 2: Alignment of human, chimp, gorilla, orangutan, rhesus macaque, and mouse amino acid sequences of FOXP2 cDNA. The vertical boxes indicate differences among the species. Image from article(subscription required to view full-text article) in Nature. Permission pending. The researchers found these results of a change in FOXP2's shape interesting because FOXP2 is the first gene known to be related to speech and language development, so they decided to investigate the variation of FOXP2 among modern humans. To test if the amino acids that comprise FOXP2 are polymorphic, they sequenced the chromosomal region where the two amino acid differences from chimpanzees are found, on exon 7 of the gene, from 22 people(44 chromosomes) from all 7 continents. Additionally they sequenced the same exon in 91 unrelated individuals(182 chromosomes) of predominantly European descent. Among these 226 sequenced chromosomes they found no amino-acid replacements in the two amino-acid variants that differs in human FOXP2 and chimpanzee FOXP2, suggesting that these variants are fixed in humans. With the use of various statistical tests, the authors also estimated that the fixation of the two amino acid variants in the human population probably occurred around 200,000 years ago, which also corresponds with the known existence of anatomically modern humans. I found the BBC's article to be relatively straightforward, and not as misleading as many other popular press articles I have read in the past giving news involving the discovery of a gene. This article specifically says that the authors of the scientific paper believe there to be many genes related to language and speech. While the article may not impress many in the scientific community because of its use of layman's terms for genetic words and for not detailing the methods the group used to analyze sequences and compare sequences of the FOXP2 gene among different genes, it does a good job of explaining what the scientists discovered about FOXP2 and the mutations it has undergone in different species without misleading readers by making oversimplified claims about the gene. The scientific article is much more complex and detailed than the popular press version. It gives more specifics as to the genetic location of the amino acid variants among orangutan and human orthologues of FOXP2. It's discussion of the results from the sequencing procedures involves the use of several statistical tests to deduce when human FOXP2 underwent it's last two mutations, as the author's estimated 200,000 years ago. Thus, each source is writing with different audiences in mind. The BBC article is obviously going to read be by more non-scientists than scientists, so less detail is used in it's description of Enard et al's discoveries. In contrast, the Nature Article is written for those familiar with scientific terminology and theories, so more use of details and methods are included to communicate the results. I would still recommend that one read the scientific journal's article (even if it is a review article of the original scientific paper), if one is capable of understanding most of such scientific vocabulary, because in converting the scientific paper's findings into everyday terms some facts may be lost or misconstrued by journalists within the popular press. As for the discovery of a gene related to the development of language or speech skills I find it very intriguing. The fact that certain changes in FOXP2's stucture occurred around the time humans speciated from other primates seems to make logical sense as to why humans speak and chimps do not(though as noted I'm sure the reasons are more complex than the presence or absence of the human version of FOXP2. I would be predict that in the future more people with language development disorders will be genetically tested to try and determine if other genes aside from FOXP2 are related to language. Other intriguing factors related to the gene's discovery is the possibility to insert this gene into species besides humans, and if this would then allow humans to directly communicate with animals. Would the FOXP2 gene's, along with other potentially found language-related genes', presence in an organism's genetic material allow that organism to develop language and speech abilities? Are there other genes related to an organism's ability to think that are also necessary for one to speak? I'm sure, if deemed ethical, more investigation into these questions would take place in the scientific community, and if not at the very least it might just make for more believable situations in the plots of movies or television shows that involve animals capable of talking. BBC Science News. 2002 Aug. 14. First language gene discovered. <http://news.bbc.co.uk/1/hi/sci/tech/2192969.stm> . Accessed 2004 Sept. 11. Enard W, Przeworski M, Fisher SE, Lai CS, Wiebe V, Kitano T, Monaco AP, Paabo S. 2002. Molecular evolution of FOXP2, a gene involved in speech and language. Nature 418: 869-72. Lai CS, Fisher SE, Hurst JA, Vargha-Khademm F, Monaco AP. 2001. A forkhead-domain gene is mutated in a severe speech and language disorder. Nature 413: 519-523. [OMIM] Online Mendelian Inheritance in Man. 2003. FOXP2. <http://www.ncbi.nlm.nih.gov/entrez/dispomim.cgi?id=605317>. Accessed 2004 Sept. 11. Questions, Comments? E-mail John<\|endoftext\|>	3.703125
86	# How do you simplify 4(3x + 7)? ##### 1 Answer Oct 19, 2015 The answer is $12 x + 28$. #### Explanation: $4 \left(3 x + 7\right)$ Distribute the $4$. $4 \cdot 3 x + 4 \cdot 7 =$ $12 x + 28$<\|endoftext\|>	4.4375
1,113	It is no secret that human activities contribute to threatening the existence of some animal species. Some of these activities are a simple by product of human overpopulation while others are intentional. Shark Finning is one of those intentional human activities that has been wreaking havoc on shark populations around the world. The International Union for the Conservation of Nature has 12 shark species listed on its list of endangered species, and shark finning is a major contributing factor. Number of Shark Species on Endangered List Shark finning is the brutal practice of capturing sharks from the ocean and cutting off their fins. The bodies of the sharks are then thrown back into the ocean alive, without any concern for the shark’s well-being. Once a shark’s fins removed, the shark cannot survive. Most of these sharks will bleed to death or will be attacked and devoured by other sea predators. Sharks may also suffocate to death because they cannot swim without their fins, and they will often sink to the floor of the ocean and die. Shark finning results in extreme suffering for these sharks, and they die in a cruel and painful way. The fishing crews who perform shark finning do so as a strategy to increase their profits. Since shark fins are the most valuable part of a shark, the fishing crews try to avoid having to carry heavy shark bodies that do not have much financial value. Instead, they slice the fins off and toss the bodies back. That way, they are able to get more shark fins on board their vessels without the weight of the shark bodies. This also increases the number of sharks that are killed on any vessel’s trip. According to the Sea Shepard Conservation Society, shark fins consist of only 4% of a shark’s weight, meaning that 96% of the shark is tossed back into the ocean to die. - Amount of EACH Shark that is tossed back into the ocean to Die! 96% One may be wondering what shark fins are used for. Well, shark fins are mainly considered a delicacy. They are used to prepare shark fin soup, a very popular and expensive Asian dish. Historically, the soup was often prepared for high class guests or served at important functions like weddings but is nowadays readily available to anyone who can afford it. The soup is also available in countries outside Asia, including some restaurants in the United States in states where shark fins are not banned. The Sea Shepard Conservation Society states that the soup is actually tasteless and has zero nutritional value, and is only popular for prestige purposes since it is expensive, retailing up to $400 a bowl. In some cultures, the fins are also considered to have medicinal value. In addition to being cruel, shark finning results in a rapid decline of shark populations. The Shark Trust gave an estimate that some 73 million sharks are killed per year through shark finning. The decline in shark populations caused by this practice has a lot of negative consequences on the ocean ecosystems. Sharks play an important role in their ecosystems. They are predators and help to keep the oceans clean by feeding on sea animals that are deceased or ones that are sickly or defective. As sharks are destroyed at a fast rate, undesirable animal populations will grow unchecked in the ocean. Also, sharks are food for some human populations, especially in developing countries. Those communities make use of the whole shark, not just the fins. However, the wasteful practice of cutting off fins and throwing away shark bodies means that there is a decreased possibility that those communities who depend on shark meat will be able to. These are some of the negative consequences of the wasteful and cruel practice of shark finning. Number of Sharks Killed Annually There are many countries around the world that have taken steps to combat the practice of shark finning, although a lot more still needs to be done. The United States, through the 2010 Shark Conservation Act, has made it illegal for vessels to bring only fins into the country. The vessels are required to bring the entire shark, with the fins attached. This means that each vessel can only kill so many sharks at a time because there is limited amount of weight each vessel can accommodate at a time. Some individual states have gone further and outlawed the use of shark fins within their borders. In those states, it is illegal for a restaurant to serve shark fin soup. A few other countries have also adopted similar laws. In an effort to reduce the popularity of shark fin soup, China has banned shark fin soup at official government functions. There is a lot that we can do to protect sharks and preserve marine ecosystems. It is important for people around the world to lobby their governments for stricter laws against shark finning. These may include a complete ban of shark fin products or a requirement that sharks be brought on shore with the fins attached to the body. Also, people need to seriously consider avoiding shark fin products altogether. This would mainly mean boycotting shark fin soup. If shark fins stopped being profitable, then there would be reduced incentive for the practice of shark finning. It is also important to join the efforts of conservation groups in combatting the practice and in monitoring shark populations. Conservation groups also carry out educational campaigns to inform people about the dangers of shark finning, and it is important to support these campaigns. Ultimately, shark finning is a cruel practice that has led to drastic declines in shark populations and damages to the marine ecosystem. It is important for people around the world to pull together to combat this practice and protect shark populations.<\|endoftext\|>	3.703125
5,044	Texas Go Math Grade 8 Lesson 2.2 Answer Key Scientific Notation with Negative Powers of 10 Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 2.2 Answer Key Scientific Notation with Negative Powers of 10. Texas Go Math Grade 8 Lesson 2.2 Answer Key Scientific Notation with Negative Powers of 10 Essential Question How can you use scientific notation to express very small quantities? You can use what you know about writing very large numbers in scientific notation to write very small numbers in scientific notation. A typical human hair has a diameter of 0.000025 meter. Write this number in scientific notation. A. Notice how the decimal point moves in the list below. Complete the list. B. Move the decimal point in 0.000025 to the right as many places as necessary to find a number that is greater than or equal to 1 and less than 10. What number did you find? ____ D. Combine your answers to B and c to represent 0.000025 in scientific notation. ____ Reflect Question 1. When you move the decimal point, how can you know whether you are increasing or decreasing the number? When the decimal point moves to the right, the number is increasing and if it moves to the left, the is decreasing. Question 2. Explain how the two steps of moving the decimal and multiplying by a power of 10 leave the value of the original number unchanged. The power of 10 represents the number of times the decimal is moved, hence multiplying will leave the value of the original number unchanged. Reflect Question 3. Critical Thinking When you write a number that is less than 1 in scientific notation, how does the power of 10 differ from when you write a number greater than 1 in scientific notation? When we write a number that is less than 1 in scientific notation, the power of 10 is negative, whereas when we write a number that is greater than 1 in scientific notation, the power of 10 is positive Write each number in scientific notation. Question 4. 0.0000829 0.0000829 Given 8.29 Place the decimal point 5 Count the number of places the decimal point is moved. 8.29 × 10-5 Multiply 8.29 times a power of 10. Since 0.0000829 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 8.29 × 10-5 Question 5. 0.000000302 0.000000302 Given 3.02 Place the decimal point 7 Count the number of places the decimal point is moved. 3.02 × 10-7 Multiply 3.02 times a power of 10. Since the original number is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 3.02 × 10-7 Question 6. A typical red blood cell in human blood has a diameter of approximately 0.000007 meter. Write this diameter in scientific notation. ________________________ 0.000007 Given 7.0 Place the decimal point 6 Count the number of places the decimal point is moved. 7.0 × 10-6 Multiply 7.0 times a power of 10. Since the original number is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 7.0 × 10-6 Reflect Question 7. Justify Reasoning Explain whether 0.9 × 10-5 is written in scientific notation. If not, write the number correctly in scientific notation. In order for a number to be written in scientific notation they must be of the format: c × 10n (1) where c is a decimal number greater or equal to 1 and less than 10 and n is an integer Since in the given number, 0.9 < 1 this number is not written in scientific notation. A correct way to write that number in scientific notation would be to move the decimal point one place to the right because of that the exponent would also decrease by one: 9× 10-6 (2) Question 8. Which number is larger, 2 × 10-3 or 3 × 10-2? Explain. 2 × 10-3 Given and 3 × 10-2 3 × 10-2 is larger when comparing exponents, -2 > -3 Write each number in standard notation. Question 9. 1.045 × 10-6 1.045 × 10-6 Given 6 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.000001045 Place the decimal point Since you are going to write a number less than 1.045, move the decimal to the left. Add placeholder zeros if necessary. The number 1.045 × 10-6 in standard notation is 0.000001045 Question 10. 9.9 × 10-5 9.9 × 10-5 Given 5 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.000099 Place the decimal point Since you are going to write a number less than 9.9, move the decimal to the left. Add placeholder zeros if necessary. The number 9.9 × 10-5 in standard notation is 0.000099 Question 11. Jeremy measured the length of an ant as 1 × 10-2 meter. Write this length in standard notation. 1 × 10-2 Given 2 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.01 Place the decimal point Since you are going to write a number less than 1, move the decimal to the left. Add placeholder zeros if necessary. The number 1 × 10-2 in standard notation is 0.01 meter 0.01 Write each number in scientific notation. (Explore Activity and Example 1) Question 1. 0.000487 Hint: Move the decimal right 4 places. 0.000487 Given 4.87 Place the decimal point 4 Count the number of places the decimal point is moved. 4.87 × 10-4 Multiply 4.87 times a power of 10. Since 0.000487 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 4.87 × 10-4 Question 2. 0.000028 Hint: Move the decimal right 5 places. Move the decimal point as many places as necessary to find a number that is greater that or equal to 1 and less than 10: 2.8 Place the decimal point (1) 5 places Count the number of places you moved the decimal point (2) 2.8 × 10-5 Multiply the number from step (1) with 10-5 (3) Note: in step (3) you multiplied 2.8 with 10-5 because you moved the decimal point 5 places to the right 2.8 × 10-5 Question 3. 0.000059 0.000059 Given 5.9 Place the decimal point 5 Count the number of places the decimal point is moved. 5.9 × 10-5 Multiply 5.9 times a power of 10. Since 0.000059 is less than 1, the decimal point moves to the right and the exponent on 10 is negative 5.9 × 10-5 Question 4. 0.0417 0.0417 Given 4.17 Place the decimal point 2 Count the number of places the decimal point is moved. 4.17 × 10-2 Multiply 4.17 times a power of 10. Since 0.0417 is less than 1, the decimal point moves to the right and the exponent on 10 is negative 4.17 × 10-2 Question 5. Picoplankton can be as small as 0.00002 centimeter. 0.00002 Given 2.0 Place the decimal point 5 Count the number of places the decimal point is moved. 2 × 10-5 Multiply 2 times a power of 10. Since 0.00002 is less than 1, the decimal point moves to the right and the exponent on 10 is negative 2 × 10-5 centimeter Question 6. The average mass of a grain of sand on a beach is about 0.000015 gram. 0.000015 gram Given 1.5 Place the decimal point 5 Count the number of places the decimal point is moved. 1.5 × 10-5 Multiply 1.5 times a power of 10. Since 0.000015 is less than 1, the decimal point moves to the right and the exponent on 10 is negative 1.5 × 10-5 gram Write each number in standard notation. (Example 2) Question 7. 2 × 10-5 Hint: Move the decimal left 5 places. 2 × 10-5 Given 5 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.00002 Place the decimal point. Since you are going to write a number less than 2, move the decimal to the left. Add placeholder zeros if necessary. The number 2 × 10-5 in standard notation is 0.00002 Question 8. 3.582 × 10-6 Hint: Move the decimal left 6 places. 3.582 × 10-6 Given 6 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.000003582 Place the decimal point. Since you are going to write a number less than 3.582, move the decimal to the left. Add placeholder zeros if necessary. The number 3.582 × 10-6 in standard notation is 0.000003582 Question 9. 8.3 × 10-4 8.3 × 10-4 Given 4 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.00083 Place the decimal point. Since you are going to write a number less than 8.3, move the decimal to the left. Add placeholder zeros if necessary. The number 8.3 × 10-4 in standard notation is 0.00083 Question 10. 2.97 × 10-2 2.97 × 10-2 Given 2 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.0279 Place the decimal point. Since you are going to write a number less than 2.97, move the decimal to the left. Add placeholder zeros if necessary. The number 2.7 × 10-2 in standard notation is 0.0297 Question 11. 9.06 × 10-5 9.06 × 10-5 Given 5 Use the exponent of the power of 10 to see how many places to move the decimal point. places 0.0000906 Place the decimal point. Since you are going to write a number less than 9.06, move the decimal to the left. Add placeholder zeros if necessary. The number 9.06 × 10-5 in standard notation is 0.0000906 Question 12. 4 × 10-5 (1) First, use the exponent of the power of 10 to see how many places to move the decimal point. Since we have 10-5 we have to move the decimal point 5 places. (2) Place the decimal point, since you are going to write a number less than 4, move the decimal point to the left Add placeholder zeros if necessary: 4 × 10-5 = 0.00004. 0.00004 Question 13. The average length of a dust mite is approximately 0.0001 meter. Write this number in scientific notation. (Example 1) Move the decimal point as many places as necessary to find a number that is greater than or equal to 1 and less than 10: 1 Place the decimal point (1) 4 places Count the number of places you moved the decimal point (2) 1 × 10-4 Multiply the number from step (1) with 10-4 (3) Note: in step (3) you multiplied 1 with 10-4 because you moved the decimal point 5 places to the right because 0.0001 is smaller than 1. 1 × 10-4 meter. Question 14. The mass of a proton is about 1.7 × 10-24 gram. Write this number in standard notation. (Example 2) 1.7 × 10-24 gram Given 24 Use the exponent of the power of 10 to see how many places to move the decimal point places 0.0000000000000000000000017 Place the decimal point Since you are going to write a number less than 1.7, move the decimal to the left Add placeholder zeros if necessary. The number 1.7 × 10-24 in standard notation is 0.0000000000000000000000017 gram Essential Question Check-In Question 15. Describe how to write 0.0000672 in scientific notation. (1) Place the decimal point such that the new number is larger or equal to 1 but less than 10. 0.0000672 ⇒ 6.72 (2) Count the number of places you moved the decimal. point: 5 places. (3) Multiply 6.72 by 10-5 (because you moved the decimal 5 places to the right the exponent is negative) 6.72 × 10-5 Use the table for problems 16-21. Write the diameter of the fibers in scientific notation. Question 16. Alpaca __________ 0.00277 Fiber diameter of Alpaca 2.77 Place the decimal point 3 Count me number 0f places the decimal point is moved. 2.77 × 10-3 Multiply 2.77 times a power of 10. Since 0.00277 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 2.77 × 10-3 Question 17. Angora rabbit _________ 0.0013 Fiber diameter of Angora rabbit 1.3 Place the decimal point 3 Count me number 0f places the decimal point is moved. 1.3 × 10-3 Multiply 1.3 times a power of 10. Since 0.0013 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 1.3 × 10-3 Question 18. Llama _____ 0.0035 Fiber diameter of Llama 3.5 Place the decimal point 3 Count me number 0f places the decimal point is moved. 3.5 × 10-3 Multiply 3.5 times a power of 10. Since 0.0035 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 3.5 × 10-3 Question 19. Angora goat ______ 0.0045 Fiber diameter of Angora goat 4.5 Place the decimal point 3 Count me number 0f places the decimal point is moved. 4.5 × 10-3 Multiply 4.5 times a power of 10. Since 0.0045 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 4.5 × 10-3 Question 20. Orb web spider 0.015 Fiber diameter of Orb web spider 1.5 Place the decimal point 2 Count me number 0f places the decimal point is moved. 1.5 × 10-2 Multiply 1.5 times a power of 10. Since 0.0045 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 1.5 × 10-2 Question 21. Vicuna 0.0008 Fiber diameter of Vicuna 8.0 Place the decimal point 4 Count me number 0f places the decimal point is moved. 8.0 × 10-4 Multiply 8.0 times a power of 10. Since 0.0008 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 8.0 × 10-4 Question 22. Make a Conjecture Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning. Both the thickness of a dog hair and the radius of a period on a page are very small, numbers (Lengths), something that we tend to write in scientific notation (because it is easier to work with when written in scientific notation). On the other hand, it is likely that the number of ounces in a cup of milk is a number that we can easily write and work with, so we don’t need to write it in scientific notation, so it is reasonable to suspect that it is the least likely to be written in scientific notation. The number of ounces in a cup of milk Question 23. Multiple Representations Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation. There are 100 centimeters in 1 meter, or 1 centimeter = 0.01 meters Because of this 7 centimeters = 0.07 meters Write 0.07 in scientific notation by moving the decimal point two places to the right: 0.07 = 7 × 10-2 meter Since 7 centimeters is a number already greater or equal to 1 and less than 10 to get a scientific notation we just multiply it with 100 = 1 7 = 7 × 100 centimeters Finally we can see that when in scientific notation the first factor is the same but the exponents of the second factor differ by 2. In scientific notation the first factor is the same but the exponents of the second factor differ by 2. Question 24. Draw Conclusions A graphing calculator displays 1.89 × 10-12 as 1.89E12. How do you think it would display 1.89 × 10-12? What does the E stand for? We are told that 1.89. 1012 will be displayed as 1.89e12 in a scientific calculator. We can conclude that here e stands for exponent of 10 and 1.89 . 10-12 will be displayed as 1.89e -12 Question 25. Communicate Mathematical Ideas When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1 ? In a scientific notation, if the exponent of 10 is negative, the number is smaller than 1. Otherwise, it is greater or equal to 1. Question 26. The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation. 0.000047 liter Given 4.7 Place the decimal point 5 Count the number of places the decimal point is moved. 4.7 × 10-5 Multiply 4.7 times a power of 10. Since 0.000047 is less than 1, the decimal point moves to the right and the exponent on 10 is negative. 4.7 × 10-5 Question 27. Justify Reasoning If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response. Since the weight of a ladybug in ounces ¡s a number smaller than 1 we would need to move the decimal point to the right. Therefore the exponent of the 10 would be a negative number. Negative, since the weight of a ladybug is smaller than 1. Physical Science The table shows the length of the radii of several very small or very large items. Complete the table. Question 28. (1) Move the decimal point to the left and remove extra zeros: 1,740,000 ⇒ 1.74 (2) Count the number of places you moved the decimal. point: 6 places (3) Multiply 1.74 by 106 (because we moved the decimal point 6 places to the left 1.74 × 106 1.74 × 106 meters Question 29. Question 30. Question 31. Question 32. Question 33. Question 34. List the items in the table in order from the smallest to largest. Compare the powers of 10 and when they are same, compare the first factor. Based on this rule, the items are arranged from smallest to largest. Texas Go Math Grade 8 Lesson 2.2 H.O.T. Focus On Higher Order Thinking Answer Key Question 35. Analyze Relationships Write the following diameters from least to greatest. 1.5 × 10-2m 1.2 × 102m 5.85 × 10-3 m 2.3 × 10-2 m 9.6 × 10-1 m Compare the powers of 10 (smaller the exponent the smaller the number), if some are the same then compare the first factors: 5.85 × 10-3 < 1.5 × 10-2 < 2.3 × 10-2 < 9.6 × 10-1 < 1.2 × 102 5.85 × 10-3, 1.5 × 10-2, 2.3 × 10-2, 9.6 × 10-1, 1.2 × 102 Question 36. Critique Reasoning Jerod’s friend Al had the following homework problem: Express 5.6 × 10-7 in standard form. Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it? 5.6 × 10-7 Given Error is that the decimal is moved in the moved in right direction and not left direction. Since the exponent of 10 is -7, the decimal should be moved to left. 7 Use the exponent of the power of 10 to see how many places to move the decimal point places 0.00000056 Place the decimal point Since you are going to write a number less than 5.6, move the decimal to the left. Add placeholder zeros if necessary. Question 37. Make a Conjecture Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer. $$\frac{a^{m}}{a^{n}}$$ = am-n $$\frac{a \times 10^{m}}{b \times 10^{-n}}$$ = $$\frac{a}{b}$$ × 10m+n<\|endoftext\|>	4.625
2,759	Standardized Tests Courses / Course / Chapter # Multiplying & Dividing Decimals : Concept, Practice & Rules Christian Killian, Yuanxin (Amy) Yang Alcocer • Author Christian Killian Christian has a bachelor's degree in business administration, a master's degree in media communication and psychology, and credits toward a PhD in social psychology. They excel at math and science and enjoy explaining things to others. They are also OSHA 30 certified. • Instructor Yuanxin (Amy) Yang Alcocer Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted. Learn about the operations of multiplying and dividing decimals. Discover decimal multiplication and division word problems with walkthroughs and solutions. Updated: 12/31/2021 ## Multiplying and Dividing Decimals Multiplying and dividing decimals is similar to multiplying and dividing whole numbers with one small difference: decimal placement, which is the place value in a number where the decimal is. For both multiplication and division, there is a strategy to place the decimal that adds one simple step to the process. For multiplication, that extra step is placing the decimal the same number of places as the decimal placements in all factors combined. Because of this, the decimal placement can be determined before calculating the answer. If we have one factor with two decimal places and another with one, our answer will have three decimal places. Example: 3.8 x .007. 38 x 7 is 266. 3.8 has one decimal place and 0.007 has three, giving us four total. Since 266 only has three decimal placements available, we will need to add a zero as the fourth. Therefore, 3.8 x 0.007 = 0.0266. Division is similar, only we move the decimal placements before the calculation. Decimal placements are removed until the number we are dividing by (the divisor) is a whole number. Then, we use the same decimal place of the number into which we are dividing (the dividend) for our answer. Example: 50.22 / 3.1. 3.1 is a decimal, so we'll move everything over one decimal place to make it a whole number. This gives us 502.2 / 31. Without any decimal, our answer is 162. Since 502.2 has one decimal place, so will our answer. 50.22 / 3.1 = 16.2. ## Multiplying Decimals Decimal numbers, those numbers with a decimal point in them, aren't scary or difficult to work with if you know the shortcuts that I will show you. In this video, we are specifically talking about multiplication and division of decimal numbers. I will show you that multiplying and dividing decimal numbers is exactly the same as multiplying and dividing whole numbers with just one difference. In the real world, multiplying and dividing decimals happens all the time, so being able to multiply and divide decimals is an essential skill to have. For example, when you are shopping, you may need to quickly find out how much tax you can expect to pay for your purchase so that you can keep an eye out so you are not overcharged. Let's first look at multiplying decimal numbers. Let's say we want to multiply 1.25 and 3.5. What do we do? We proceed by first multiplying the two numbers while ignoring the decimal. So we go ahead and multiply 125 with 35. We get 4,375. Now this is where the one difference comes in and where we can apply our trick. The trick here is to count the number of decimal places we have in total. We have two from 1.25 and we have one from 3.5. We have a total of three decimal places, so that tells us that we need to count three decimal places, and that is where we put our decimal point. So our decimal goes between the 4 and the 3. Our final answer is 4.375. An error occurred trying to load this video. Try refreshing the page, or contact customer support. Coming up next: How to Estimate with Decimals to Solve Math Problems ### You're on a roll. Keep up the good work! Replay Your next lesson will play in 10 seconds • 0:01 Multiplying Decimals • 1:35 Sales Tax: A… • 2:41 Dividing Decimals • 3:43 Sharing Burgers: A… • 4:29 Lesson Summary Save Save Want to watch this again later? Timeline Autoplay Autoplay Speed Speed ## Multiplying Decimals from Word Problems The most common use of multiplying decimals, outside of statistics and advanced mathematics, is calculating money. Tax, interest, dividends, and even everyday prices use decimals. In United States currency, dollars are whole numbers and cents are decimals. Multiplying one price by another involves multiplying decimals. Example: Maia lost her job and deferred her student loans for one year. She owes \$59,847.19 with an interest rate of 5.875% (0.05875). How much interest will be added by the end of her deferment? The equation we need to solve here is 59,847.19 x .05875. Using our strategy for multiplying decimals, we will first multiply the numbers without decimals. 59,84719 x 5875 is 35,160,224,125. Then we add the decimal places. \$59,847.19 has two decimal places and .05875 has five. This gives us seven decimal places for our answer. 59,847.19 x .05875 = 3,516.0224125. Rounded to the hundredths place for cents, Maia's added interest will be \$3,156.02. ## Dividing Decimals from Word Problems Dividing decimals typically occurs when we have to split a decimal amount by another decimal amount. It can apply to money, distance, time, food; anything that can be divided can also be divided again. Example: Trey made chili for his big game night. After everyone has had their fill, there are 12.5 cups left in his crockpot. How many 2.5 cup containers can he fill for leftovers? This equation is 12.5 / 2.5. Our strategy tells us to move the decimal until both numbers are whole. Since 12.5 and 2.5 both have one decimal place, we end up with 125 / 25, which is 5. Trey will have 5 containers of leftovers. ## Sales Tax: A Multiplication Example Now let's look at a real-world example. You are shopping, and you want to buy two pairs of shoes. The total for the two shoes is \$62.18. The sales tax in your area is 6 percent. How much should you expect to pay in sales tax? To solve this problem, you need to multiply your total of \$62.18 with the sales tax of 0.06 percent. Yes, we are multiplying the two decimals together. Remember what we just learned? First, go ahead and multiply as if there are no decimals. We multiply 6,218 with 6 to get 37,308. Then we count the total number of decimal places we have between our two numbers. We have four. Now we count four decimal places in our answer to find where we should put our decimal point. We place it between the beginning 3 and the 7. Our final answer is 3.7308. We can expect to pay around \$3.73 in sales tax. ## Dividing Decimals Let's move onto division now. Division is also similar to dividing whole numbers. The only difference here is that if the number we are dividing by is a decimal, then we will want to convert it to a whole number before we divide. If we are dividing 7.24 by 0.2, we would first change the 0.2 to a 2. To do that, we move the decimal place over one space to the right. To unlock this lesson you must be a Study.com Member. Video Transcript ## Multiplying Decimals Decimal numbers, those numbers with a decimal point in them, aren't scary or difficult to work with if you know the shortcuts that I will show you. In this video, we are specifically talking about multiplication and division of decimal numbers. I will show you that multiplying and dividing decimal numbers is exactly the same as multiplying and dividing whole numbers with just one difference. In the real world, multiplying and dividing decimals happens all the time, so being able to multiply and divide decimals is an essential skill to have. For example, when you are shopping, you may need to quickly find out how much tax you can expect to pay for your purchase so that you can keep an eye out so you are not overcharged. Let's first look at multiplying decimal numbers. Let's say we want to multiply 1.25 and 3.5. What do we do? We proceed by first multiplying the two numbers while ignoring the decimal. So we go ahead and multiply 125 with 35. We get 4,375. Now this is where the one difference comes in and where we can apply our trick. The trick here is to count the number of decimal places we have in total. We have two from 1.25 and we have one from 3.5. We have a total of three decimal places, so that tells us that we need to count three decimal places, and that is where we put our decimal point. So our decimal goes between the 4 and the 3. Our final answer is 4.375. ## Sales Tax: A Multiplication Example Now let's look at a real-world example. You are shopping, and you want to buy two pairs of shoes. The total for the two shoes is \$62.18. The sales tax in your area is 6 percent. How much should you expect to pay in sales tax? To solve this problem, you need to multiply your total of \$62.18 with the sales tax of 0.06 percent. Yes, we are multiplying the two decimals together. Remember what we just learned? First, go ahead and multiply as if there are no decimals. We multiply 6,218 with 6 to get 37,308. Then we count the total number of decimal places we have between our two numbers. We have four. Now we count four decimal places in our answer to find where we should put our decimal point. We place it between the beginning 3 and the 7. Our final answer is 3.7308. We can expect to pay around \$3.73 in sales tax. ## Dividing Decimals Let's move onto division now. Division is also similar to dividing whole numbers. The only difference here is that if the number we are dividing by is a decimal, then we will want to convert it to a whole number before we divide. If we are dividing 7.24 by 0.2, we would first change the 0.2 to a 2. To do that, we move the decimal place over one space to the right. To unlock this lesson you must be a Study.com Member. #### What is an example of a decimal word problem? The laptop KC wants is \$809.97. Their state tax is 6%, or \$1.06 times the sale price. How much should KC expect to spend total? The equation here is \$809.97 x \$1.06. The strategy for multiplying decimals tells us to multiply without decimals, so 80997 x 106 = 8,585,682. Next, we add the number of decimal places in the factors. Both 809.97 and 1.06 have two decimal places, giving us four total. Then, we move the decimal point four places in on our previous product. 809.97 x 1.06 = 858.5682. KC can expect to spend \$858.57 on their new laptop. #### How do you divide decimals in word problems? First, change the decimal placement in both numbers so that the divisor is a whole number. Then, divide like usual. The decimal place in the dividend will be the same in the answer. #### How do you solve word problems involving multiplying decimals? First, determine the equation using the numbers and context provided in the problem. Then, multiply the factors without decimal points. Next, add the decimal places of all factors together. Finally, move the decimal point on the previous answer the same number of spaces as the added total. ### Register to view this lesson Are you a student or a teacher? Back ### Resources created by teachers for teachers Over 30,000 video lessons & teaching resources‐all in one place. Video lessons Quizzes & Worksheets Classroom Integration Lesson Plans I would definitely recommend Study.com to my colleagues. Itâ€™s like a teacher waved a magic wand and did the work for me. I feel like itâ€™s a lifeline. Jennifer B. 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11,356	Ask most people what they associate with astronomy and they will mention planets, stars, galaxies, etc. Ask them what they associate with astronomers, however, and the answer will probably be telescopes – but it is less than four hundred years since the invention of the telescope. By contrast, human awareness of the heavens goes back millennia; almost certainly deep into prehistoric times. Until comparatively recently, astronomers have studied the heavens with nothing more sophisticated than the naked eye. From earliest times, they would have tried to make sense of the great complexity of what can be seen in the skies. Even the most casual observer will be aware the skies present a differing appearance from hour to hour, as the Sun, Moon and stars march steadily across the heavens. They will realise, however, that the stars are fixed in relation to one another, often making distinctive patterns in the sky. They will also realise that the Moon changes its appearance from night to night, sometimes waxing sometimes waning. But why does the Sun rise and set in different places at different times of the year? Why is the Moon sometimes visible in broad daylight? Why do some stars remain close to the same point in the sky and never set? And what is to be made of the bright star-like objects that do not remain in a fixed position in relation to their neighbours, but move at differing speeds across the starry background, always keeping to the roughly same plane as the Moon in its wanderings. Today, we know that the Moon goes round the Earth and the Earth and other planets go round the Sun. However, this does not even begin to tell the full picture and the achievements of people such as the Maya and the ancient Babylonians – who were not even armed with these basic facts – cannot be overstated. The workings of the celestial clockwork make the achievements of the finest Swiss watchmaker pale into insignificance, yet from a modern perspective they are not difficult to understand; the object of this short (9,000 word) work is to give the reader just such an understanding. The Celestial Sphere Astronomers often use a model known as a celestial sphere to illustrate the movements of the Sun, Moon, planets and stars as seen from Earth. This can be thought of as being similar to a geographer’s globe, except it surrounds the Earth and we look at it from the inside. Another way of thinking of it is as a grid system projected onto the heavens to help us find our way around. To demonstrate such a grid would at one time have required the facilities of a planetarium, but there are now many ‘augmented reality’ smartphone apps available that can achieve almost the same effect. Let’s first consider the finer points of the celestial sphere itself. Like the geographer’s globe, it will have two poles, an equator and lines of latitude and longitude. The celestial North and South Poles and the celestial equator (usually referred to simply as the “equator”) are projections of their terrestrial counterparts onto the grid system. The coordinate system used to define a point on the grid system differs slightly to that used by geographers. The position of an object to the north or south of the equator is given by declination (Dec.). Like latitude, it is measured in degrees, but instead of suffixing the declination with an N or an S, northern declinations are given positive values and southern declinations negative values. The equivalent of longitude is right ascension (R.A.), which has a zero line similar to the Greenwich meridian that passes through a point known as the vernal point (to be discussed shortly) or first point of Ares. Right ascension is measured eastwards from the first point of Ares. It can be measured in degrees but is usually measured in hours, minutes and seconds. An hour corresponds to 15 degrees, so 24 hours is equivalent to 360 degrees. From anywhere in the world, our field of view will be bounded by the horizon, which divides the celestial sphere exactly into two. The point directly above us is known as the zenith. Running through both poles and the zenith is a great circle known as the meridian. The angle of elevation of any object above the horizon is known as the altitude; the angular distance around the horizon, measured clockwise from due North is known as the azimuth. Let us take an imaginary or smartphone augmented reality trip to the (geographical) North Pole. The celestial North Pole is directly overhead, at the zenith, and the equator lies exactly on the horizon. Between the equator and the Pole are a series of concentric circles, growing ever smaller the nearer they are to the Pole. These represent differing declinations. A series of lines extend upwards from the equator, converging at the Pole. These are the lines of right ascension. Note that because the declination circles are parallel to the horizon, we can see them in their entirety. However, we cannot see anything south of the equator at all. If we observe the sky for a few hours, the grid and stars will appears to revolve in a clockwise around the celestial North Pole. This is happening because the Earth is rotating on its axis, in a west to east direction, or anticlockwise as viewed from “above” the North Pole. This motion is known as diurnal motion. During the course of its diurnal motion, a star will reach its highest point in the sky when it is on the meridian. When a star reaches the meridian, it is said to culminate. If we look close to the Pole, we will see a bright star. This is Polaris, the Pole Star. It appears to be almost fixed while everything else wheels around it. The further a star is from the Pole, the larger the circle in which it moves, with those near the equator moving in the largest circle of all. Note that no star ever rises or sets; we can see half of the stars the whole of the time. Before we move on, how long do you think it takes the stars to make a complete circuit of the skies? The answer is of course the same length of time it takes the Earth to make a complete turn on its axis – but this is not 24 hours. The Earth makes a complete turn on its axis once every 23 hours 56 minutes and four seconds. This period of time is known as the sidereal day. However, we reckon time by the Sun rather than the stars, and as we shall see, the solar day is slightly longer. Now we will travel to London, which is at latitude 51.5 degrees N. The first thing we notice is that the grid now appears tipped over towards one side. The celestial North Pole is no longer at the zenith; in fact its altitude will be equal to the geographical latitude. The declination circles are now no longer parallel to the horizon, so only those close to the Pole will be visible in their entirety and the others will be visible only as ever-decreasing arcs. However, we can now partially at least see declination circles that are south of the equator. Exactly half of the equator is visible, and it cuts the horizon at points due east and due west. The celestial North Pole lies due north. There is a declination circle whose southernmost point just touches the northern horizon, and only the stars within this circle now remain permanently above the horizon or, as Homer put it in The Odyssey, “never bathe in Ocean’s stream”. Such stars are said to be circumpolar from that latitude (at either Pole, as we have seen, all the stars are circumpolar). Stars further south do spend increasing amounts of time below the horizon, though those North of the equator are still up for more than half of a sidereal day. Stars lying directly on the equator are visible for exactly half of a sidereal day. Note that these stars rise due east and set due west. Stars located still further south are visible for less than half of a sidereal day. Finally, on the southern horizon, is the northernmost point of a declination circle that lies entirely below the horizon. Stars lying within this circle are permanently out of view and include those making up the Southern Cross and our nearest stellar neighbour, Alpha Centauri. Next let us go down to the equator. The grid will now appear to be completely tipped onto its side, and the declination circles will now lie at right-angles to the horizon. Exactly half of each circle will be visible, but our observer can now see all of them. The celestial North Pole lies exactly on the northern horizon and, 180 degrees away, the celestial South Pole has come into view. Every single star will be above the horizon for half of the sidereal day. It will now be clear that were we to continue south, the celestial North Pole would dip below the horizon and the process we have just witnessed would occur in reverse until upon reaching the geographical South Pole, we would see the celestial South Pole at the zenith. The Sun and Seasons Now let us observe the Sun and stars as a sidereal day passes. At the end of one sidereal day, all the stars will be back where they started – but the Sun will be lagging behind. In fact, it will take approximately four minutes for the Sun to catch up. This is because while the Earth has been spinning on its axis, it has also been moving in its orbit around the Sun, completing a 1/365.24th of a circuit. Like the axial rotation, the orbital motion is west to east, or anticlockwise. (Astronomers refer to such motion as direct; clockwise motion (which is rare in the Solar System) is said to be retrograde.) The Sun, as viewed from Earth, appears to have changed its position slightly with respect to the stars. A solar day is defined as the time between successive crossings of the meridian by the Sun, but because the Earth’s orbital speed varies slightly over the course of a year, the solar day is not constant in length. It is only over the course of a year that it averages out to the familiar 24 hours. The result of the solar day being about four minutes longer than the sidereal day is that any given star will rise four minutes earlier each (solar) day. This is why the constellations visible at a given time vary over the course of the year. After a year has passed, the solar and sidereal days come back into step. Our star will rise at the same time that it did on the corresponding day a year ago. If we follow the Sun’s apparent movement over the course of a year, we will see that it will make a complete circuit of the celestial sphere. The path it traces out is known as the ecliptic. The ecliptic represents the plane of the Earth’s orbit around the Sun and it is inclined to the equator at an angle of 23.5 degrees. The reason for this is that the Earth’s axis of rotation is not perpendicular to the plane of its orbit, but inclined at an angle of 23.5 degrees. This inclination is known as the obliquity of the ecliptic. Constellations straddling the ecliptic are said to be zodiacal. These include the familiar twelve signs of the Zodiac, but to make matters confusing there is actually a thirteenth zodiacal constellation, Ophiuchus the Serpent Bearer, that has been ignored by astrologers and is not considered to be part of the Zodiac. The two points at which the ecliptic and the equator intersect are known as the vernal point (which we have already encountered) and the autumnal point. What effect will the Sun’s movement along the ecliptic have over the course of a year? The ecliptic is inclined to the equator, so the Sun will spend half of the year north of the equator and the other half of the year south of it. Recall that for the Northern Hemisphere, if a star is north of the equator it will be up for more than half of the sidereal day, but if it is south of the equator it will be up for less than half of the sidereal day. The same applies to anything on the celestial sphere, and this includes the Sun. Thus for half of the year, day will be longer than night and for the other half it will be shorter. On the two days of the year known as equinoxes, day and night will be of equal length; this will occur when the Sun is at the vernal or autumnal point. The vernal point is known as the Sun’s ascending node because this is where it crosses into the northern hemisphere from the south. Similarly, the autumnal point is known as the descending node. Mid-way between the equinox points, the Sun will reach its most northerly and most southerly positions; these points are known as the solstices. In the northern hemisphere, the summer solstice occurs when the Sun is at the most northerly point on the celestial sphere and the winter solstice when it is at the most southerly point. This is the explanation for the seasons. In lower latitudes, the Sun attains greater elevations above the horizon. When it is at either equinox, the Sun will be directly overhead at midday along the equator. At the summer solstice, it will be directly overhead at midday in the latitude defined by the Tropic of Cancer and at the winter solstice it will be directly overhead at midday in the latitude defined by Tropic of Capricorn. This is why it gets rather hot in these parts of the world. Conversely, the Sun is circumpolar during the summer months within the Arctic Circle, but never rises at all during the winter months. This is the explanation for the famous ‘Midnight Sun’. The situation is reversed for the Antarctic Circle. Most people think of the Sun rising in the East and setting in the West. But the rising point of the Sun is actually only due East on two days of the year, those when it is at one or other of the equinox points. In the summer months, the azimuth of rising point moves north, reaching its maximum extent at the summer solstice, before returning south. In the winter months, the azimuth of the rising point moves south reaching its maximum extent at the winter solstice, before returning north. Around the solstices, the rising appears to stand still for a few days; the word ‘solstice’ is derived from this phenomenon. The setting points move in the same manner, with the winter sunset limit lying opposite the summer sunrise limit, and the summer sunset limit lying opposite the winter sunrise limit. These seasonal variations in the rising and setting points vary with latitude, being more pronounced in higher latitudes. The Earth’s Orbit Let us now consider the Earth’s orbit around the Sun in a little more detail. The orbit is not circular but elliptical, meaning that distance between the Earth and the Sun isn’t constant but varies over the course of the year, ranging from 147 million km (when Earth is said to be at perihelion) to 152 million km (aphelion). The Earth’s orbital speed is at its greatest at perihelion and at its least at aphelion. This is because its movements are governed by Kepler’s Laws of Planetary Motion, which we will examine in more detail presently. The mean distance is 149.6 million km (93 million miles) and this distance is referred to as the astronomical unit (AU). The departure of the orbit from a perfect circle is known as the orbital eccentricity. Perihelion does not occur in the same place each year, but advances by 11.64 seconds of an arc on each orbit. This is due to gravitational effects of other planets in the Solar System. Precession, Nutation and other cycles There are in addition a number of more gradual motions which are only significant over a long term. The most important of these is precession. In addition to rotating on its axis, the Earth also oscillates like a spinning top, each oscillation taking about 25,800 years and causing the Earth’s spatial orientation to gradually change. This motion is due chiefly to the pull of the Sun and the Moon, though the planets make a small contribution. The observable effect is to make the nodes of the ecliptic gradually move westwards at about 50 seconds of an arc per year. The stars will remain fixed in relation to the ecliptic, but as our celestial sphere grid system uses the Earth as its frame of reference, they will appear to move very slightly against it. This means that star maps have to be calibrated for a particular epoch which since 1984 has been Epoch 2000.0, the start of the year 2000. The effect, though small, is cumulative and just about visible to the naked eye over a lifetime (a good amateur telescope on a suitable mount could show it in a matter of weeks if not days). More significant effects are experienced over longer periods – in antiquity, for example, the Southern Cross could be seen from Greece and the ancient Greeks included it in their star charts. 14,000 years from now it will be visible all over Britain. The precessional motion is not smooth but slightly wavy. This irregularity is known as nutation and it is the result of a slight nodding of the Earth due to variations in the distances and positions of the Sun and the Moon. In addition to these effects, the obliquity of the ecliptic varies with time, chiefly due to nutation, though the gravitational effects of other planets also play a part. Finally the eccentricity also varies, albeit very gradually. Again, this is due to the gravitational effects of other planets. The precessional cycle and the cyclical changes in the obliquity and orbital eccentricity are now known as the Milanković cycles. They are named for the Serbian mathematician Milutin Milanković, who proposed a link between them and cyclical changes in Earth’s climate while interned in Budapest during World War I. Climatologists now accept that Milanković was right, but his views attracted little interest in his lifetime. Four types of year Up until now, we have used the term ‘year’ rather loosely. Most people think of a ‘year’ as being the time it takes the Earth to go once round the Sun. That is certainly a type of year, but not the only type. The Earth in fact has four different types of year. The first is the sidereal year (365.256 days), and this is indeed the time it takes the Earth to go once round the Sun – but this is not the ‘year’ we base our calendar on. The Gregorian calendar, which is now used throughout the Christian world, is actually based on the tropical year (365.243 days), which is defined as the time between the Sun making two passages through the vernal point. The vernal point is moving slowly in the opposite direction to the Sun along the ecliptic as a result of precession, so the Sun ‘arrives’ there about twenty minutes before it completes its circuit of the celestial sphere. Hence the tropical year is slightly shorter than the sidereal year. However, the progression of seasons is dictated by the former, so the calendar is based upon it. The third type of year is the anomalistic year (365.260 days), which is defined as the time between the Earth making two returns to perihelion. As we have seen, the perihelion advances with each circuit, the Earth requires a bit longer to ‘catch up’; hence the anomalistic year is fractionally longer than the sidereal year. Finally there is the eclipse year (346.620 days), which we shall encounter presently. The Moon and its Orbit Most people will think nothing if they happen to see the Moon in the night sky, but are often surprised to see it in broad daylight. In fact, the Moon spends on average as much of its time above the horizon in day time as it does in night time. The most singular feature of the Moon is, in fact, something which most of us take completely for granted. This is that it appears almost exactly the same size as the Sun. The explanation is simple – the Sun is about four hundred times larger in diameter than the Moon, but it is also about four hundred times further away. The odds against this happening – if not quite astronomical – are pretty low. The Moon’s orbit around the Earth is inclined at an angle of 5 degrees to the ecliptic. The Moon’s apparent path around the celestial sphere intersects the ecliptic at two points again known as nodes; as with the intersections between the ecliptic and equator there is an ascending node and a descending node. These nodes do not remain fixed, but move westwards on the celestial sphere at 19 degrees per year due to perturbation by the Sun, taking 18.61 years to complete a nodal cycle. This phenomenon is known as the regression of nodes. The orbit itself is rather more elliptical than that of the Earth around the Sun. Distance from Earth (centre to centre) varies from between 356,410 kilometres (minimum distance, or perigee) to 406,697 kilometres (maximum distance, or apogee). The Moon’s orbital speed increases at perigee, and it decreases at apogee. Like the Earth, this is due to Kepler’s Laws of Planetary Motion, which apply to all orbiting bodies. The eccentricity of the orbit is quite pronounced, so the effect is quite noticeable in terms of nightly movement on the celestial sphere, and this has been known since ancient times. In a manner similar to the Earth’s perihelion, the Moon’s perigee advances with each orbit, taking 8.85 years to complete a cycle. Phases of the Moon The most noticeable feature of the Moon is that its appearance changes from night to night. These phases are due to differing portions of its day-lit side being presented to us as it moves around the Earth. At the start of the cycle it cannot be seen because it lies in the same direction as the Sun and its illuminated side faces away from us. A few days later it will have moved eastwards away from the Sun and will be seen as a slim crescent in the evening sky. As the days pass, the Moon is seen ever higher in the evening sky as it continues to grow or wax. After seven to eight days the Moon will be 90 degrees east of the Sun in the sky; this point is known as the first quarter and the right half of the Earth-facing side is illuminated. At around fifteen days, the Moon’s distance from the Sun reaches 180 degrees. At this point the Moon rises at sunset. The entire Earth-facing side is now illuminated and we see a full Moon. Thereafter, the Moon begins to wane, going through its phases in reverse as its angular distance from the Sun begins to decrease once more. After about 22 days the Moon is 90 degrees west of the Sun; this point is known as the third quarter and the left half of the Earth-facing side is illuminated. Subsequently the Moon becomes an increasingly slim crescent, moving ever closer to the Sun and appearing only just before sunrise. Finally, after 29.531 days on average, it disappears from view and the cycle begins again. Five types of month Most people think of this cycle of 29.531 days as being a month, but they also think of the Moon going round the Earth once a month. In fact, the Moon takes only 27.321 days to go round the Earth. So which ‘month’ is right? Well actually both are. It all depends on what is meant by a month. As we have seen, the Earth has four types of year; the Moon, not to be outdone, has five types of month. The most obvious, perhaps, is the time the moon takes to go once round the Earth. This is known as the sidereal month and as we have seen, it is 27.321 days. But because the Earth is moving round the Sun at the same time the Moon is moving round the Earth, it takes the Moon a bit longer than a sidereal month to return to the same position with respect to the Sun and the Earth. As it is this which governs the phases, it takes more than a sidereal month to go through a complete cycle or lunation. The time for a lunation is known as the synodic month. The Earth’s orbital speed varies slightly over the course of a year, so the synodic month is not fixed. 29.531 days is only the average figure. The tropical month is slightly shorter than the sidereal month. It is defined as the time from one lunar equinox to another. The lunar equinox occurs when the Moon crosses the equator; this takes slightly less than a sidereal month due to the effects of precession (c.f. tropical year). Next is the anomalistic month, the time taken for the Moon to go from perigee to perigee. The perigee advances, so this is longer than the sidereal month and is 27.554 days. Finally we have the draconic month of 27.212 days. This is the time between successive passages by the Moon through the same node. The nodes are moving westwards and the Moon is moving eastwards along the celestial sphere, so it takes less than a complete orbit for the Moon to return to the node, and thus the draconic month is shorter than the sidereal month. The word ‘draconic’ refers to a mythical dragon thought to devour the Sun and the Moon during solar and lunar eclipses; the eclipse year is also sometimes referred to as the draconic year for this reason. The interrelationship of various types of month and year are of great importance when it comes to predicting eclipses, and these cycles may have been understood as far back as prehistoric times. Lunar and Solar Calendars In the Western world, we have long been accustomed to a year of 365 days, with a leap day inserted into February every fourth year. The Gregorian calendar, now the most widely used civil calendar in the world, does have exceptions to this leap year every fourth year rule, but the last such ‘non-leap year’ was in 1900 and the next will not occur until 2100. Most of us will live our lives without ever having been troubled by such details, but they are important. The Gregorian calendar is an example of a solar calendar, or one based on the tropical year. Since this is not an exact number of days, a leap day must be intercalated (inserted) at intervals, and the Gregorian calendar provides for an extra day in February if the year is divisible by four. An exception to the rule is made if the year is divisible by 100 but not by 400, as is the case for 1900 and 2100, but not 2000. The Gregorian calendar was introduced in 1582 in the time of Pope Gregory XIII. It was a refinement to the earlier Julian calendar, which inserted the leap day every fourth year without exception. This gave a year of 365.25 days, which is slightly longer than the tropical year of 365.243 days. The Julian calendar was introduced by Julius Caesar in 46 BC and the error, though small, had amounted to several days by the sixteenth century. The Gregorian was not immediately adopted everywhere, due to resistance in Protestant countries to a Catholic innovation. In Britain, the changeover did not occur until 1752, by which time the correction amounted to eleven days and so Wednesday, 2 September was followed by Thursday, 14 September. The story that this led to riots by people demanding the return of their eleven days is probably apocryphal. In Russia the new system was not adopted until early in the communist era, by which time thirteen days had to be dropped from the calendar. An ironic consequence was that the date of the Great October Socialist Revolution was shifted into November. Solar calendars follow the seasons, but the months do not follow the phases of the Moon because there are not an exact number of synodic months in a tropical year. A lunar calendar is one based on the phases of the Moon and examples include the Islamic calendar, which comprises twelve synodic months and therefore lags the solar calendar by 11 to 12 days each (tropical) year. The Islamic calendar is the official calendar of Saudi Arabia, but elsewhere in the Islamic world it is used mainly for religious purposes. To get round the problem of a lunar calendar fairly rapidly drifting out of synch with the tropical year, some calendrical systems insert an intercalary month every so often, though various calendars use different systems for determining how and when these occur. Such systems are known as lunisolar; examples include the Hebrew and Chinese calendars. Lunar and solar calendars generally come into line every 19 years. This is because 19 tropical years are almost exactly 235 synodic months; thus every 19 years the Moon will have the same phase on the same day of the year. This 19-year cycle is known as the Metonic cycle after the Greek philosopher Meton of Athens (ca 440 BC) who noticed it, though it was undoubtedly known earlier. The Metonic cycle formed the basis of the Greek calendar until 46 BC, when the Julian calendar was adopted. Moonrise, Moonset and lunar movements Like the Sun, the Moon does not rise and set in exactly the same place every day. The azimuth of the rising and setting points varies cyclically over the course of a sidereal month between northern and southern limits and, as with the Sun, these variations are more pronounced in higher latitudes. Note that the ‘month’ in question here is the sidereal rather than synodic month hence the Moon will not be at the same phase at two successive risings or settings at a particular point. Another way of looking at this is to consider only the azimuth of rising and setting of the full Moon, which will vary between the same limits over the course of a year. However, these limits themselves open out and close up over the course of the 18.61 year nodal cycle. In the Northern hemisphere, the variation reaches a maximum when the ascending node is co-incident with the summer solstice; these are the major standstill points. When the descending node reaches this point, the variation is at a minimum; these are the minor standstill points. Between these limits, the standstill points gradually close up and then re-open. The situation is reversed in the Southern Hemisphere. In simpler terms, at the major standstill the Moon’s 5 degree orbital inclination is added to the effect of the Earth’s axial tilt; at the minor standstill it is subtracted. Thus the variation exceeds that of the Sun at major standstill, but is less than it at the minor standstill. As we have seen, the cycle is driven by the sidereal and not the synodic month, so different phases of the Moon will be best observed at different times of the year. The full Moon, for example, rides majestically high in the winter skies, but in summer its performance is decidedly lacklustre. It struggles into the sky, staggers wearily along the southern horizon for a few hours before giving up and disappearing again. The explanation is straightforward enough: when full the Moon is in the opposite part of the sky to the Sun, so in winter it behaves as the Sun in summer, and vice-versa. In spring, the waxing first quarter Moon is most favourably presented for observation, and in autumn it is the turn of the waning last quarter Moon. The waxing crescent is best seen in mid-spring; the waning crescent in mid-summer. These rules hold in both hemispheres, because the seasons are reversed in the Southern Hemisphere. As with the standstill points, these effects are accentuated and diminished over the course of the 18.61 year nodal cycle. The Dark side of the Moon When people refer to ‘the dark side of the Moon’ they really mean the side that cannot be seen from here on Earth. As is correctly pointed out in the eponymous Pink Floyd album, there is no dark side of the Moon and both sides experience equal portions of day and night. It is, however, true that the Moon’s sidereal day is exactly one sidereal month, so in the main one side permanently faces the Earth. However, it is not strictly speaking true to say that we can only see one side from Earth. The orbital speed is not constant, so the orbit and rotation get slightly out of step at times, which causes a slightly different face to be presented. This effect is known as libration in longitude. In addition, because the Moon’s axis is inclined by 6.5 degrees to its orbit, it appears to ‘nod’ back and forth over the course of a month – this is libration in latitude. Finally, parallax effects result in slightly different faces being presented to the observer at different times of the day; in total 59 percent of the Moon’s surface may be seen from Earth (though of course no more than 50 percent at any one time). The word ‘planet’ comes from the Greek word planetes, meaning ‘wanderer’. Long before the time of the Classical Greek civilisation, man would have been aware of bright star-like objects that did not did not remain fixed in relation to the stars but moved in roughly the same narrow band to which the Sun and Moon are constrained. Five planets (excluding the Earth) have been known since prehistoric times – Mercury, Venus, Mars, Jupiter and Saturn. They fall into two groups, the inferior planets, whose orbits lie close to the sun that of the Earth (Mercury and Venus) and the superior planets whose orbits whose orbits lie further away from the Sun (all the other planets, excluding Earth). The distance of each planet from the Sun is often given in astronomical units. Incidentally, the terms ‘superior’ and ‘inferior’ do not mean that the superior planets are ‘better’ planets. The motion of each planet around the Sun is governed by Kepler’s Laws of Planetary Motion, which were formulated by the German mathematician Johannes Kepler between 1609 and 1618 and they apply not just to planets but all orbiting bodies, such as the Moon, satellites of other planets and even artificial Earth satellites. The First Law states that the orbit of any planet around the Sun will be an ellipse, with the Sun at one focus. (If you add the distances of any point on an ellipse from each of the two foci you will always get the same result. By comparison, if you measure the distance of any point on a circle from the centre of that circle, you will always get the same result. In fact these properties define circles and ellipses, which are both examples of what are termed conic sections by mathematicians. The Second Law states that the movement of any planet in its orbit is such that its radius vector (an imaginary line joining the planet to the Sun) sweeps out equal areas in equal times. This explains why the Earth and other planets move faster when they are close to perihelion and why the Moon moves faster when it is close to perigee. The sector swept out in, say, 24 hours, is shorter at these times, but because the Earth (or Moon) is moving faster, it is also ‘fatter’ and these two effects exactly cancel out. The Third Law states that the square of a planet’s orbital period in years is equal to the cube of its mean distance from the Sun in astronomical units. More generally, the square of the orbital period of any orbiting body is proportional to its mean distance from the body it orbits. These laws arise naturally from Newton’s Law of Universal Gravitation, which states that between any two objects, there exists an attractive force that is proportional to their masses multiplied together and divided by the square of their distance apart. Objects under consideration can be stars, planets, satellites or even the apocryphal apple that is said to have given Newton the idea in the first place. Aspects of the planets As seen from the Earth, certain positions of the planets relative to the Sun are known as aspects. For superior planets the two principal aspects are opposition and conjunction. At opposition, a planet is opposite to the Sun in the sky, i.e. they are 180 degrees apart. It will be visible throughout the night and will reach the meridian it midnight. Opposition is the best time to observe a superior planet, because it is at its closest to the Earth. At conjunction, a superior planet is on the opposite side of the Sun to the Earth. It will not be visible from earth at this time, being lost in the Sun’s glare. When a planet is at either opposition or conjunction (i.e. it, the Earth and the Sun are in a straight line) it is said to be at syzgy. The Moon is at syzgy when it is both new and full. When a planet is at an angle of 90 degrees from the Sun as seen from Earth, it is said to be at quadrature. We see a half-Moon when it is at quadrature. Inferior planets cannot reach opposition or quadrature, but have two types of conjunction, inferior conjunction, when they lie between the Earth and the Sun and superior conjunction, when they are on the far side of the Sun. When an inferior planet is at its greatest angular separation from the Sun it is at greatest elongation. At its greatest elongation west it will appear in the morning sky; at greatest elongation east it will appear in the evening sky. An inferior planet can never be seen throughout the night. The inferior planets display phases like the Moon but when best seen (i.e. at elongation) they are crescent. They will be at full phase at superior conjunction and “new” at inferior conjunction, but cannot be seen at these times. The superior planets show very little phase effect; only Mars shows a pronounced gibbous phase when it is at quadrature. Movements of the planets As seen from the Earth, the planets normally appear to move from west to east. However, around opposition a superior planet can appear to halt and then move briefly in an east to west direction before resuming its normal progress. This retrograde motion, so beloved of astrologers, occurs because the Earth, which is moving more rapidly, catches up and overtakes the planet in question. The points where the planet halts before changing direction are known as stationary points. The planets all keep fairly close to the ecliptic, but all have orbits that are slightly inclined to it. Orbits are defined in terms of six elements or quantities. These are the semi-major axis (a) or mean distance from the Sun; the eccentricity (e); the inclination to the ecliptic (i); the longitude of the ascending node (Ω); the argument of perihelion (ω) which is angular displacement from Ω; and the time of perihelion passage (T). A planet’s ‘year’ is known as its sidereal period, corresponding to the Earth’s sidereal year. The time taken for a planet to return to a particular aspect (such as opposition) as seen from Earth is known as the synodic period (c.f. the Moon’s synodic month). There is little doubt that a total eclipse of the Sun is one of the most awesome spectacles of Nature available anywhere in the Solar System. On no other planet is there such an exact match between the apparent size of the Sun and the apparent size of a satellite – despite some planets having upwards on fifty of the latter to choose from, while we on Earth have to make do with just the one. Not quite as spectacular, perhaps, but still noteworthy is the sight of the Moon turning a deep blood-red as it enters the Earth’s shadow during a lunar eclipse. The phenomena are related, but strictly speaking a solar eclipse is an occultation or hiding of a self-luminous body (in this case the Sun) by the Moon. In principle there is no difference between this and the occultation of stars that occur throughout the month as the Moon pursues its course around the Earth. By contrast, a lunar eclipse entails the Moon being cut off from the source of its illumination as it enters the Earth’s shadow. Unlike point sources (such as a distant searchlight), extended luminous objects such as the Sun do not cast sharp shadows. A shadow will of course be cast when an object is interposed between the observer and the light source, but it will have two regions: the umbra in which the light source is wholly obscured and the penumbra in which it is only partially obscured. For a disc such as the Moon, the Earth as seen from the Moon’s surface or a hot-air balloon drifting in front of the Sun as seen by an observer on the ground, the umbra will be cone-shaped, converging to a point; the umbra will be fan-shaped and diverging. Types of Solar eclipse The Moon’s umbra under favourable conditions will just reach the Earth. It does not remain stationary but races across the Earth’s surface as the Moon moves in its orbit. The path it follows is known as the track. Observers inside the umbra will see a total solar eclipse; those outside it but still within the penumbra will see a partial solar eclipse; those completely outside the Moon’s shadow will see nothing. The degree of obscuration of the Sun by the Moon or magnitude will increase the closer an observer is to the zone of totality. Magnitude ranges for 0 (no obscuration) to 1 (totality) and it refers to the solar diameter covered, not area. A 0.5 magnitude eclipse is one in which half the Sun’s diameter is covered, but a little geometry will show that only 40 percent of the Sun’s area will actually be hidden by such an eclipse. The actual duration of totality for any eclipse varies and is dictated by three factors: the distance of the Earth from the Sun when the eclipse occurs; the distance of Moon from the Earth when the eclipse occurs; and the latitude at which the eclipse occurs. If the Earth is at its maximum distance from the Sun its apparent diameter will be diminished and if the Moon is at its minimum distance from Earth its apparent diameter will be increased; these factors favour long eclipses. The Earth is rotating in the same direction as the Moon’s shadow is moving, and this has the effect of prolonging the time the latter will linger over a particular region. The speed the Earth’s surface is moving depends on latitude – at 40 degrees north or south of the equator, the west to east motion is 1,270 kilometres per hour but at the equator it is 1,670 kilometres per hour. The relative speed of the Moon’s shadow is thus lower at lower latitudes and thus eclipses that take place in tropical latitudes tend to be of greater duration than those occurring in temperate latitudes. If the Moon is at or close to its maximum distance from Earth, even if it passes directly in front of the Sun the umbra will not quite reach Earth and a ring of sunlight is left showing. Such eclipses are said to be annular. Total and annular eclipses are referred to as central eclipses, and annular eclipses are the slightly more frequent of the two types. Occasionally, an eclipse is just total at mid-track, but due to the curvature of the Earth the umbra doesn’t touch the end-points. The result is a hybrid total/annular eclipse with observers at mid-track experiencing a total eclipse but those at either end-point viewing only an annular eclipse. Finally in about one third of all solar eclipses only the penumbra reaches the Earth with the umbra missing it altogether. Such eclipses are partial only; nowhere on Earth is a total eclipse seen. Stages of a Solar eclipse The key events in a solar eclipse as viewed from a particular site are known as contacts. First Contact occurs when the Moon’s western edge begins to slide across the Sun and is the point at which the penumbra first begins to move across the site. It is abbreviated to P1, for first penumbral contact. Second Contact occurs when the Moon’s eastern edge touches the Sun’s eastern edge. The marks the onset of totality or annularity, and for a total eclipse is the point at which the umbra begins to move across the site. It is abbreviated to U1 for first umbral contact (though strictly speaking this term is only appropriate for a total eclipse). Third Contact occurs when the Moon’s western edge leaves the Sun’s western edge. This marks the end of totality or annularity and is the point at which the umbra leaves the site. It is abbreviated to U1. Finally Fourth Contact, abbreviated to P2, marks the departure of the penumbra from the site and the end of the eclipse. In a partial eclipse, only P1 and P2 occur. Types of Lunar eclipse Whereas the Moon’s umbra will affect only a small portion of the Earth, the Earth’s umbra is large enough to fully immerse the Moon. During a lunar eclipse, the Moon never entirely disappears from view but appears reddish. This is due to refraction or bending of sunlight by the Earth atmosphere into the umbra; red light is more easily refracted. There are three types of lunar eclipse; total, when the whole of the Moon enters the umbra; partial when only a portion does; and penumbral when the Moon just grazes the penumbra. The latter type generally results in only a slight dimming of a portion of the Moon and is often undetectable to the naked eye. Unlike a solar eclipse, a lunar eclipse may be viewed anywhere on Earth where the Moon is above the horizon. Stages of a Lunar eclipse As with solar eclipses, the key stages of a lunar eclipse are referred to as contacts though unlike a solar eclipse these are the same from any point on Earth. P1 occurs when the Moon begins to enter the Earth’s penumbra. U1 is the point at which the Moon begins to enter the umbra; U2 is the point at which it is fully inside the umbra, marking the onset of totality. U3 is the point at which the Moon begins to leave the umbra, marking the end of totality; U4 is the point at which the Moon leaves the umbra altogether. P2 is the point at which the Moon leaves the penumbra and the eclipse ends. U2 and U3 do not occur in a partial eclipse. In a penumbral eclipse, U1 and U4 do not occur either. When do solar eclipses occur? As you might have inferred, a solar eclipse can only occur at new Moon – but why don’t they occur at every new Moon, i.e. once every lunation? Recall that the Moon’s orbit is inclined at about 5 degrees to the ecliptic. So the Moon usually ‘misses’ the Sun. Recall that there are two nodes where the Moon’s path crosses the ecliptic. Only when the Sun is close to a node at new Moon can an eclipse occur, although it does not have to be exactly at a node for an eclipse to occur; for the two discs to touch in a ‘grazing’ encounter will at minimum produce a partial eclipse. The region the Sun has to occupy at new Moon to produce an eclipse is known as the eclipse limit. This varies, depending on the distance of the Moon from Earth at the time the new Moon occurs, and that of the Earth from the Sun. It ranges from between 30.70 degrees to 37.02 degrees in total, or from 15.35 to 18.51 degrees each side of the node. For a central eclipse to occur, the limit is less, ranging from 9.92 to 11.83 degrees each side of the node. With the Sun moving along the celestial sphere at just under one degree per day, it will be apparent that it will take it more than a synodic month of 29.53 days to traverse even the minimum distance. In other words, the Sun will never be able to get through one of these ‘danger zones’ without the Moon catching up with it at some stage and causing an eclipse. Furthermore, if the Sun has only just entered the eclipse limit when the Moon comes around, the latter will have time to cause a second eclipse before the Sun can get out of the way. The time period during which the Sun is within the eclipse limit is known as an eclipse season. Eclipses can only occur during an eclipse season and as we have just seen, at least one must occur. How many eclipses will occur in a calendar year, given at least one must occur whenever the Sun approaches a node? Recall the nodes are moving along the celestial sphere in the opposite direction to the Sun, completing a complete cycle every 18.61 years. It will therefore take the Sun slightly less than a year to make successive passages through the same node. This is the ‘fourth kind of year’, the eclipse year mentioned earlier, of 346.62 days. There will be two eclipse seasons in each eclipse year and a minimum of two solar eclipses and a maximum of four. The calendar year is longer than an eclipse year and so the eclipse year will end at different times of the calendar year. Normally there will only be two eclipse seasons (and hence a minimum of two eclipses) in a calendar year, but if an eclipse year ends in December, a portion of a third eclipse season can be squeezed into that calendar year, meaning that a maximum of five solar eclipses could occur. Unfortunately, you will have to wait until 2206 before this next happens. When do lunar eclipses occur? Just as solar eclipse can only occur at new Moon, so a lunar eclipse can only occur when the Moon is full. The condition for a lunar eclipse is for the Moon to pass through the opposite node to the one through which the Sun is passing during an eclipse season. As with a solar eclipse, this must happen at least once during an eclipse season, and can happen twice. The maximum number of both types combined in an eclipse season is only three, because it would take 1 ½ lunations to produce two solar and two lunar eclipses, which is longer than the maximum length of an eclipse season. However, there will always be at least one of each. This rule does include penumbral lunar eclipses, which many authorities omit from eclipse statistics. The word ‘saros’ is taken from an ancient Babylonian word meaning ‘repetitive’ and was adopted by Sir Edmund Halley to describe an 18-year cycle of eclipses first recorded by the Babylonians in 400 BC, though it may well have been known much earlier. The saros results from a series of coincidences of nature: 223 synodic months (6585.32 days) is almost exactly the same length of time as 19 eclipse years (6585.78 days) and also coincides with 239 anomalistic months (6585.54). The net effect is that at the conclusion of 223 synodic months from the time of an eclipse, not only are the Sun and Moon in the same places in the sky (thus giving rise to another eclipse) but the Moon will be at the same distance from the Earth as for the previous eclipse and the eclipse limit will thus be the same. This latter factor is equally important because were the Moon to be at a greater distance from Earth than previously, the eclipse limit would be smaller and an eclipse might not occur at all. However, there is one important difference. 223 synodic months does not contain a whole number of days. The odd 0.32 of a day means the second eclipse will occur at a longitude of 0.32 times 360 degrees, i.e. approximately 115 degrees west of the first eclipse due to the Earth’s additional rotation. Eclipses occur more frequently than every 18 years, so there are a number of saros cycles in operation at any one time, and each one is given a number. Saros cycles involving the Moon’s descending node receive even numbers and those involving the ascending node receive odd numbers. Each saros cycle evolves and has a finite life. For a saros cycle involving the Moon’s descending node, the series begins with an eclipse at the South Pole. Each successive eclipse then has a track more northerly than the last, until a final eclipse at the North Pole concludes the cycle. For a saros cycle involving the Moon’s ascending node, the reverse happens, with the series beginning at the North Pole and concluding at the South Pole. At any one time there will be 43 saros cycles in operation and as soon as one concludes at one Pole another one will begin at the other Pole. The length of a cycle ranges from between 1,206 to 1,442 years. This all happens because 19 eclipse years are actually 0.46 days longer than 223 synodic months, and the Sun will not be in exactly the same place for successive eclipses. Given the Sun moves approximately one degree per day along the celestial sphere, each eclipse will occur about 0.46 degrees west of its predecessor. The Moon of course will also be 0.46 degrees further west than before. For the descending node, this will additionally put the Moon slightly further north than before; for the ascending node, the Moon is slightly further south than before.<\|endoftext\|>	3.84375
5,388	# Lessons In Electric Circuits -- Volume IV (Digital) - Chapter 2 PreviousContentsNext # BINARY ARITHMETIC ## Numbers versus numeration It is imperative to understand that the type of numeration system used to represent numbers has no impact upon the outcome of any arithmetical function (addition, subtraction, multiplication, division, roots, powers, or logarithms). A number is a number is a number; one plus one will always equal two (so long as we're dealing with real numbers), no matter how you symbolize one, one, and two. A prime number in decimal form is still prime if it's shown in binary form, or octal, or hexadecimal. π is still the ratio between the circumference and diameter of a circle, no matter what symbol(s) you use to denote its value. The essential functions and interrelations of mathematics are unaffected by the particular system of symbols we might choose to represent quantities. This distinction between numbers and systems of numeration is critical to understand. The essential distinction between the two is much like that between an object and the spoken word(s) we associate with it. A house is still a house regardless of whether we call it by its English name house or its Spanish name casa. The first is the actual thing, while the second is merely the symbol for the thing. That being said, performing a simple arithmetic operation such as addition (longhand) in binary form can be confusing to a person accustomed to working with decimal numeration only. In this lesson, we'll explore the techniques used to perform simple arithmetic functions on binary numbers, since these techniques will be employed in the design of electronic circuits to do the same. You might take longhand addition and subtraction for granted, having used a calculator for so long, but deep inside that calculator's circuitry all those operations are performed "longhand," using binary numeration. To understand how that's accomplished, we need to review to the basics of arithmetic. Adding binary numbers is a very simple task, and very similar to the longhand addition of decimal numbers. As with decimal numbers, you start by adding the bits (digits) one column, or place weight, at a time, from right to left. Unlike decimal addition, there is little to memorize in the way of rules for the addition of binary bits: `0 + 0 = 01 + 0 = 10 + 1 = 11 + 1 = 101 + 1 + 1 = 11` Just as with decimal addition, when the sum in one column is a two-bit (two-digit) number, the least significant figure is written as part of the total sum and the most significant figure is "carried" to the next left column. Consider the following examples: `. 11 1 <— Carry bits —--> 11. 1001101 1001001 1000111. + 0010010 + 0011001 + 0010110. ——— ——— ———. 1011111 1100010 1011101` The addition problem on the left did not require any bits to be carried, since the sum of bits in each column was either 1 or 0, not 10 or 11. In the other two problems, there definitely were bits to be carried, but the process of addition is still quite simple. As we'll see later, there are ways that electronic circuits can be built to perform this very task of addition, by representing each bit of each binary number as a voltage signal (either "high," for a 1; or "low" for a 0). This is the very foundation of all the arithmetic which modern digital computers perform. ## Negative binary numbers With addition being easily accomplished, we can perform the operation of subtraction with the same technique simply by making one of the numbers negative. For example, the subtraction problem of 7 - 5 is essentially the same as the addition problem 7 + (-5). Since we already know how to represent positive numbers in binary, all we need to know now is how to represent their negative counterparts and we'll be able to subtract. Usually we represent a negative decimal number by placing a minus sign directly to the left of the most significant digit, just as in the example above, with -5. However, the whole purpose of using binary notation is for constructing on/off circuits that can represent bit values in terms of voltage (2 alternative values: either "high" or "low"). In this context, we don't have the luxury of a third symbol such as a "minus" sign, since these circuits can only be on or off (two possible states). One solution is to reserve a bit (circuit) that does nothing but represent the mathematical sign: `. 1012 = 510 (positive).. Extra bit, representing sign (0=positive, 1=negative). \|. 01012 = 510 (positive).. Extra bit, representing sign (0=positive, 1=negative). \|. 11012 = -510 (negative)` As you can see, we have to be careful when we start using bits for any purpose other than standard place-weighted values. Otherwise, 11012 could be misinterpreted as the number thirteen when in fact we mean to represent negative five. To keep things straight here, we must first decide how many bits are going to be needed to represent the largest numbers we'll be dealing with, and then be sure not to exceed that bit field length in our arithmetic operations. For the above example, I've limited myself to the representation of numbers from negative seven (11112) to positive seven (01112), and no more, by making the fourth bit the "sign" bit. Only by first establishing these limits can I avoid confusion of a negative number with a larger, positive number. Representing negative five as 11012 is an example of the sign-magnitude system of negative binary numeration. By using the leftmost bit as a sign indicator and not a place-weighted value, I am sacrificing the "pure" form of binary notation for something that gives me a practical advantage: the representation of negative numbers. The leftmost bit is read as the sign, either positive or negative, and the remaining bits are interpreted according to the standard binary notation: left to right, place weights in multiples of two. As simple as the sign-magnitude approach is, it is not very practical for arithmetic purposes. For instance, how do I add a negative five (11012) to any other number, using the standard technique for binary addition? I'd have to invent a new way of doing addition in order for it to work, and if I do that, I might as well just do the job with longhand subtraction; there's no arithmetical advantage to using negative numbers to perform subtraction through addition if we have to do it with sign-magnitude numeration, and that was our goal! There's another method for representing negative numbers which works with our familiar technique of longhand addition, and also happens to make more sense from a place-weighted numeration point of view, called complementation. With this strategy, we assign the leftmost bit to serve a special purpose, just as we did with the sign-magnitude approach, defining our number limits just as before. However, this time, the leftmost bit is more than just a sign bit; rather, it possesses a negative place-weight value. For example, a value of negative five would be represented as such: `Extra bit, place weight = negative eight. \|. 10112 = 510 (negative).. (1 x -810) + (0 x 410) + (1 x 210) + (1 x 110) = -510` With the right three bits being able to represent a magnitude from zero through seven, and the leftmost bit representing either zero or negative eight, we can successfully represent any integer number from negative seven (10012 = -810 + 110 = -110) to positive seven (01112 = 010 + 710 = 710). Representing positive numbers in this scheme (with the fourth bit designated as the negative weight) is no different from that of ordinary binary notation. However, representing negative numbers is not quite as straightforward: `zero 0000positive one 0001 negative one 1111positive two 0010 negative two 1110positive three 0011 negative three 1101positive four 0100 negative four 1100positive five 0101 negative five 1011positive six 0110 negative six 1010positive seven 0111 negative seven 1001. negative eight 1000` Note that the negative binary numbers in the right column, being the sum of the right three bits' total plus the negative eight of the leftmost bit, don't "count" in the same progression as the positive binary numbers in the left column. Rather, the right three bits have to be set at the proper value to equal the desired (negative) total when summed with the negative eight place value of the leftmost bit. Those right three bits are referred to as the two's complement of the corresponding positive number. Consider the following comparison: `positive number two's complement————— —————-001 111010 110011 101100 100101 011110 010111 001` In this case, with the negative weight bit being the fourth bit (place value of negative eight), the two's complement for any positive number will be whatever value is needed to add to negative eight to make that positive value's negative equivalent. Thankfully, there's an easy way to figure out the two's complement for any binary number: simply invert all the bits of that number, changing all 1's to 0's and vice versa (to arrive at what is called the one's complement) and then add one! For example, to obtain the two's complement of five (1012), we would first invert all the bits to obtain 0102 (the "one's complement"), then add one to obtain 0112, or -510 in three-bit, two's complement form. Interestingly enough, generating the two's complement of a binary number works the same if you manipulate all the bits, including the leftmost (sign) bit at the same time as the magnitude bits. Let's try this with the former example, converting a positive five to a negative five, but performing the complementation process on all four bits. We must be sure to include the 0 (positive) sign bit on the original number, five (01012). First, inverting all bits to obtain the one's complement: 10102. Then, adding one, we obtain the final answer: 10112, or -510 expressed in four-bit, two's complement form. It is critically important to remember that the place of the negative-weight bit must be already determined before any two's complement conversions can be done. If our binary numeration field were such that the eighth bit was designated as the negative-weight bit (100000002), we'd have to determine the two's complement based on all seven of the other bits. Here, the two's complement of five (00001012) would be 11110112. A positive five in this system would be represented as 000001012, and a negative five as 111110112. ## Subtraction We can subtract one binary number from another by using the standard techniques adapted for decimal numbers (subtraction of each bit pair, right to left, "borrowing" as needed from bits to the left). However, if we can leverage the already familiar (and easier) technique of binary addition to subtract, that would be better. As we just learned, we can represent negative binary numbers by using the "two's complement" method and a negative place-weight bit. Here, we'll use those negative binary numbers to subtract through addition. Here's a sample problem: `Subtraction: 710 - 510 Addition equivalent: 710 + (-510)` If all we need to do is represent seven and negative five in binary (two's complemented) form, all we need is three bits plus the negative-weight bit: `positive seven = 01112negative five = 10112` `. 1111 <— Carry bits. 0111. + 1011. ——. 10010. \|. Discard extra bit.. Answer = 00102` Since we've already defined our number bit field as three bits plus the negative-weight bit, the fifth bit in the answer (1) will be discarded to give us a result of 00102, or positive two, which is the correct answer. Another way to understand why we discard that extra bit is to remember that the leftmost bit of the lower number possesses a negative weight, in this case equal to negative eight. When we add these two binary numbers together, what we're actually doing with the MSBs is subtracting the lower number's MSB from the upper number's MSB. In subtraction, one never "carries" a digit or bit on to the next left place-weight. Let's try another example, this time with larger numbers. If we want to add -2510 to 1810, we must first decide how large our binary bit field must be. To represent the largest (absolute value) number in our problem, which is twenty-five, we need at least five bits, plus a sixth bit for the negative-weight bit. Let's start by representing positive twenty-five, then finding the two's complement and putting it all together into one numeration: `+2510 = 0110012 (showing all six bits) One's complement of 110012 = 1001102One's complement + 1 = two's complement = 1001112 -2510 = 1001112` Essentially, we're representing negative twenty-five by using the negative-weight (sixth) bit with a value of negative thirty-two, plus positive seven (binary 1112). Now, let's represent positive eighteen in binary form, showing all six bits: `. 1810 = 0100102. . Now, let's add them together and see what we get:.. 11 <— Carry bits. 100111. + 010010. ——--. 111001` Since there were no "extra" bits on the left, there are no bits to discard. The leftmost bit on the answer is a 1, which means that the answer is negative, in two's complement form, as it should be. Converting the answer to decimal form by summing all the bits times their respective weight values, we get: `(1 x -3210) + (1 x 1610) + (1 x 810) + (1 x 110) = -710 ` Indeed -710 is the proper sum of -2510 and 1810. ## Overflow One caveat with signed binary numbers is that of overflow, where the answer to an addition or subtraction problem exceeds the magnitude which can be represented with the alloted number of bits. Remember that the place of the sign bit is fixed from the beginning of the problem. With the last example problem, we used five binary bits to represent the magnitude of the number, and the left-most (sixth) bit as the negative-weight, or sign, bit. With five bits to represent magnitude, we have a representation range of 25, or thirty-two integer steps from 0 to maximum. This means that we can represent a number as high as +3110 (0111112), or as low as -3210 (1000002). If we set up an addition problem with two binary numbers, the sixth bit used for sign, and the result either exceeds +3110 or is less than -3210, our answer will be incorrect. Let's try adding 1710 and 1910 to see how this overflow condition works for excessive positive numbers: `. 1710 = 100012 1910 = 100112.. 1 11 <— Carry bits. (Showing sign bits) 010001. + 010011. ——-- . 100100 ` The answer (1001002), interpreted with the sixth bit as the -3210 place, is actually equal to -2810, not +3610 as we should get with +1710 and +1910 added together! Obviously, this is not correct. What went wrong? The answer lies in the restrictions of the six-bit number field within which we're working Since the magnitude of the true and proper sum (3610) exceeds the allowable limit for our designated bit field, we have an overflow error. Simply put, six places doesn't give enough bits to represent the correct sum, so whatever figure we obtain using the strategy of discarding the left-most "carry" bit will be incorrect. A similar error will occur if we add two negative numbers together to produce a sum that is too low for our six-bit binary field. Let's try adding -1710 and -1910 together to see how this works (or doesn't work, as the case may be!): `. -1710 = 1011112 -1910 = 1011012.. 1 1111 <— Carry bits. (Showing sign bits) 101111. + 101101. ——--. 1011100 . \| . Discard extra bit.FINAL ANSWER: 0111002 = +2810` The (incorrect) answer is a positive twenty-eight. The fact that the real sum of negative seventeen and negative nineteen was too low to be properly represented with a five bit magnitude field and a sixth sign bit is the root cause of this difficulty. Let's try these two problems again, except this time using the seventh bit for a sign bit, and allowing the use of 6 bits for representing the magnitude: `. 1710 + 1910 (-1710) + (-1910) .. 1 11 11 1111. 0010001 1101111. + 0010011 + 1101101. ——— ———. 01001002 110111002. \|. Discard extra bit.. ANSWERS: 01001002 = +3610. 10111002 = -3610` By using bit fields sufficiently large to handle the magnitude of the sums, we arrive at the correct answers. In these sample problems we've been able to detect overflow errors by performing the addition problems in decimal form and comparing the results with the binary answers. For example, when adding +1710 and +1910 together, we knew that the answer was supposed to be +3610, so when the binary sum checked out to be -2810, we knew that something had to be wrong. Although this is a valid way of detecting overflow, it is not very efficient. After all, the whole idea of complementation is to be able to reliably add binary numbers together and not have to double-check the result by adding the same numbers together in decimal form! This is especially true for the purpose of building electronic circuits to add binary quantities together: the circuit has to be able to check itself for overflow without the supervision of a human being who already knows what the correct answer is. What we need is a simple error-detection method that doesn't require any additional arithmetic. Perhaps the most elegant solution is to check for the sign of the sum and compare it against the signs of the numbers added. Obviously, two positive numbers added together should give a positive result, and two negative numbers added together should give a negative result. Notice that whenever we had a condition of overflow in the example problems, the sign of the sum was always opposite of the two added numbers: +1710 plus +1910 giving -2810, or -1710 plus -1910 giving +2810. By checking the signs alone we are able to tell that something is wrong. But what about cases where a positive number is added to a negative number? What sign should the sum be in order to be correct. Or, more precisely, what sign of sum would necessarily indicate an overflow error? The answer to this is equally elegant: there will never be an overflow error when two numbers of opposite signs are added together! The reason for this is apparent when the nature of overflow is considered. Overflow occurs when the magnitude of a number exceeds the range allowed by the size of the bit field. The sum of two identically-signed numbers may very well exceed the range of the bit field of those two numbers, and so in this case overflow is a possibility. However, if a positive number is added to a negative number, the sum will always be closer to zero than either of the two added numbers: its magnitude must be less than the magnitude of either original number, and so overflow is impossible. Fortunately, this technique of overflow detection is easily implemented in electronic circuitry, and it is a standard feature in digital adder circuits: a subject for a later chapter. ## Bit groupings The singular reason for learning and using the binary numeration system in electronics is to understand how to design, build, and troubleshoot circuits that represent and process numerical quantities in digital form. Since the bivalent (two-valued) system of binary bit numeration lends itself so easily to representation by "on" and "off" transistor states (saturation and cutoff, respectively), it makes sense to design and build circuits leveraging this principle to perform binary calculations. If we were to build a circuit to represent a binary number, we would have to allocate enough transistor circuits to represent as many bits as we desire. In other words, in designing a digital circuit, we must first decide how many bits (maximum) we would like to be able to represent, since each bit requires one on/off circuit to represent it. This is analogous to designing an abacus to digitally represent decimal numbers: we must decide how many digits we wish to handle in this primitive "calculator" device, for each digit requires a separate rod with its own beads. A ten-rod abacus would be able to represent a ten-digit decimal number, or a maxmium value of 9,999,999,999. If we wished to represent a larger number on this abacus, we would be unable to, unless additional rods could be added to it. In digital, electronic computer design, it is common to design the system for a common "bit width:" a maximum number of bits allocated to represent numerical quantities. Early digital computers handled bits in groups of four or eight. More modern systems handle numbers in clusters of 32 bits or more. To more conveniently express the "bit width" of such clusters in a digital computer, specific labels were applied to the more common groupings. Eight bits, grouped together to form a single binary quantity, is known as a byte. Four bits, grouped together as one binary number, is known by the humorous title of nibble, often spelled as nybble. A multitude of terms have followed byte and nibble for labeling specfiic groupings of binary bits. Most of the terms shown here are informal, and have not been made "authoritative" by any standards group or other sanctioning body. However, their inclusion into this chapter is warranted by their occasional appearance in technical literature, as well as the levity they add to an otherwise dry subject: • Bit: A single, bivalent unit of binary notation. Equivalent to a decimal "digit." • Crumb, Tydbit, or Tayste: Two bits. • Nibble, or Nybble: Four bits. • Nickle: Five bits. • Byte: Eight bits. • Deckle: Ten bits. • Playte: Sixteen bits. • Dynner: Thirty-two bits. • Word: (system dependent). The most ambiguous term by far is word, referring to the standard bit-grouping within a particular digital system. For a computer system using a 32 bit-wide "data path," a "word" would mean 32 bits. If the system used 16 bits as the standard grouping for binary quantities, a "word" would mean 16 bits. The terms playte and dynner, by contrast, always refer to 16 and 32 bits, respectively, regardless of the system context in which they are used. Context dependence is likewise true for derivative terms of word, such as double word and longword (both meaning twice the standard bit-width), half-word (half the standard bit-width), and quad (meaning four times the standard bit-width). One humorous addition to this somewhat boring collection of word-derivatives is the term chawmp, which means the same as half-word. For example, a chawmp would be 16 bits in the context of a 32-bit digital system, and 18 bits in the context of a 36-bit system. Also, the term gawble is sometimes synonymous with word. Definitions for bit grouping terms were taken from Eric S. Raymond's "Jargon Lexicon," an indexed collection of terms -- both common and obscure -- germane to the world of computer programming. Lessons In Electric Circuits copyright (C) 2000-2006 Tony R. Kuphaldt, under the terms and conditions of the Design Science License. PreviousContentsNext GW1NGL NA7KR Kevin Roberts Ham Radio Page last updated on 09/10/2012 by Kevin Roberts NA7KR a colection of Ham Radio and Electronic Information<\|endoftext\|>	4.375
8,223	A humanoid robot is a robot with its body shape built to resemble the human body. The design may be for functional purposes, such as interacting with human tools and environments, for experimental purposes, such as the study of bipedal locomotion, or for other purposes. In general, humanoid robots have a torso, a head, two arms, and two legs, though some forms of humanoid robots may model only part of the body, for example, from the waist up. Some humanoid robots also have heads designed to replicate human facial features such as eyes and mouths. Androids are humanoid robots built to aesthetically resemble humans. Humanoid robots are now used as research tools in several scientific areas. Researchers study the human body structure and behavior (biomechanics) to build humanoid robots. On the other side, the attempt to simulate the human body leads to a better understanding of it. Human cognition is a field of study which is focused on how humans learn from sensory information in order to acquire perceptual and motor skills. This knowledge is used to develop computational models of human behavior and it has been improving over time. It has been suggested that very advanced robotics will facilitate the enhancement of ordinary humans. See transhumanism. Although the initial aim of humanoid research was to build better orthosis and prosthesis for human beings, knowledge has been transferred between both disciplines. A few examples are powered leg prosthesis for neuromuscularly impaired, ankle-foot orthosis, biological realistic leg prosthesis and forearm prosthesis. Besides the research, humanoid robots are being developed to perform human tasks like personal assistance, through which they should be able to assist the sick and elderly, and dirty or dangerous jobs. Humanoids are also suitable for some procedurally-based vocations, such as reception-desk administrators and automotive manufacturing line workers. In essence, since they can use tools and operate equipment and vehicles designed for the human form, humanoids could theoretically perform any task a human being can, so long as they have the proper software. However, the complexity of doing so is immense. They are also becoming increasingly popular as entertainers. For example, Ursula, a female robot, sings, plays music, dances and speaks to her audiences at Universal Studios. Several Disney theme park shows utilize animatronic robots that look, move and speak much like human beings. Although these robots look realistic, they have no cognition or physical autonomy. Various humanoid robots and their possible applications in daily life are featured in an independent documentary film called Plug & Pray, which was released in 2010. Humanoid robots, especially those with artificial intelligence algorithms, could be useful for future dangerous and/or distant space exploration missions, without having the need to turn back around again and return to Earth once the mission is completed. Sensors can be classified according to the physical process with which they work or according to the type of measurement information that they give as output. In this case, the second approach was used. Proprioceptive sensors sense the position, the orientation and the speed of the humanoid's body and joints. In human beings the otoliths and semi-circular canals (in the inner ear) are used to maintain balance and orientation. In addition humans use their own proprioceptive sensors (e.g. touch, muscle extension, limb position) to help with their orientation. Humanoid robots use accelerometers to measure the acceleration, from which velocity can be calculated by integration; tilt sensors to measure inclination; force sensors placed in robot's hands and feet to measure contact force with environment; position sensors, that indicate the actual position of the robot (from which the velocity can be calculated by derivation) or even speed sensors. Arrays of tactels can be used to provide data on what has been touched. The Shadow Hand uses an array of 34 tactels arranged beneath its polyurethane skin on each finger tip. Tactile sensors also provide information about forces and torques transferred between the robot and other objects. Vision refers to processing data from any modality which uses the electromagnetic spectrum to produce an image. In humanoid robots it is used to recognize objects and determine their properties. Vision sensors work most similarly to the eyes of human beings. Most humanoid robots use CCD cameras as vision sensors. Sound sensors allow humanoid robots to hear speech and environmental sounds, and perform as the ears of the human being. Microphones are usually used for this task. Actuators are the motors responsible for motion in the robot. Humanoid robots are constructed in such a way that they mimic the human body, so they use actuators that perform like muscles and joints, though with a different structure. To achieve the same effect as human motion, humanoid robots use mainly rotary actuators. They can be either electric, pneumatic, hydraulic, piezoelectric or ultrasonic. Hydraulic and electric actuators have a very rigid behavior and can only be made to act in a compliant manner through the use of relatively complex feedback control strategies. While electric coreless motor actuators are better suited for high speed and low load applications, hydraulic ones operate well at low speed and high load applications. Piezoelectric actuators generate a small movement with a high force capability when voltage is applied. They can be used for ultra-precise positioning and for generating and handling high forces or pressures in static or dynamic situations. Ultrasonic actuators are designed to produce movements in a micrometer order at ultrasonic frequencies (over 20 kHz). They are useful for controlling vibration, positioning applications and quick switching. Pneumatic actuators operate on the basis of gas compressibility. As they are inflated, they expand along the axis, and as they deflate, they contract. If one end is fixed, the other will move in a linear trajectory. These actuators are intended for low speed and low/medium load applications. Between pneumatic actuators there are: cylinders, bellows, pneumatic engines, pneumatic stepper motors and pneumatic artificial muscles. Planning and control In planning and control, the essential difference between humanoids and other kinds of robots (like industrial ones) is that the movement of the robot has to be human-like, using legged locomotion, especially biped gait. The ideal planning for humanoid movements during normal walking should result in minimum energy consumption, as it does in the human body. For this reason, studies on dynamics and control of these kinds of structures has become increasingly important. The question of walking biped robots stabilization on the surface is of great importance. Maintenance of the robot’s gravity center over the center of bearing area for providing a stable position can be chosen as a goal of control. To maintain dynamic balance during the walk, a robot needs information about contact force and its current and desired motion. The solution to this problem relies on a major concept, the Zero Moment Point (ZMP). Another characteristic of humanoid robots is that they move, gather information (using sensors) on the "real world" and interact with it. They don’t stay still like factory manipulators and other robots that work in highly structured environments. To allow humanoids to move in complex environments, planning and control must focus on self-collision detection, path planning and obstacle avoidance. Humanoid robots do not yet have some features of the human body. They include structures with variable flexibility, which provide safety (to the robot itself and to the people), and redundancy of movements, i.e. more degrees of freedom and therefore wide task availability. Although these characteristics are desirable to humanoid robots, they will bring more complexity and new problems to planning and control. The field of whole-body control deals with these issues and addresses the proper coordination of numerous degrees of freedom, e.g. to realize several control tasks simultaneously while following a given order of priority. Timeline of developments This section's factual accuracy is disputed. (June 2010) (Learn how and when to remove this template message) \|c. 250 BC\|\|The Liezi described an automaton.\| \|c. 50 AD\|\|Greek mathematician Hero of Alexandria described a machine that automatically pours wine for party guests.\| \|1206\|\|Al-Jazari described a band made up of humanoid automata which, according to Charles B. Fowler, performed "more than fifty facial and body actions during each musical selection." Al-Jazari also created hand washing automata with automatic humanoid servants,[verification needed] and an elephant clock incorporating an automatic humanoid mahout striking a cymbal on the half-hour. His programmable "castle clock" also featured five musician automata which automatically played music when moved by levers operated by a hidden camshaft attached to a water wheel.\| \|1495\|\|Leonardo da Vinci designs a humanoid automaton that looks like an armored knight, known as Leonardo's robot.\| \|1738\|\|Jacques de Vaucanson builds The Flute Player, a life-size figure of a shepherd that could play twelve songs on the flute and The Tambourine Player that played a flute and a drum or tambourine.\| \|1774\|\|Pierre Jacquet-Droz and his son Henri-Louis created the Draughtsman, the Musicienne and the Writer, a figure of a boy that could write messages up to 40 characters long.\| Nikola Tesla publicly demonstrates his "automaton" technology by wirelessly controlling a model boat at the Electrical Exposition held at Madison Square Garden in New York City during the height of the Spanish–American War. \|1921\|\|Czech writer Karel Čapek introduced the word "robot" in his play R.U.R. (Rossum's Universal Robots). The word "robot" comes from the word "robota", meaning, in Czech and Polish, "labour, drudgery".\| \|1927\|\|The Maschinenmensch ("machine-human"), a gynoid humanoid robot, also called "Parody", "Futura", "Robotrix", or the "Maria impersonator" (played by German actress Brigitte Helm), perhaps the most memorable humanoid robot ever to appear on film, is depicted in Fritz Lang's film Metropolis.\| \|1928\|\|The electrical robot Eric opens an exhibition of the Society of Model Engineers at London's Royal Horticultural Hall in London, and tours the world\| \|1941-42\|\|Isaac Asimov formulates the Three Laws of Robotics, used in his robot science fiction stories, and in the process of doing so, coins the word "robotics".\| \|1948\|\|Norbert Wiener formulates the principles of cybernetics, the basis of practical robotics.\| \|1961\|\|The first digitally operated and programmable non-humanoid robot, the Unimate, is installed on a General Motors assembly line to lift hot pieces of metal from a die casting machine and stack them. It was created by George Devol and constructed by Unimation, the first robot manufacturing company.\| \|1967 to 1972\|\|Waseda University initiated the WABOT project in 1967, and in 1972 completed the WABOT-1, the world's first full-scale humanoid intelligent robot. It was the first android, able to walk, communicate with a person in Japanese (with an artificial mouth), measure distances and directions to the objects using external receptors (artificial ears and eyes), and grip and transport objects with hands.\| \|1969\|\|D.E. Whitney publishes his article "Resolved motion rate control of manipulators and human prosthesis".\| \|1970\|\|Miomir Vukobratović has proposed Zero Moment Point, a theoretical model to explain biped locomotion.\| \|1972\|\|Miomir Vukobratović and his associates at Mihajlo Pupin Institute build the first active anthropomorphic exoskeleton.\| \|1980\|\|Marc Raibert established the MIT Leg Lab, which is dedicated to studying legged locomotion and building dynamic legged robots.\| \|1983\|\|Using MB Associates arms, "Greenman" was developed by Space and Naval Warfare Systems Center, San Diego. It had an exoskeletal master controller with kinematic equivalency and spatial correspondence of the torso, arms, and head. Its vision system consisted of two 525-line video cameras each having a 35-degree field of view and video camera eyepiece monitors mounted in an aviator's helmet.\| \|1984\|\|At Waseda University, the Wabot-2 is created, a musician humanoid robot able to communicate with a person, read a normal musical score with his eyes and play tunes of average difficulty on an electronic organ.\| \|1985\|\|Developed by Hitachi Ltd, WHL-11 is a biped robot capable of static walking on a flat surface at 13 seconds per step and it can also turn.\| \|1985\|\|WASUBOT is another musician robot from Waseda University. It performed a concerto with the NHK Symphony Orchestra at the opening ceremony of the International Science and Technology Exposition.\| \|1986\|\|Honda developed seven biped robots which were designated E0 (Experimental Model 0) through E6. E0 was in 1986, E1 – E3 were done between 1987 and 1991, and E4 - E6 were done between 1991 and 1993.\| \|1989\|\|Manny was a full-scale anthropomorphic robot with 42 degrees of freedom developed at Battelle's Pacific Northwest Laboratories in Richland, Washington, for the US Army's Dugway Proving Ground in Utah. It could not walk on its own but it could crawl, and had an artificial respiratory system to simulate breathing and sweating.\| \|1990\|\|Tad McGeer showed that a biped mechanical structure with knees could walk passively down a sloping surface.\| \|1993\|\|Honda developed P1 (Prototype Model 1) through P3, an evolution from E series, with upper limbs. Developed until 1997.\| \|1995\|\|Hadaly was developed in Waseda University to study human-robot communication and has three subsystems: a head-eye subsystem, a voice control system for listening and speaking in Japanese, and a motion-control subsystem to use the arms to point toward campus destinations.\| \|1995\|\|Wabian is a human-size biped walking robot from Waseda University.\| \|1996\|\|Saika, a light-weight, human-size and low-cost humanoid robot, was developed at Tokyo University. Saika has a two-DOF neck, dual five-DOF upper arms, a torso and a head. Several types of hands and forearms are under development also. Developed until 1998.\| \|1997\|\|Hadaly-2, developed at Waseda University, is a humanoid robot which realizes interactive communication with humans. It communicates not only informationally, but also physically.\| \|2000\|\|Honda creates its 11th bipedal humanoid robot, able to run, ASIMO.\| \|2001\|\|Sony unveils small humanoid entertainment robots, dubbed Sony Dream Robot (SDR). Renamed Qrio in 2003.\| \|2001\|\|Fujitsu realized its first commercial humanoid robot named HOAP-1. Its successors HOAP-2 and HOAP-3 were announced in 2003 and 2005, respectively. HOAP is designed for a broad range of applications for R&D of robot technologies.\| \|2002\|\|HRP-2, biped walking robot built by the Manufacturing Science and Technology Center (MSTC) in Tokyo.\| \|2003\|\|JOHNNIE, an autonomous biped walking robot built at the Technical University of Munich. The main objective was to realize an anthropomorphic walking machine with a human-like, dynamically stable gait.\| \|2003\|\|Actroid, a robot with realistic silicone "skin" developed by Osaka University in conjunction with Kokoro Company Ltd.\| \|2004\|\|Persia, Iran's first humanoid robot, was developed using realistic simulation by researchers of Isfahan University of Technology in conjunction with ISTT.\| \|2004\|\|KHR-1, a programmable bipedal humanoid robot introduced in June 2004 by a Japanese company Kondo Kagaku.\| \|2005\|\|The PKD Android, a conversational humanoid robot made in the likeness of science fiction novelist Philip K Dick, was developed as a collaboration between Hanson Robotics, the FedEx Institute of Technology, and the University of Memphis.\| \|2005\|\|Wakamaru, a Japanese domestic robot made by Mitsubishi Heavy Industries, primarily intended to provide companionship to elderly and disabled people.\| \|2005\|\|The Geminoid series is a series of ultra-realistic humanoid robots or Actroid developed by Hiroshi Ishiguro of ATR and Kokoro in Tokyo. The original one, Geminoid HI-1 was made at its image. Followed Geminoid-F in 2010 and Geminoid-DK in 2011.\| \|2006\|\|Nao is a small open source programmable humanoid robot developed by Aldebaran Robotics, in France. Widely used by worldwide universities as a research platform and educational tool.\| \|2006\|\|RoboTurk is designed and realized by Dr Davut Akdas and Dr Sabri Bicakci at Balikesir University. This Research Project Sponsored By The Scientific And Technological Research Council Of Turkey (TUBITAK) in 2006. RoboTurk is successor of biped robots named "Salford Lady" and "Gonzalez" at university of Salford in the UK. It is the first humanoid robot supported by Turkish Government.\| \|2006\|\|REEM-A was the first fully autonomous European biped humanoid robot, designed to play chess with the Hydra Chess engine. The first robot developed by PAL Robotics, it was also used as a walking, manipulation, speech and vision development platform.\| \|2006\|\|iCub, a biped humanoid open source robot for cognition research.\| \|2006\|\|Mahru, a network-based biped humanoid robot developed in South Korea.\| \|2007\|\|TOPIO, a ping pong playing robot developed by TOSY Robotics JSC.\| \|2007\|\|Twendy-One, a robot developed by the WASEDA University Sugano Laboratory for home assistance services. It is not biped, as it uses an omni-directional mobile mechanism.\| \|2008\|\|Justin, a humanoid robot developed by the German Aerospace Center (DLR).\| \|2008\|\|KT-X, the first international humanoid robot developed as a collaboration between the five-time consecutive RoboCup champions, Team Osaka, and KumoTek Robotics.\| \|2008\|\|Nexi, the first mobile, dexterous and social robot, makes its public debut as one of TIME magazine's top inventions of the year. The robot was built through a collaboration between the MIT Media Lab Personal Robots Group, UMass Amherst and Meka robotics.\| \|2008\|\|Salvius, The first open source humanoid robot built in the United States is created.\| \|2008\|\|REEM-B, the second biped humanoid robot developed by PAL Robotics. It has the ability to autonomously learn its environment using various sensors and carry 20% of its own weight.\| \|2008\|\|Surena, This robot was introduced in December 13, 2008. It had a height of 165 centimetres and weight of 60 kilograms, and is able to speak according to predefined text. It also has remote control and tracking ability.\| \|2009\|\|HRP-4C, a Japanese domestic robot made by National Institute of Advanced Industrial Science and Technology, shows human characteristics in addition to bipedal walking.\| \|2009\|\|Turkey's first dynamically walking humanoid robot, SURALP, is developed by Sabanci University in conjunction with Tubitak.\| \|2009\|\|Kobian, a robot developed by WASEDA University can walk, talk and mimic emotions.\| \|2009\|\|DARwIn-OP, an open source robot developed by ROBOTIS in collaboration with Virginia Tech, Purdue University, and University of Pennsylvania. This project was supported and sponsored by NSF.\| \|2010\|\|NASA and General Motors revealed Robonaut 2, a very advanced humanoid robot. It was part of the payload of Shuttle Discovery on the successful launch February 24, 2011. It is intended to do spacewalks for NASA.\| \|2010\|\|Researchers at Japan's National Institute of Advanced Industrial Science and Technology demonstrate their humanoid robot HRP-4C singing and dancing along with human dancers.\| \|2010\|\|In September the National Institute of Advanced Industrial Science and Technology also demonstrates the humanoid robot HRP-4. The HRP-4 resembles the HRP-4C in some regards but is called "athletic" and is not a gynoid.\| \|2010\|\|REEM, a humanoid service robot with a wheeled mobile base. Developed by PAL Robotics, it can perform autonomous navigation in various surroundings and has voice and face recognition capabilities.\| \|2011\|\|Robot Auriga was developed by Ali Özgün HIRLAK and Burak Özdemir in 2011 at University of Cukurova. Auriga is the first brain controlled robot, designed in Turkey. Auriga can service food and medicine to paralysed people by patient's thoughts. EEG technology is adapted for manipulation of the robot. The project was supported by Turkish Government.\| \|2011\|\|In November Honda unveiled its second generation Honda Asimo Robot. The all new Asimo is the first version of the robot with semi-autonomous capabilities.\| \|2012\|\|In April, the Advanced Robotics Department in Italian Institute of Technology released its first version of the COmpliant huMANoid robot COMAN which is designed for robust dynamic walking and balancing in rough terrain.\| \|2013\|\|On December 20–21, 2013 DARPA Robotics Challenge ranked the top 16 humanoid robots competing for the US$2 million cash prize. The leading team, SCHAFT, with 27 out of a possible score of 30 was bought by Google. PAL Robotics launches REEM-C the first humanoid biped robot developed as a robotics research platform 100% ROS based.\| \|2014\|\|Manav – India's first 3D printed humanoid robot developed in the laboratory of A-SET Training and Research Institutes by Diwakar Vaish (head Robotics and Research, A-SET Training and Research Institutes).\| \|2014\|\|After the acquisition of Aldebaran, SoftBank Robotics releases the Pepper robot available for everyone.\| \|2014\|\|Nadine is a female humanoid social robot designed in Nanyang Technological University, Singapore, and modelled on its director Professor Nadia Magnenat Thalmann. Nadine is a socially intelligent robot which returns greetings, makes eye contact, and remembers all the conversations it has had.\| \|2015\|\|Sophia is a humanoid robot developed by "Hanson Robotics", Hong Kong, and modelled after Audrey Hepburn. Sophia has artificial intelligence, visual data processing and facial recognition.\| \|2016\|\|OceanOne, developed by a team at Stanford University, led by computer science professor Oussama Khatib, completes its first mission, diving for treasure in a shipwreck off the coast of France, at a depth of 100 meters. The robot is controlled remotely, has haptic sensors in its hands, and artificial intelligence capabilities.\| \|2017\|\|PAL Robotics launches TALOS, a fully electrical humanoid robot with joint torque sensors and EtherCAT communication technology that can manipulate up to 6Kg payload in each of its grippers.\| Humanoid Robots portrayed in 21st-century films and television shows This section contains information of unclear or questionable importance or relevance to the article's subject matter. (August 2018) (Learn how and when to remove this template message) In the selected 21st-century films and television shows, humanoid robots (sometimes also referred to as "synthetic humans" or "replicants") are portrayed which are virtually indistinguishable from real humans. \|TV Show\|\|Release Date\|\|Rotten Tomatoes\|\|Metacritic\|\|Seasons\| \|Humans\|\|14 June 2015\|\|91%\|\|73%\|\|3 (as of 19/05/2018)\| \|Altered Carbon\|\|2 February 2018\|\|65%\|\|64%\|\|2 (as of 16/01/2019)\| \|Film\|\|Release Date\|\|Rotten Tomatoes\|\|Metacritic\|\|CinemaScore\| \|I, Robot\|\|7 July 2004\|\|57% (203 reviews)\|\|59% (38 reviews)\|\|A-\| \|Ex Machina\|\|7 May 2015\|\|92% (251 reviews)\|\|78% (42 reviews)\|\|N/A\| \|Blade Runner 2049\|\|5 October 2017\|\|87% (380 reviews)\|\|81% (54 reviews)\|\|A-\| \|Prometheus\|\|7 June 2012\|\|73% (292 reviews)\|\|64% (43 reviews)\|\|B\| - "A Ping-Pong-Playing Terminator". Popular Science. Archived from the original on 2011-03-29. - "Best robot 2009". www.gadgetrivia.com. Archived from the original on July 24, 2010. - "Shadow Robot Company: The Hand Technical Specification". Archived from the original on 2008-07-08. Retrieved 2009-04-09. - Bazylev D.N.; et al. (2015). "Approaches for stabilizing of biped robots in a standing position on movable support". Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 15 (3): 418. Archived from the original on 2015-07-08. - Dietrich, A., Whole-Body Impedance Control of Wheeled Humanoid Robots, ISBN 978-3-319-40556-8, Springer International Publishing, 2016, "Archived copy". Archived from the original on 2017-09-07. Retrieved 2017-08-31.CS1 maint: Archived copy as title (link) - Joseph Needham (1986), Science and Civilization in China: Volume 2, p. 53, England: Cambridge University Press - Hero of Alexandria; Bennet Woodcroft (trans.) (1851). Temple Doors opened by Fire on an Altar. Pneumatics of Hero of Alexandria. London: Taylor Walton and Maberly (online edition from University of Rochester, Rochester, NY). Retrieved on 2008-04-23. - Fowler, Charles B. (October 1967), "The Museum of Music: A History of Mechanical Instruments", Music Educators Journal 54 (2): 45-9 - Rosheim, Mark E. (1994). Robot Evolution: The Development of Anthrobotics. Wiley-IEEE. pp. 9–10. ISBN 0-471-02622-0. - "Ancient Discoveries, Episode 11: Ancient Robots". History Channel. Archived from the original on 2014-03-01. Retrieved 2008-09-06. - "MegaGiant Robotics". megagiant.com. Archived from the original on 2007-08-19. - "Archived copy". Archived from the original on 2005-09-12. Retrieved 2005-09-12.CS1 maint: Archived copy as title (link) - "Archived copy". Archived from the original on 2006-05-22. Retrieved 2005-11-15.CS1 maint: Archived copy as title (link) - "Humanoid History -WABOT-". www.humanoid.waseda.ac.jp. Archived from the original on 1 September 2017. Retrieved 3 May 2018. - Zeghloul, Saïd; Laribi, Med Amine; Gazeau, Jean-Pierre (21 September 2015). "Robotics and Mechatronics: Proceedings of the 4th IFToMM International Symposium on Robotics and Mechatronics". Springer. Retrieved 3 May 2018 – via Google Books. - "Historical Android Projects". androidworld.com. Archived from the original on 2005-11-25. - Robots: From Science Fiction to Technological Revolution, page 130 - Duffy, Vincent G. (19 April 2016). "Handbook of Digital Human Modeling: Research for Applied Ergonomics and Human Factors Engineering". CRC Press. Retrieved 3 May 2018 – via Google Books. - Resolved motion rate control of manipulators and human prostheses DE Whitney - IEEE Transactions on Man-Machine Systems, 1969 - [permanent dead link] - "Electric Dreams - Marc Raibert". robosapiens.mit.edu. Archived from the original on 8 May 2005. Retrieved 3 May 2018. - "Archived copy". Archived from the original on 2005-10-19. Retrieved 2005-11-15.CS1 maint: Archived copy as title (link) - "Honda｜ASIMO｜ロボット開発の歴史". honda.co.jp. Archived from the original on 2005-12-29. - "droidlogic.com". droidlogic.com. Archived from the original on January 22, 2008. - "Research & Development". Archived from the original on 2008-05-09. - "Humanoid Robotics". Archived from the original on 2016-02-09. - "Archived copy". Archived from the original on 2006-06-15. Retrieved 2007-12-07.CS1 maint: Archived copy as title (link) - "新サイトへ". kokoro-dreams.co.jp. Archived from the original on 2006-10-23. - "Humanoid Robot - Dynamics and Robotics Center". Archived from the original on 2016-09-19. - "PKD Android". pkdandroid.org. Archived from the original on 2009-10-01. Retrieved 2019-01-29. - "Archived copy". Archived from the original on 2007-07-01. Retrieved 2007-07-02.CS1 maint: Archived copy as title (link) - "Aldebaran Robotics". Aldebaran Robotics. Archived from the original on 2010-06-14. - Dr Davut Akdas, "Archived copy". Archived from the original on 2012-07-13. Retrieved 2013-07-10.CS1 maint: Archived copy as title (link), RoboTurk, - Eduard Gamonal. "PAL Robotics — advanced full-size humanoid service robots for events and research world-wide". pal-robotics.com. Archived from the original on 2012-01-04. - "iCub.org". icub.org. Archived from the original on 2010-07-16. - Erico Guizzo. "Humanoid Robot Mahru Mimics a Person's Movements in Real Time". ieee.org. Archived from the original on 2012-10-20. - Roxana Deduleasa (5 December 2007). "I, the Ping-Pong Robot!". softpedia. Archived from the original on 2 February 2009. - 早稲田大学理工学部機械工学科菅野研究室 TWENDYチーム. "TWENDY-ONE". twendyone.com. Archived from the original on 2012-12-21. - redaktion dlr.de; db. "DLR Portal - Der Mensch im Mittelpunkt - DLR präsentiert auf der AUTOMATICA ein neues Chirurgie-System". dlr.de. Archived from the original on 2014-04-29. - "Archived copy". Archived from the original on 2010-01-06. Retrieved 2009-09-05.CS1 maint: Archived copy as title (link) - "Best Inventions Of 2008". Time. 2008-10-29. Archived from the original on 2012-11-07. - "Personal Robots Group". Archived from the original on 2010-04-14. - "Meka Robotics LLC". Archived from the original on 2011-01-02. - "Archived copy". Archived from the original on 2010-04-19. Retrieved 2010-04-27.CS1 maint: Archived copy as title (link) - Eduard Gamonal. "PAL Robotics — advanced full-size humanoid service robots for events and research world-wide". pal-robotics.com. Archived from the original on 2012-03-09. - "humanoid robot project". sabanciuniv.edu. Archived from the original on 2010-04-22. - "Japanese Humanoid Robot, Kobian, Walks, Talks, Crys and Laughs (VIDEO)". The Inquisitr News. Archived from the original on 2011-11-23. - "Say Hello to Robonaut2, NASA's Android Space Explorer of the Future". Popular Science. Archived from the original on 2010-02-07. - "How to Make a Humanoid Robot Dance". Archived from the original on 2010-11-07. - Eduard Gamonal. "PAL Robotics — advanced full-size humanoid service robots for events and research world-wide". pal-robotics.com. Archived from the original on 2011-03-13. - "'Türkler yapmış arkadaş' dedirttiler". MILLIYET HABER - TÜRKIYE'NIN HABER SITESI. 14 January 2012. Archived from the original on 6 January 2015. - "COmpliant HuMANoid Platform (COMAN)". iit.it. Archived from the original on 2012-12-05. Retrieved 2018-12-17. - "Home". theroboticschallenge.org. Archived from the original on 2015-06-11. - Menezes, Beryl. "Meet Manav, India's first 3D-printed humanoid robot". www.livemint.com. Archived from the original on 2015-09-29. Retrieved 2015-09-30. - J. Zhang J, N. Magnenat Thalmann and J. Zheng, Combining Memory and Emotion With Dialog on Social Companion: A Review, Proceedings of the ACM 29th International Conference on Computer Animation and Social Agents (CASA 2016), pp. 1-9, Geneva, Switzerland, May 23–25, 2016 - Berger, Sarah (2015-12-31). "Humanlike, Social Robot 'Nadine' Can Feel Emotions And Has A Good Memory, Scientists Claim". International Business Times. Retrieved 2016-01-12. - "How did a Stanford-designed 'humanoid' discover a vase from a Louis XIV shipwreck?". montereyherald.com. Archived from the original on 21 October 2017. Retrieved 3 May 2018. - TALOS: A new humanoid research platform targeted for industrial applications - "HUMANS". Rotten Tomatoes. Retrieved 2018-11-16. - "Altered Carbon". Rotten Tomatoes. Retrieved 2018-11-16. - "Ex Machina". Rotten Tomatoes. Retrieved 2018-11-16. - "Blade Runner 2049". Rotten Tomatoes. Retrieved 2018-11-16. - "Prometheus". Rotten Tomatoes. Retrieved 2018-11-16. - Asada, H. and Slotine, J.-J. E. (1986). Robot Analysis and Control. Wiley. ISBN 0-471-83029-1. - Arkin, Ronald C. (1998). Behavior-Based Robotics. MIT Press. ISBN 0-262-01165-4. - Brady, M., Hollerbach, J.M., Johnson, T., Lozano-Perez, T. and Mason, M. (1982), Robot Motion: Planning and Control. MIT Press. ISBN 0-262-02182-X. - Horn, Berthold, K. P. (1986). Robot Vision. MIT Press. ISBN 0-262-08159-8. - Craig, J. J. (1986). Introduction to Robotics: Mechanics and Control. Addison Wesley. ISBN 0-201-09528-9. - Everett, H. R. (1995). Sensors for Mobile Robots: Theory and Application. AK Peters. ISBN 1-56881-048-2. - Kortenkamp, D., Bonasso, R., Murphy, R. (1998). Artificial Intelligence and Mobile Robots. MIT Press. ISBN 0-262-61137-6. - Poole, D., Mackworth, A. and Goebel, R. (1998), Computational Intelligence: A Logical Approach. Oxford University Press. ISBN 0-19-510270-3. - Russell, R. A. (1990). Robot Tactile Sensing. Prentice Hall. ISBN 0-13-781592-1. - Russell, S. J. & Norvig, P. (1995). Artificial Intelligence: A Modern Approach. Prentice-Hall. Prentice Hall. ISBN 0-13-790395-2. http://www.techentice.com/manav-indias-first-3d-printed-robot-from-iit-mumbai/ http://www.livemint.com/Industry/rc86Iu7h3rb44087oDts1H/Meet-Manav-Indias-first-3Dprinted-humanoid-robot.html - Carpenter, J., Davis, J., Erwin‐Stewart, N., Lee. T., Bransford, J. & Vye, N. (2009). Gender representation in humanoid robots for domestic use. International Journal of Social Robotics (special issue). 1 (3), 261‐265.The Netherlands: Springer. - Carpenter, J., Davis, J., Erwin‐Stewart, N., Lee. T., Bransford, J. & Vye, N. (2008). Invisible machinery in function, not form: User expectations of a domestic humanoid robot. Proceedings of 6th conference on Design and Emotion. Hong Kong, China. - Williams, Karl P. (2004). Build Your Own Human Robots: 6 Amazing and Affordable Projects. McGraw-Hill/TAB Electronics. ISBN 0-07-142274-9. ISBN 978-0-07-142274-1. \|Wikimedia Commons has media related to Humanoid robots.\|<\|endoftext\|>	3.921875
866	Aug 01, 2018 by editorial team Do you know how to help your child with their social skills? As a parent, you have a unique opportunity to teach your child social skills even adults struggle with, like building friendships, managing conflict, and handling rejection. Developing social skills early is a critical part of your child’s future happiness. Adolescents who have strong social skills are more likely to be accepted by their peers, develop friendships, be viewed as effective problem solvers and perform better academically. It is the foundation for success in all areas of life! Skills that all children should learn While some children will always be more outgoing than others, it's helpful for all children to learn basic social skills. According to the University of Memphis, “following directions, holding a proper conversation, listening, giving compliments, proper behavior during transition times, teasing, bullying, or just "hanging out" with friends” are skills that children of all ages can learn. The goal is not to make your child into a social butterfly, but to help them form meaningful bonds with other people, interact with others in an appropriate way, and know how to handle uncomfortable social situations. Social skills for preschool and elementary school kids Researchers at Vanderbilt University found that these skills are essential to have for elementary school children: Listening to others Following the steps Following the rules Asking for help Taking turns when you talk Getting along with others Staying calm with others Being responsible for your behavior Doing nice things for others These are fundamental skills and behaviors that will help your child succeed in a world that values social interaction, and will help them build more advanced social skills later on. Skills for pre-teens and teens As social interactions get more complex, your child’s social skills may need to increase. According to James Windell, teens can increase their social and emotional intelligence, if they learn how to: Set personal goals Identify and change self-defeating behaviors Be assertive about their needs Have feelings for others Handle anger constructively Resolve conflicts peacefully Easier said than done, right? Here is how you can help your child develop appropriate social skills at any age. Model positive social skills Your children may not always listen to what you say, but they are always watching what you do. Show confidence, be friendly to strangers and treat your child with respect, no matter how badly they are behaving at any given moment. Don’t label your child as “shy” Labeling your child as “shy” has the potential to become a self-fulfilling prophecy. Instead, acknowledge their feelings and show them how they can overcome their fears. Give them opportunities to practice Supporting and encouraging your child's' friendships is very important. In addition to driving them to dance practice and organizing playdates, you can let your children practice their social skills in a digital environment. Helping your child develop social skills throughout their life can be a critical part of your child’s happiness. Limiting their time on technology so that they have plenty of real-life experiences to develop social skills is key. They will learn more from what they witness than from what you tell them, so being mindful of what you model and being present and available to discuss what else your children see is valuable in developing versatile social skills. For more, check out our Family Detox Guide. And please register below for our free newsletter! Would you like to join a community of like-minded conscious consumers as well? Do you want to read less about practical nontoxic choices and just see checklists of thoughtful options? Then join the D-Tox Academy! Each month, we will "meditate" on a body part or system. The goal is to connect with our body, senses, and symptoms to rely on this curiosity and "listening" as guidance for a gentle, detox journey. Access Sophia's shopping list for her household staples. They're her favorite low toxic items that she can't live without. Also see which EMF protection products she uses.<\|endoftext\|>	3.71875
419	We found 709 reviewed resources for physical fitness Applying Physical & Health Literacy 3 mins 6th - 12th CCSS: Adaptable Raise teenagers' awareness about the importance of living an active life and staying physically fit by using this short health video. Starting with a series of facts about the negative consequences of physical inactivity and unbalanced... Exploring Physical & Health Literacy 3 mins 4th - 8th CCSS: Adaptable Help students discover the importance of physical fitness with this health literacy video. Beginning with a comparison of the choices made by two fictional teenagers, this resource continues on to explore the basic physical and thinking... The Physics of Sports: An 8th Grade Physical Science Project 6th - 12th CCSS: Designed Explore the relationship between sports and physics in a cross-curricular lesson. Middle and high schoolers prepare a multimedia presentation based on a chosen sport. They answer five physics vocabulary questions about how the laws of... Introduction to Physical & Health Literacy 3 mins K - 5th CCSS: Adaptable Support children in living fun, active, and healthy lives with a video on physical and health literacy. Whether its learning basic skills like running, jumping, and throwing, or making healthy choices about the foods they eat, students... Do Something about… Eating Healthy - Day 9: Fitting in Fitness 9th - 12th Lesson 9 in this 10-lesson unit put together by Do Something, Inc. is about fitting fitness into daily activivty. First off though, the class needs to take a look at what kind, and how much, activity they do in a day, in a week, etc.... Other popular searches - Indoor Physical Fitness - Physical Fitness Components - Physical Fitness Unit Plan - Physical Fitness Test - Physical Fitness Activities - Physical Fitness Lessons - Flexibility Physical Fitness - Aspects of Physical Fitness - Health and Physical Fitness - Physical Fitness Programs - Physical Fitness Games - Physical Fitness Year Plan<\|endoftext\|>	4
365	# Angles in triangles The three angles in any triangle add up to 180°. You can use this information to find a missing angle. ## Scalene triangle A scalene triangle has three different angles. To find a missing angle you need to know the other two angles. $x^\circ = 180^\circ - (70^\circ + 50^\circ )$ $= 180^\circ - 120^\circ = 60^\circ$ ## Isosceles triangle An isosceles triangle is one with two sides equal in length and two equal angles. You only need to know one angle in an isosceles triangle to work out the other two. Question In the diagram below the triangle is isosceles. What is the value of $$a^\circ$$? $a^\circ = 180^\circ - (70^\circ + 70^\circ )$ $= 180^\circ - 140^\circ = 40^\circ$ Question What is the size of angle $${p}$$? This is an isosceles triangle, so both the bottom angles are $${p}$$. The angles in a triangle add up to $${180}^\circ$$, so: $p + p + 40 = 180$ $2p + 40 = 180$ $2p = 140$ $p = 70$ So the missing angles are both $$70^\circ$$. ## Equilateral triangle An equilateral triangle is one in which all three angles are equal. The angles add up to 180°, so each angle is 60°. You don't need to be told any angles in an equilateral to find a missing angle. The angles are always 60°.<\|endoftext\|>	4.5625
1,149	Watch the video or read the steps below: How to find the mean of the probability distribution : Overview Finding the mean of a probability distribution is easy in probability and statistics—if you know how. This how to will guide you through a few simple steps necessary to find the mean of the probability distribution or binomial distribution. You’ll often find these types of questions in textbook chapters on binomial probability distributions. The binomial distribution is just a simple trial where there are two outcomes: success or failure. For example, if you are counting how many times you draw an Ace from a deck of cards, you could assign “Success” to “Drawing an Ace” and “Failure” to drawing any other card. You can find the mean of the probability distribution by creating a probability table. How to find the mean of the probability distribution: Steps Sample question: “A grocery store has determined that in crates of tomatoes, 95% carry no rotten tomatoes, 2% carry one rotten tomato, 2% carry two rotten tomatoes, and 1% carry three rotten tomatoes. Find the mean number of rotten tomatoes in the crates.” - Step 1: Convert all the percentages to decimal probabilities. For example: 95% = .95 2% = .02 2% = .02 1% = .01 - Step 2: Construct a probability distribution table. (If you don’t know how to do this, see how to construct a probability distribution).) - Step 3: Multiply the values in each column. (In other words, multiply each value of X by each probability P(X).) Referring to our probability distribution table: 0 × .95 = 0 1 × .02 = .02 2 × .02 = .04 3 × .01 = .03 - Step 4: Add the results from step 3 together. 0 + .02 + .04 + .03 = .09 is the mean. You’re done finding the mean for a probability distribution! A binomial distribution represents the results from a simple experiment where there is “success” or “failure.” For example, if you are polling voters to see who is voting Democrat, the voters that say they will vote Democrat is a “success” and anything else is a failure. One of the simplest binomial experiments you can perform is a coin toss, where “heads” could equal “success” and “tails” could equal “failure.” The mean of binomial distribution is much like the mean of anything else. It answers the question “If you perform this experiment many times, what’s the likely (the average) result?. Formula for Mean of Binomial Distribution The formula for the mean of binomial distribution is: μ = n p Where “n” is the number of trials and “p” is the probability of success. For example: if you tossed a coin 10 times to see how many heads come up, your probability is .5 (i.e. you have a 50 percent chance of getting a heads and 50 percent chance of a tails) and “n” is how many trials — 10. Therefore, the mean of this particular binomial distribution is: 10 .5 = 5. This makes sense: if you toss a coin ten times you would expect heads to show up on average, 5 times. Mean for a Binomial Distribution on the TI-83 Sample problem: Find the mean for a binomial distribution with n = 5 and p = 0.12. Again, the TI 83 doesn’t have a function for this. But if you know the formula (np), it’s pretty easy to enter it on the home screen. Step 1: Multiply n by p. 5 .12 ENTER Hey, that was easy! Something to think about: You may be wondering why it was so easy to calculate the mean. After all being asked to “calculate the mean for a binomial distribution” sounds scary. If you think about what a mean (or average) is, then you’ll see why it was so easy. In the sample question, n = 5 and p = 0.12. What is “n”? That’s the number of items. So imagine a list of 5 items with a certain score: 1 = 0.12 2 = 0.12 3 = 0.12 4 = 0.12 5 = 0.12 If you were asked to find the average score for those five items, you wouldn’t even have to do the math: it’s just 0.12, right? Finding the mean for a binomial distribution is just a little different: you add up all of the probabilities (0.12 + 0.12 + 0.12 + 0.12 + 0.12). Or a faster way, just multiply n by p. Check out our Youtube channel for more help and stats tips!------------------------------------------------------------------------------ Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free! Comments? Need to post a correction? Please post a comment on our Facebook page.<\|endoftext\|>	4.21875
540	ZIMSEC O Level Geography Notes: River deposition - Deposition occurs when a river no longer as sufficient energy to transport its load. - When its velocity begins to fall and has less energy, a river’s competence (maximum size of material which a river is capable of transporting) and capacity ( maximum amount of load that a river is capable of transporting) falls and therefore deposition begins. - Deposition occurs when: - Discharge is reduced after a period of low precipitation. - Velocity is reduced upon the river reaching the dam, lake,sea or ocean resulting in the formation of deltas. - Shallow water occurs on the inside section of a meander for example. - The load is suddenly increased for example in the event of a landslide for instance when a portion of bank collapses into the river. - When the river overflows its banks so that the velocity outside the channel is reduced resulting in the formation of a floodplain. - During floods, especially in the lower course rivers spread to the sides of the channel. - Frictional drag and the reduced gradient slow down the flowing water resulting in deposition. - Deposition occurs along the entire course of the river: - On the channel bed. - The river valley floor especially during floods. - On the river’s banks as in a meander. - At the river’s mouth when it empties into the sea. NB Deposition occurs at any part of a river’s course depending on a river’s energy and velocity. The division of a river into stages is therefore useful but by no means conclusive. - When the river loses its energy to any of the reasons pointed out above the following happens. - The heaviest material/load is deposited first this is why rivers are littered with boulders in the upper course. - This is because traction load and siltation loads require more energy to transport. - The finest material is deposited last and may reach the sea where it is deposited onto and to form deltas. - The dissolved load which is in solution water is deposited at all but transported to the sea where it maintains the saltiness of oceans. - The deposition of sand and silt leads to the development of a gently sloping plain known as a flood plain. - Deposition can result in aggredation where the river’s bed and gradient are increased. This can happen at deltas and on alluvial fans. To learn more about the features formed by river erosion and to access more topics go to the Geography Notes page.<\|endoftext\|>	3.90625
850	We study how viral RNA polymerases work and contribute to RNA virus pathogenicity. Emerging RNA viruses diseases have a severe impact on our economy (approximately $90 billion in the USA per year), but they also pose an unpredictable threat to our health. For instance, influenza A virus outbreaks in 1918, 1957, 1968 and 2009 have caused millions of deaths. Unfortunately, we know still relatively little about how pathogens like the influenza A virus cause lethal disease and how they amplify themselves. However, we do know that both are strongly linked to the activity of the viral enzyme that copies the viral genome. Clearly, we need to make an effort to study this enzyme in more detail. RNA viruses use RNA as the genetic material of their genome. This RNA can be single-stranded (ssRNA) or double-stranded (dsRNA). In addition, the single-stranded RNA can be of positive sense (+RNA), which means that it is infectious in a cell, or negative sense -RNA), which means that it needs to be associated with viral proteins in order to be infectious. Important human viruses belong to ssRNA viruses, such as the influenza A virus and Ebolavirus (both -RNA) or the SARS-coronavirus and Denguevirus (both +RNA). The influenza A virus Influenza A virus strains are identified by the glycoproteins that reside on the outside of the virus: the haemagglutinin (HA) protein and the neuraminidase (NA). For simplicity, these different HA and NA proteins are numbered (e.g. H1N1). For humans, the H1N1, H2N2, and H3N2 subtypes have historically been the most important. In addition to subtyping, human influenza A viruses can be classified into seasonal, pandemic or zoonotic influenza A viruses. The outcome of infections with these three viruses is strikingly different. Seasonal \| Infections with seasonal influenza viruses usually lead to mild upper respiratory tract disease. This affects 5-10% of the adult population and 20-30% of children every year. However, in up to 5 million cases seasonal influenza infections require hospitalisation and in up to half a million of these cases may be fatal. Zoonotic \| By contrast, infections with zoonotic influenza A strains, such as the avian H5N1 or H7N9 viruses, is much more severe. Typically, these infections lead to severe pneumonia and are currently estimated to have a 38% (for H7N9) to 50% (for H5N1) or mortality rate. Pandemic \| Infections with pandemic influenza A viruses vary from relatively mild (the 2009 pandemic) to devastating (the 1918 Spanish Flu). The latter pandemic has been estimated to have killed 50 million people worldwide. An RNA viruses genome can only be replicated and transcribed by a viral RNA polymerase. This is a specialised enzyme that is able to RNA as template and make more RNA using nucleotides as substrate. Domains \| The heart of the RNA polymerase is the RNA polymerase domain, a protein fold that is conserved among many RNA viruses. This domain can be flanked by additional domains that contribute other enzymatic functions, such as cap-snatching in the influenza A virus polymerase. The domains may also allow the RNA polymerase to interact with key cellular proteins. Error rate \| Any mistakes made by the RNA polymerase lead to mutations in the viral genome. Although these mutations frequently have deleterious effects, they can sometimes also help the virus escape antiviral or immune response pressures. It is thus beneficial for RNA viruses to have an RNA polymerase than introduces errors with a relatively high frequency (approx. one mutation per genome copy made), so they can evolve quickly. Manipulation of the RNA virus mutation is essential for medicine: life attenuated viruses often have a lower mutation frequency and a large number of antiviral drugs works by increasing the error rate of the polymerase, thereby forcing the virus to mutate essential enzymatic functions and effectively killing itself.<\|endoftext\|>	3.875
847	# A block is gently placed on a conveyor belt moving horizontal with constant speed After t=4s the velocity of the block becomes equal to velocity of the belt If the coefficient of friction between the block and the belt is μ=0.2, then the velocity of the conveyor belt is . A 8ms1 B 4ms1 C 6ms1 D 18ms1 Video Solution Text Solution Generated By DoubtnutGPT ## To solve the problem, we need to analyze the motion of the block placed on the conveyor belt. Here’s a step-by-step breakdown of the solution:Step 1: Understanding the ProblemThe block is placed on a conveyor belt that is moving with a constant speed. Initially, the block is at rest relative to the ground. The block starts to accelerate due to the friction between it and the conveyor belt until its velocity matches that of the belt after a time t=4seconds.Step 2: Identify the Forces Acting on the BlockThe only horizontal force acting on the block is the frictional force, which provides the necessary acceleration for the block to catch up with the conveyor belt. The frictional force can be expressed as:$$F_f = \mu \cdot m \cdot g$$where:- $$\mu = 0.2$$ (coefficient of friction)- $$m$$ is the mass of the block (which we will see cancels out)- $$g = 10 \, \text{m/s}^2$$ (acceleration due to gravity)Step 3: Calculate the Acceleration of the BlockThe acceleration $$a$$ of the block can be calculated from the frictional force:$$a = \frac{F_f}{m} = \frac{\mu \cdot m \cdot g}{m} = \mu \cdot g$$Substituting the values:$$a = 0.2 \cdot 10 = 2 \, \text{m/s}^2$$Step 4: Use the Kinematic EquationSince the block starts from rest, we can use the kinematic equation to find the final velocity $$v$$ of the block after time $$t = 4 \, \text{s}$$:$$v = u + at$$where:- $$u = 0$$ (initial velocity of the block)- $$a = 2 \, \text{m/s}^2$$- $$t = 4 \, \text{s}$$Substituting the values:$$v = 0 + (2 \cdot 4) = 8 \, \text{m/s}$$Step 5: ConclusionThe final velocity of the block after 4 seconds is equal to the velocity of the conveyor belt, which is:$$v = 8 \, \text{m/s}$$Thus, the velocity of the conveyor belt is 8 m/s.--- \| Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams. Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation<\|endoftext\|>	4.75
882	An important aspect of understanding what happens as gases are heated and cooled also relates to the air around them and its temperature. Watching two videos on how carbon dioxide warms up and cools down, the gas was contained in a rubber balloon. It's obvious from our previous knowledge that the balloon expanded and shrank. But the air around each of the balloons was doing something, too, which allowed it to do so. Our two dimensional models and ball-and-stick models do not show how the molecules in a substance are in constant motion. But a computer simulation will! Using an online program, students saw what happens to molecules as they are heated. How does heating a substance affect its motion? How does it affect the volume of a substance? Take a look for yourself... But it seems like when there is more energy in a substance, the molecules move faster and take up more space. Is this why I smell a hot pizza quicker across a room than a cold pizza? Hmmmmm.... Our findings: It took significantly less time for the warm ammonia to change the indicator paper than the cold ammonia. How will this help us explain how we can possibly change a substance in order to smell it? Only time will tell! So as students uncovered that air is comprised of different molecules, we're uncovering how this idea can be connected to different odors. How are odors different from one another? By smelling various smells, students are also seeing the atoms that make up a molecule of each odor. We'll be looking for patterns to distinguish one smell from another, as well as see if the molecules that determine similar smells are somehow related, too. From our investigation, we discovered three big ideas! 1. Different substances can be made of both the same types of atoms or of different atoms. 2. Substances that are made of the same types of atoms differ in two ways: either the number of atoms or the ways in which they are arranged. Comprised of many different gases, and each gas is actually a molecule, or a combination of at least two atoms. Since every gas has different properties, our idea behind particles being different makes perfect sense! We built gas molecules out of different colored marshmallows, recognizing their differences. While these models help us to see gases as different, they are limited. They don't know movement and are so incredibly small for what they really represent. I wonder where this will take us in our study! Sixth graders recently looked at various metals, all of which are defined as elements. Elements are all made of the same atom throughout. Therefore, gold is made of gold atoms, and hydrogen is made of hydrogen atoms. We briefly looked at the Periodic Table to recognize patterns between the elements, and ultimately, tried to see similarities between all the elements near one another. It's getting us thinking for sure! As we dig deeper into understanding the basics of chemistry, students are recognizing that everything has different properties because it is made of different particles. These particles are really atoms, which make up all matter. Since different atoms are structured differently, this is what must make each element unique. All this learning is helping us relate back to our driving question of "How Can We Smell From a Distance?" and our smaller sub-question now of "How Is One Odor Different from Another?" It's all starting to come together now... So if different odors are made of different particles, and odors are gases, is it fair to say that other states of matter are made of different particles, too? Using various metals (zinc, aluminum, iron, and copper), students are recognizing that these metals have unique properties that define them, too. Between their color, hardness, and malleability, they are all unique. What must this tell us about the particles that make up each of them? Hmmmm....I wonder! ;-) After quite a few students shared their thinking with the whole class, we worked through what our "consensus" should be. Great work (below is Mrs. Brinza's journal for what we agreed upon). As students complete their States of Matter Concept Maps for our technology integration class, they'll be adding these ideas to their concept maps. Here's a sneak peak into some of their work!<\|endoftext\|>	4.5625
745	The Chemical Earth K188.8.131.52 identify the difference between elements, compounds and mixtures in terms of particle theory: The particle theory: 1. Matter is made of tiny particles 2. Particles of matter are in constant motion 3. Particles of matter are held together by very strong electric forces 4. There are empty spaces between the particles of matter that are very large compared to the particles themselves. 5. Each substance has unique particles that are different from the particles of other substances 6. Temperature affects the speed of the particles. The higher the temperature, the faster the speed of the particles. K184.108.40.206 identify that the ...view middle of the document... It has a varied composition (i.e., salt and fresh water), and is mainly made up of the compound water (H2O) but also contains many water soluble sulfates and soluble carbonates, and elements like chlorine and sodium dissolved as ions. K220.127.116.11 identify and describe procedures that can be used to separate naturally occurring mixtures of: • solids of different sizes, • solids and liquids dissolved solids in liquids • liquids and gases K18.104.22.168 assess separation techniques for their suitability in separating examples of earth materials, identifying the differences in properties which enable these separations Separation Procedures: - usually separations require a number of processes or procedures to separate the constituent substances from a mixture. These separations are physical separations. • Filtration – separates undissolved solids from liquids or gases by passing the mixture through a screen such as filter paper which is fine enough to collect the particles of the solid. • Solution – usually used in combination with another separation method, and is based on the fact that some constituents in a mixture dissolve in a solvent such as water more readily than others. That is, a mixture is added to a solvent and can be separated through the fact that one constituent will dissolve more readily than the others (although to fully separate the mixture, another method such as filtration would have to be used). • Evaporation – relies upon the varying evaporating points of the constituent substances within a mixture. A mixture is heated (in an evaporating basin and the process is usually sped up using a Bunsen burner) and one substance will evaporate, leaving the other substance behind. • Crystallisation - depends on the components of the mixture having different solubilities in a selected liquid (usually water) at different temperatures. For example, a mixture of salt and baking powder are both soluble in hot water, but when the hot water (with the mixture dissolved in it) is cooled, the baking soda will crystallise because it is much less soluble at cooler temperatures. • Sedimentation – occurs when solid particles are allowed to settle from water (or other liquids) or air. This occurs most readily when the solvent is not moving. • Decantation – the process of pouring off a liquid above a solid which has been allowed to settle by sedimentation. • Sieving – the process of separating solid particles of various sizes. • Centrifugation – involves a centrifuge that spins and separates solids and liquids. • Distillation – is effective where the constituent substances of a mixture have very different boiling points. The mixture is heated and the substance with the lowest boiling point boils, is cooled in a condenser and is collected as a pure liquid. The components with the higher boiling points remain in the distilling flask. Fractional distillation is also used to...<\|endoftext\|>	3.9375
253	## Intermediate Algebra for College Students (7th Edition) $$2^{-2} + \frac{1}{2}x^{0} =\frac{3}{4}$$ $$2^{-2} + \frac{1}{2}x^{0}$$ Recall the negative exponent rule: $a^{−n}=\frac{1}{a^{n}}$ and $\frac{1}{a^{-n}} = a^{n}$ and the zero exponent rule: $b^{0}=1$ Thus, $$2^{-2} + \frac{1}{2}x^{0}$$ can be simplified as $$\frac{1}{2^{2}} + \frac{1}{2}$$ $$\frac{1}{4} + \frac{1}{2}$$ Find the least common multiple (LCM) of the factors $4$ and $2$: $4: 2 \times 2$ $2: 2$ Adjust the fractions using the LCM: $$\frac{1}{4} + \frac{1}{2}$$ $$=\frac{1}{4} + \frac{2}{4}$$ $$=\frac{3}{4}$$<\|endoftext\|>	4.4375
719	\|Abstract (english)\|\| \| Bullying is commonly defined as repeated aggressive behavior in which there is an imbalance of power or strength between the two parties (Nansel et al., 2001 ; Olweus, 1993). Bullying behaviors may be direct or overt (e.g., hitting, kicking, name-calling, or taunting) or more subtle or indirect in nature (e.g., rumor-spreading, social exclusion, friendship manipulation, or cyberbullying ; Espelage & Swearer, 2004 ; Olweus, 1993 ; Rigby, 2002). In the last 10 years, studies on bullying have focused on different integrative approaches of this complex problem. One of the most investigated approaches is the social- ecological perspective which takes account of reciprocal interplay between individuals involved in the bully/victim continuum and his complex contexts (Bronfenbrenner, 1979 ; Espelage & Swearer, 2004 ; Olweus, 1993). Bullying does not occur in isolation. This phenomenon is encouraged and/or inhibited as a result of the complex relationships between the individual, family, peer group, school, community, and culture. This study examined an ecological perspective on bullying behaviors. The aim of this study was to investigate differences between individual characteristics, family, peer, school and neighborhood contexts of victims, bullies and noninvolved children. A total of 880 primary school children (10 to 16 years old) participated in the investigation during one school semester. For testing the differences between groups we used the one way ANOVA for independent samples. Overall, the results of this study suggested statistically significant differences between bullies, victims and noninvolved children for individual characteristics and all aforementioned contexts. On individual characteristics there were statistically significant differences in empathy level, where bullies had lower levels of empathy than victims and noninvolved children, impulsivity, where bullies had higher levels of impulsivity than victims and noninvolved children, and time spent on media, where bullies had more time spent on media than victims and noninvolved children. However, there were no statistically significant differences in sex and age between these three groups. In family context, especially on parents’ behavior, we also found statistically significant differences. Bullies and victims had parents who used more negative discipline and psychological control in raising their children and showed less acceptance of their children than parents of noninvolved children. Bullies also had parents who gave them less autonomy and less supervision than parents of victims and noninvolved children. The parents’ positive discipline and permissiveness were not statistically significant. In peer context, there was only one statistically significant variable ; peer acceptance where noninvolved children were the most accepted by peers, bullies were a little less accepted and victims were the least accepted by peers. The difference in the number of friends was not statistically significant. In the school context, all measured variables were statistically significant. Noninvolved children have better school grades than victims and they also feel more safety in school than victims and bullies. For bullies, the school climate was perceived as the most negative, for victims it was less negative, and for noninvolved children it was positive. For the last investigated context, neighborhood, we found statistically significant differences. The bullies perceived the neighborhood as the most dangerous ; by the victims it was perceived as less dangerous, and for noninvolved children as the least dangerous. We can conclude that there are major differences in individual characteristics as well as in multiple contexts between children with a different bullying status. Knowing these differences, we can direct our efforts at developing focused intervention programs for all children involved in bullying behavior.<\|endoftext\|>	3.671875
1,293	If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. ### Course: Precalculus (Eureka Math/EngageNY)>Unit 4 Lesson 4: Topic B: Trigonometry and triangles # Solving for an angle with the law of sines Sal is given a triangle with two side lengths and one angle measure, and he finds all the missing angle measures using the law of sines. Created by Sal Khan. ## Want to join the conversation? • At , why do we take the inverse sine instead of dividing both sides by sine to get rid of the sine of the right side? • You can't divide the sines of two different angles. On scene , our angles are theta and 40 degrees. Although dividing by sin(theta) would remove the sine from the right side, you would only be left dividing the sine of 40 degrees and the sine of theta on the left side. However, you can use inverse sine and remove the sines that way, because the inverse sine of the sine of 40 degrees is 40 degrees and the inverse sine of the sine of theta is theta. • I didn't understand the inverse sin step. Can anyone please explain to me what it is? • The inverse sin gets him to theta by itself • Is there a Law Of Tangents? • Yes there is. The Law of Tangents is a statement involving the tangents of two angles in a triangle and the lengths of opposite sides. The Law of Tangents state: a-b/a+b = tan[1/2(A-B)]/tan[1/2(A+B)] • If you search the law of sines on the internet, it'll mostly give you A/sin(a) = B/sin(b) = C/sin(c). But, to find a missing angle, it's best to use sin(a)/A = sin(b)/B = sin(c)/C. Is there a way to use A/sin(a) = B/sin(b) = C/sin(c) to find a missing angle? • If you have A/sin(a) = B/sin(b), you can just solve for the angle using Algebra. A = B * sin(a) / sin(b) A * sin(b) / B = sin(a) a = arcsin(A * sin(b) / B) • Hi, I have a question. I have an hp35s calculator. 1. I did the following question on law of sines- sin14°13'36"/803.94=sinX/1879.28 I get the answer 35°03'48". Which is correct, but the answer could also be 144°56'12". Why is that? I only ask because a question I was doing required me to know that. 2.Your example of the law of sines. sin40/30=sinx/40 I got an answer of 58°59'13" I think I did this as precise as I could because the question I did above came out correctly and is the same format. I'm sure you weren't getting into that precise of an answer, but in my situation, I must have it that precise for a test. So, my question for this one- would my answer be correct in terms of pricision? Thank you, Jerry • 145 degrees is just the same as 35 degrees but pointing in the opposite direction. The vertical height of the triangle is the same. • What is the difference between degree mode and radian mode? Thanks • You said mode in your question, so I suppose you are refering to calculators. Degree mode will set up your calculator to work with degrees.So in any operation that involves an angle amplitude or trig ratio, your results will be in degrees. In radian mode, your calculator will set up to work with radians, that means, a different unit of amplitude. Always remember to switch modes acordingly to the unit you are using, else you will get some pretty weird answers. Hope this helps. • Why Sal doesn't consider that theta angle might be 180 - 58.9869695348 ? There's the second solution: 18.9869695348 • You are correct. Since the triangle is in the form SSA there is ambiguity as sin(180-theta) = sin(theta). I have notified Khan Academy of this mistake. Hopefully it will get rectified in the future. • Does it matter that much at whether a capital "A" is used or a lowercase "a" is used in the low of sines. Example; sinA/a= sinB/b=sinC/c if the triangle angle are the capital 'A"'s • Nope- the letters don't matter- they only stand for words. as long as the values are being divided correctly (AKA, the angle of sine are divided by the corresponding side length), you could call one angle "Marshmallow" and another angle "Cheeseburger" and it wouldn't make one smidgeon of a difference. • I'm not sure if this is just another way to do it, or it worked this once by luck, but about into the video I paused it because something clicked when he was talking about the constant ratio between all sides and angles in a triangle, and I immediately went to google calculator, found the constant ratio, multiplied it by 40(the measure of the side, not the angle) and then did the inverse sine of that, and I unpaused the video to see if I got it correct(I did). Will this always work? • 👍Of course, it is right! you can test this in the practice btw! This is pure genius since you put it constant ratio. Great Job! Hope this helped you understand and get encouraged! #YouKhanLearnAnything.💪 • I was doing the problems, I got an answer, but it was incorrect. I checked what the solution was and it said that `sin(theta) = sin(180 - ANSWER)` Can anyone explain how this works? It was not explained in the video.<\|endoftext\|>	4.53125
585	ESL Flashcard Game for Kindergarten: The Circle Many ESL teachers who are asked to teach in kindergartens sometimes assume that English lessons with small children should be energetic and full of surprises all the time to keep young children’s interest, well this isn’t always the case. Teaching ESL in Kindergarten Although teaching ESL in kindergarten should have lots of energy, there are times when you should bring down energy levels for the kindergarten children. Teaching English in kindergarten needn’t be a million miles an hour all the time; it’s too much for you and it’s too much for the children. Follow our new YouTube channel for weekly 2-minute ESL video lessons! This ESL flashcard game for kindergarten is one such activity where the kids are reviewing the English words you’ve taught them, but in a slow and calm manner. Kindergarten kids sometimes can’t follow the rules of games, simply because they are too young, but this flashcard game is easy to follow for the kindergarten kids and they’ll review the words you taught them from the flashcards very well. What to do in your kindergarten class Pre-teach your ESL vocabulary with the kindergarten class and when you are happy that the majority of the kids are comfortable with the flashcards, lay them out on the floor. Spread them out so there is a good amount of space between each flashcard. Use the space in your kindergarten classroom. Playing the ‘The Circle’ flashcard game with your kindergarten kids Choose 4-5 students to stand up and hold hands to form a circle. Stand the circle in the middle of the flashcards. When you are ready, whisper one of the flashcard words to your seated students. The students then call out the flashcard and then circle has to move to position themselves around the correct flashcard your kindergarten children have called out. Change the circle often so all students have a turn in the middle. Like this Kindergarten ESL flashcard game? Click the games below to find something similar Volcano! - Arrange your ESL flashcards onto the kindergarten classroom floor like stepping stones over hot lava. The teacher goes first and calls out the flashcards as you step on them to avoid the hot lava on the floor! A good ESL game for kindergarten, which is simple to follow. Angry Crab! - The teacher holds up English flashcards for the kindergarten kids to say. Once the class has called out the correct word, the teacher becomes an angry crab walking from side to side and the kids must make it past to the other side of the river. Stand Up, Sit Down - A nice ESL warmer for kindergarten where the kids must follow what the teachers says or does. Perfect for the first activity of the day.<\|endoftext\|>	3.75
225	Prepare Ye the Way 300 - 200 BC, the Middle East The major inheritors of the Alexandrian empire were the generals Seleucus in Mesopotamia and Ptolemy in Egypt. Warfare continued between them and among lesser kingdoms and leagues of the fragmented world. New states appeared and others vanished into the embrace of still other conquerers. Through it all, the victor was the concept of Hellenism. The ancient world eargerly accepted the outward form and often the more subtle glories of the new civilization. New cities in the Greek style sprang into being, including Alexandria, the great seaport at the mouth of the Nile. The eastern Mediterranean world became cosmopolitan with widespread interchange of peoples and ideas. Greek became the lanuage of educated men, and more men were educated. The Romans to the west proceeded with their conquestof Italy. When their expansion touched Carthage, war broke out, continuing through the century until Rome's victory in 201 BC. Swollen with conquest, Rome was strong enough to turn toward the quarreling Greek states in the east.<\|endoftext\|>	3.71875
448	Set of 8 Boomwhackers Item # P7-7400 Demonstrate open- and closed-piped resonance with these colorful tubes. When whacked against the floor or your knee, each tube produces a clear tone. Students can even play songs! A Fun Way to Demonstrate Resonance! Demonstrate open- and closed-piped resonance with these colorful tubes. When whacked against the floor or your knee, each tube produces a clear tone. Students can even play songs! Tubes are color-coded and labeled with the musical note they produce. You'll get an 8-tone scale with 4 "Octavator" end caps, which halve the fundamental frequency, moving the tone down one octave. Experiment and lesson ideas are included. Approximately 12" to 24" long. Activities and Uses Use the Boomwhackers for many sound-related activities. Have the students graph length versus frequency produced. Download a color-coded song and play it with the class. Demonstrate resonance with a tuning fork and a Boomwhacker. Noise Filtering: Take the Boomwhackers to a noisy environment, such as a crowded hallway or cafeteria. Put a tube to your ear and listen to the sounds. The tube filters and amplifies the same tone that it produces when it is whacked. If you hold the tube tightly against your head, it becomes a closed tube and will amplify the note an octave lower. In using this product, national and states' science education standards are covered. Some examples are provided here. These are representative, however. Check with your state to find the exact standards. - Sound is produced by vibrating objects. The pitch of the sound can be varied by changing the rate of vibration. - Waves, including sound and seismic waves, waves on water, and light waves, have energy and can transfer energy when they interact with matter. - Relate characteristics of sounds that we hear to properties of sound waves. - Verify that vibrating an object can produce sound. Products being sold are not toys. They are for Educational / Labratory use only. They are not for use by children 12 and under.<\|endoftext\|>	3.65625
492	There are 11 major organ systems in the human body. For this article, there is an overview for five of these organ systems. Each contains at least one vital organ and other structures that are important for healthy body function. The nervous system is the main command system that directs function for all other systems. However, without proper functioning of the cardiovascular system and respiratory system, the nervous system shuts down within a short period of time. The nervous system sends signals throughout the body to control function and movement. It is composed of the brain, spinal cord and peripheral nervous system. It directs quick responses to stimuli, such as automatic reflexes. The nervous system works in conjunction with the endocrine system to control metabolism and other body functions. While the nervous system relies mostly on electrical signals for messaging, the endocrine system uses chemical messengers. It secretes hormones into blood and other body fluids. Water balance, body growth and responses to stress are some of the activities regulated by the endocrine system. Glands that secrete hormones include the pituitary, thyroid, adrenal, pancreas and hypothalamus. Sciencing Video Vault The cardiovascular system is occasionally referred to as the circulatory system. It contains the heart, blood vessels and blood. The blood transports nutrients, hormones, gases and waste products by using the blood vessels. The heart pumps the blood throughout the body and maintains blood pressure. Arteries pump blood away from the heart, and veins return blood toward the heart. The respiratory system contains the nasal cavities, throat areas and lungs. The pharynx is shared with the digestive tract. Air moves from the pharynx to the larynx, which protects the opening to the trachea. The trachea is the main passageway to the lungs. It acts as an air filter. Inside the lungs, oxygen is extracted from the air, and carbon dioxide is exhaled as a waste product. In the digestive system, food is absorbed and processed by the body. After being swallowed through the mouth, food moves through the esophagus and into the stomach. The stomach breaks down the food mechanically and chemically so it can be digested by the small intestine and used for nutrition. Any undigested material is then moved through the large intestine and excreted through the anus. The liver is also considered part of the digestive system. It releases bile to aid in digestion.<\|endoftext\|>	3.75
1,042	Blotting is the process by which DNA, RNA or proteins are transferred onto a membrane in order to be visualised. Credit: Paul Cowan/Shutterstock.com The first of these methods was Southern Blotting, developed in 1975 by Edward Southern, and is used to detect the sequence of DNA fragments. Since then, two other methods have been developed, called Northern and Western blotting. These are processes which are used to identify RNA and protein sequences, respectively. Methodology of Southern blotting DNA fragments are initially generated with restriction enzymes and separated by size in a process called gel electrophoresis. Alkaline is then used to denature the doubled-stranded DNA, forming single strands. The DNA is then transferred onto a nitrocellulose or nylon sheet by placing a membrane over the gel and using the flow of buffer to encourage the movement of DNA from the gel to the membrane. Capillary flow and vacuum transfer are the two most common methods used to transfer the DNA fragments. In capillary flow, the gel is placed above the level of buffer on a supporting block with a membrane placed on top. A stack of absorbent towels are then placed over the membrane and are used to absorb the buffer from beneath the gel, lifting the DNA fragments up onto the membrane. Using vacuum transfer, the membrane is placed beneath the gel and both are submerged in buffer. A vacuum is then used to create a flow which pull the DNA fragments down onto the membrane. Heating or UV radiation are then used to ensure that the DNA fragments remain attached to the membrane permanently, whilst maintaining the specific arrangement of DNA. Single-stranded labelled probes are then used to bind to target sequences. The sheet is incubated with these probes and only the complementary probes bind. The non-complementary probes are then washed from the membrane, ensuring that only the bound probes remain. These probes can then be detected by autoradiography to reveal the pattern of hybridisation on an x-ray film. Applications of Southern blotting Southern blotting has many different uses. Firstly, gene rearrangements can be analysed. For example, in immunology this method can be used to identify the clonal rearrangements of T cell receptor genes. Secondly, specific fragments of DNA can be identified from within a mixture of many other fragments. Other examples of uses include both restriction fragment length polymorphism (RFLP) and variable number tandem repeat (VNTR) analysis. RFLP uses the differences in length in homologous DNA sequences to map genomes, and can be used in forensic and paternity tests. VNTR analysis uses the differences in length of repeated nucleotide sequences to form a DNA fingerprint, a method that is commonly used in paternity and forensic testing. These methods can also be used in the diagnosis of disease caused by mutation, for example sickle cell anaemia. This genetic condition is due to a single nucleotide polymorphism (A to T) in the beta-globin gene, resulting in abnormal haemoglobin. Southern blotting for Fragile X syndrome Southern blots have been used extensively to help identify genes with amplified repeat regions. These are short, repetitive sequences in DNA that do not encode gene products. One example where southern blotting can be useful is in the diagnosis of Fragile X syndrome. This genetic condition is due to the increase in the CGG/CCG repeat region which is located within the FMR1 gene. This gene usually contains between 5-40 repeat regions. Individuals with 55-200 repeats have a FMR1 gene premutation, and individuals with >200 repeats have fragile X syndrome. The increase in repeats leads to methylation of the gene, which inhibits transcription and prevents the production of the FMRP protein. This protein is required for the normal function of the nervous system, therefore loss of protein function leads to the symptoms of fragile X syndrome. In diagnosis, methylation-sensitive restriction enzyme EclX1 and Methylation-insensitive enzyme EcoR1 are used to allow the differentiation of methylated alleles and non-methylated alleles. Methylated alleles are cut only once to give a singular DNA fragment of 5.1 kb. However non-methylated alleles are cut twice, producing a fragment of 2.8 kb. Non-methylated premutation repeats (<200) can therefore be distinguished from methylated mutation repeats of around 200 repeats long, as southern blotting can be used to identify the sizes of the DNA fragments. Overall, Southern blotting is an important method in the diagnosis and study of disease (such as fragile X syndrome and sickle cell anaemia) and analysis of DNA for other reasons (such as forensic and paternity testing). However, southern blotting is very technically complex, expensive, laborious and requires a large quantity of DNA sample. New methods are therefore slowly replacing southern blotting, for example real time PCR. This process is much easier and faster than southern blotting and only requires a very small volume of DNA. VIDEO Further Reading<\|endoftext\|>	3.78125
1,634	# Lesson 2Half Interested or More InterestingSolidify Understanding ## Jump Start Notice and Wonder Previously, you examined the following context: Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of of anti-parasite medicine is given to a dog and the medicine breaks down such that of the medicine becomes ineffective every hour. How much of the dose is still active in the dog’s bloodstream after , after , and after hours? Here are three representations of that context. List at least two things that you notice and one thing you are wondering about relative to these representations. Representation #1: Representation #2: Representation #3: ## Learning Focus Examine how the properties of exponents work with rational exponents. Write equivalent exponential functions using different growth factors. What do rational exponents and negative exponents mean in contexts? Do the laws of exponents work with rational exponents? How does the growth factor change if we focus on a month of exponential growth instead of a year? ## Open Up the Math: Launch, Explore, Discuss Carlos and Clarita, the Martinez twins, have run a summer business every year for the past years. Their first business, a neighborhood lemonade stand, earned a small profit that their father insisted they deposit in a savings account at the local bank. When the Martinez family moved a few months later, the twins decided to leave the money in the bank where it has been earning interest annually. Carlos was reminded of the money when he found the annual bank statement they had received in the mail. “Remember how Dad said we could withdraw this money from the bank when we are years old,” Carlos said to Clarita. “We have in the account now. I wonder how much that will be years from now?” ### 1. Carlos calculates the value of the account year at a time. He has just finished calculating the value of the account for the first years. Describe how he can find the next year’s balance and record that value in the table. Year Amount ### 2. Clarita thinks Carlos is silly calculating the value of the account one year at a time, and says that he could have written a formula for the year and then evaluated his formula when . Write Clarita’s formula for the year and use it to find the account balance at the end of year . ### 3. Carlos was surprised that Clarita’s formula gave the same account balance as his year-by-year strategy. Explain, in a way that would convince Carlos, why this is so. “I can’t remember how much money we earned that summer,” said Carlos. “I wonder if we can figure out how much we deposited in the account five years ago, knowing the account balance now?” ### 4. Carlos continued to use his strategy to extend his table year-by-year back years. Explain what you think Carlos is doing to find his table values one year at a time and continue filling in the table until you get to , which Carlos uses to represent “ years ago.” Year Amount Explanation: ### 5. Clarita evaluated her formula for . Again, Carlos is surprised that they get the same results. Explain why Clarita’s method works. Clarita doesn’t think leaving the money in the bank for another years is such a great idea and suggests that they invest the money in their next summer business. “We’ll have some start-up costs, and this will pay for them without having to withdraw money from our other accounts.” Carlos remarked, “But we’ll be withdrawing our money halfway through the year. Do you think we’ll lose out on this year’s interest?” “No, they’ll pay us a half-year portion of our interest,” replied Clarita. “But how much will that be?” asked Carlos. ### 6. Calculate the account balance and how much interest you think Carlos and Clarita should be paid if they withdraw their money year from now. Remember that they currently have in the account and that they earn annually. Describe your strategy. Clarita used this strategy: She substituted for in the formula and recorded this as the account balance. Carlos had some questions about Clarita’s strategy: • What numerical amount do we multiply by when we use as a factor? • What happens if we multiply by and then multiply the result by again? Shouldn’t that be a full year’s worth of interest? Is it? • If multiplying by is the same as multiplying by , what does that suggest about the value of ? ### 7. Answer each of Carlos’s questions listed as best as you can. Pause and Reflect As Carlos is reflecting on this work, Clarita notices the date on the bank statement that started this whole conversation. “This bank statement is three months old!” she exclaims. “That means the bank will owe us of a year’s interest.” “So how much interest will the bank owe us then?” asked Carlos. ### 8. Find as many ways as you can to answer Carlos’s question: How much will their account be worth in of a year (nine months) if it earns annually and is currently worth ? Carlos now knows he can calculate the amount of interest earned on an account in smaller increments than one full year. He would like to determine how much money is in an account each month that earns annually with an initial deposit of . He starts by considering the amount in the account each month during the first year. He knows that by the end of the year the account balance should be , since it increases during the year. ### 9. Complete the table showing what amount is in the account each month during the first months. Deposit $300$ $\frac{1}{12}$ $\frac{2}{12}$ $\frac{3}{12}$ $\frac{4}{12}$ $\frac{5}{12}$ $\frac{6}{12}$ $\frac{7}{12}$ $\frac{8}{12}$ $\frac{9}{12}$ $\frac{10}{12}$ $\frac{11}{12}$ $1\phantom{\rule{0.278em}{0ex}}\text{year}$ $\mathrm{}315$ ### 10. What number did you multiply the account by each month to get the next month’s balance? Carlos knows the exponential equation that gives the account balance for this account on an annual basis is . Based on his work finding the account balance each month, Carlos writes the following equation for the same account: . ### 11. Verify that both equations give the same results. Using the properties of exponents, explain why these two equations are equivalent. ### 12. What is the meaning of the in this equation? Carlos shows his equation to Clarita. She suggests his equation could also be approximated by , since . Carlos replies, “I know the in the equation means I am earning interest annually, but what does the mean in your equation?” ### 13. Answer Carlos’s question. What does the mean in ? Pause and Reflect The properties of exponents can be used to explain why . Here are some more examples of using the properties of exponents with rational exponents. For each of the following, rewrite the expression using the properties of exponents and explain what the expression means in terms of the context. ## Ready for More? Use to explain why . ## Takeaways The following properties of exponents that make sense for positive integer exponents also apply and make sense for negative integer exponents and for fractional exponents: We can interpret the negative exponential factor in as meaning: We can interpret a fractional exponential factor in as meaning: ## Lesson Summary In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents. ## Retrieval Use the rules of exponents to find three expressions that would be equivalent to the one provided. ### 2. Rewrite each of the expressions.<\|endoftext\|>	4.5
146	# Roberto is dividing his baseball cards equally between himself,his brother,and his 5 friends.Roberto was left with 6 cards.How many cards did Roberto give away?Enter and solve a division equation to solve the problem.Use x for the total number of cards. Feb 20, 2018 $\frac{x}{7} = 6$ So Roberto started with $42$ cards and gave away $36$. #### Explanation: $x$ is the total number of cards. Roberto divided those cards seven ways, ending up with six cards for himself. $6 \times 7 = 42$ So that is the total number of cards. Because he kept $6$, he gave away $36$.<\|endoftext\|>	4.4375
1,568	# Area and Perimeter Definition, Formulas \| How to find Area and Perimeter? Area and Perimeter is an important and basic topic in the Mensuration of 2-D or Planar Figures. The area is used to measure the space occupied by the planar figures. The perimeter is used to measure the boundaries of the closed figures. In Mathematics, these are two major formulas to solve the problems in the 2-dimensional shapes. Each and every shape has two properties that are Area and Perimeter. Students can find the area and perimeter of different shapes like Circle, Rectangle, Square, Parallelogram, Rhombus, Trapezium, Quadrilateral, Pentagon, Hexagon, and Octagon. The properties of the figures will vary based on their structures, angles, and size. Scroll down this page to learn deeply about the area and perimeter of all the two-dimensional shapes. ## Area and Perimeter Definition Area: Area is defined as the measure of the space enclosed by the planar figure or shape. The Units to measure the area of the closed figure is square centimeters or meters. Perimeter: Perimeter is defined as the measure of the length of the boundary of the two-dimensional planar figure. The units to measure the perimeter of the closed figures is centimeters or meters. ### Formulas for Area and Perimeter of 2-D Shapes 1. Area and Perimeter of Rectangle: • Area = l Ã— b • Perimeter = 2 (l + b) • Diagnol = âˆšlÂ² + bÂ² Where, l = length 2. Area and Perimeter of Square: • Area = s Ã— s • Perimeter = 4s Where s = side of the square 3. Area and Perimeter of Parallelogram: • Area = bh • Perimeter = 2( b + h) Where, b = base h = height 4. Area and Perimeter of Trapezoid: • Area = 1/2 Ã— h (a + b) • Perimeter = a + b + c + d Where, a, b, c, d are the sides of the trapezoid h is the height of the trapezoid 5. Area and Perimeter of Triangle: • Area = 1/2 Ã— b Ã— h • Perimeter = a + b + c Where, b = base h = height a, b, c are the sides of the triangle 6. Area and Perimeter of Pentagon: • Area = (5/2) s Ã— a • Perimeter = 5s Where s is the side of the pentagon a is the length 7. Area and Perimeter of Hexagon: • Area = 1/2 Ã— P Ã— a • Perimeter = s + s + s + s + s + s = 6s Where s is the side of the hexagon. 8. Area and Perimeter of Rhombus: • Area = 1/2 (d1 + d2) • Perimeter = 4a Where d1 and d2 are the diagonals of the rhombus a is the side of the rhombus 9. Area and Perimeter of Circle: • Area = Î rÂ² • Circumference of the circle = 2Î r Where r is the radius of the circle Î = 3.14 or 22/7 10. Area and Perimeter of Octagon: • Area = 2(1 + âˆš2) sÂ² • Perimeter = 8s Where s is the side of the octagon. ### Solved Examples on Area and Perimeter Here are some of the examples of the area and perimeter of the geometric figures. Students can easily understand the concept of the area and perimeter with the help of these problems. 1. Find the area and perimeter of the rectangle whose length is 8m and breadth is 4m? Solution: Given, l = 8m b = 4m Area of the rectangle = l Ã— b A = 8m Ã— 4m A = 32 sq. meters The perimeter of the rectangle = 2(l + b) P = 2(8m + 4m) P = 2(12m) P = 24 meters Therefore the area and perimeter of the rectangle is 32 sq. m and 24 meters. 2. Calculate the area of the rhombus whose diagonals are 6 cm and 5 cm? Solution: Given, d1 = 6cm d2 = 5 cm Area = 1/2 (d1 + d2) A = 1/2 (6 cm + 5cm) A = 1/2 Ã— 11 cm A = 5.5 sq. cm Thus the area of the rhombus is 5.5 sq. cm 3. Find the area of the triangle whose base and height are 11 cm and 7 cm? Solution: Given, Base = 11 cm Height = 7 cm We know that Area of the triangle = 1/2 Ã— b Ã— h A = 1/2 Ã— 11 cm Ã— 7 cm A = 1/2 Ã— 77 sq. cm A = 38.5 sq. cm Thus the area of the triangle is 38.5 sq. cm. 4. Find the area of the circle whose radius is 7 cm? Solution: Given, We know that, Area of the circle = Î rÂ² Î = 3.14 A = 3.14 Ã— 7 cm Ã— 7 cm A = 3.14 Ã— 49 sq. cm A = 153.86 sq. cm Therefore the area of the circle is 153.86 sq. cm. 5. Find the area of the trapezoid if the length, breadth, and height is 8 cm, 4 cm, and 5 cm? Solution: Given, a = 8 cm b = 4 cm h = 5 cm We know that, Area of the trapezoid = 1/2 Ã— h(a + b) A = 1/2 Ã— (8 + 4)5 A = 1/2 Ã— 12 Ã— 5 A = 6 cmÃ— 5 cm A = 30 sq. cm Therefore the area of the trapezoid is 30 sq. cm. 6. Find the perimeter of the pentagon whose side is 5 meters? Solution: Given that, Side = 5 m The perimeter of the pentagon = 5s P = 5 Ã— 5 m P = 25 meters Therefore the perimeter of the pentagon is 25 meters. ### FAQs on Area and Perimeter 1. How does Perimeter relate to Area? The perimeter is the boundary of the closed figure whereas the area is the space occupied by the planar. 2. How to calculate the perimeter? The perimeter can be calculated by adding the lengths of all the sides of the figure. 3. What is the formula for perimeter? The formula for perimeter is the sum of all the sides. Scroll to Top Scroll to Top<\|endoftext\|>	4.65625
884	Stars Locations are Uncertain Whether viewed dimly through the haze and lights of a city or in all their glory in a pristine wilderness, the stars that surround the Earth are magnificent, and one day Earthlings will travel to some of the new planets that astronomers are locating. However, the stars we see are not necessarily where we think they are, according to an international research team. "We know that the light from distant stars takes a very long time to reach the Earth," says Dr. Akhlesh Lakhtakia, distinguished professor of engineering science and mechanics, Penn State. "But, taking into account the distance a star will have moved while that light travels, we still may not be able to accurately locate the star.Ó Negative phase velocity media or materials with negative refractive index may be responsible for this locational uncertainty. Recently, materials researchers at the University of California San Diego, working with micro and nano materials, developed a metamaterial that had a negative refractive index for microwaves, proving that negative phase materials could exist at least in the microwave part of the electromagnetic spectrum. Their requirements for this material were that both the relative permittivity, a measure of the charge separation in a material, and the relative permeability, a measure of how electrons loop in materials, of a substance must be less than zero. While the implications for negative phase velocity media in the nano world are the creation of a perfect lens, a lens with no distortion with applications for optical transmission devices, CDs, DVDs, microwave systems, etc., in the universe at large, these media can disguise the location of a star, according to the researchers. A material with negative index of refraction transmits light or other wave energy differently than one with positive index of refraction. In all natural materials, when an energy beam Ð light, radar, microwave Ð passes through water or glass or some other material, the beam is displaced in the same direction. The amount of displacement depends upon how much the material slows the speed of the beam. In negative phase velocity media, the displacement is in the opposite direction. Lakhtakia and Tom. G. Mackay, lecturer in Mathematics, University of Edinburgh decided to look at why the permittivity and permeability had to be less than zero. They found that one or both permeability and permittivity could be less than zero and negative phase velocity would occur. They then found that both could be greater than zero and a negative index of refraction would occur but only when special relativity came into play. The researchers looked at transmission through space, where high velocities are common. "First I did the derivations with the observer moving and the energy source stationary," says Lakhtakia. " Then Mackay did the derivations with the observer stationary and the light source moving." What they found was that it depends on the state of the observer whether any particular media at any time has negative or positive index of refraction. The relative velocity of the observer changes the index of any material. "Light coming off a stellar object passes through many different regions of space filled with different media and is affected by different gravitational fields," says Lakhtakia. "When we finally see it, we cannot really know where it originated." While this may be of no consequence today, Lakhtakia believes it has important implications for when space travel is common. Because this is a direction dependent effect, it will change the telemetry of objects and spacecraft. "The business of space navigation and interpreting star maps could be a lot more complicated than we now think it is," says Lakhtakia. "Imagine mining of extrasolar asteroids. We might not want to send humans to do the mining, but robots would have to know where the asteroid is and where on its surface to mine when it left our solar system." Calculations would need to be made from Earth on an asteroid that might not be where we visually see it. The effects of negative phase velocity media would need to be taken into consideration. Another problem would be navigating from somewhere far away from the Earth in a space ship using information gathered from the Earth. Depending on the velocity of the spacecraft and the object aimed for, negative phase velocity media between the spacecraft and the destination would also need to be considered. Source: Penn State<\|endoftext\|>	3.78125
1,810	# Natural Log Rules – Brief Analysis And Examples Are you taking a college or high school math class? One of the areas you will cover is natural logs. So what exactly is a natural log? A natural log of any number is its logarithm to the base of e (mathematical constant); where e is a transcendental number that is approximately equal to 2.718281828459. Sounds complex, right? But it will no longer be complex when you understand natural log rules. In this post, we are going to take a closer look at the most important natural log rules, highlight other natural log properties, and demonstrate how to apply them with examples. We will also demonstrate the difference between natural logs and other logarithms. ## What is In in Natural Log Rules? Natural log is also referred to as In. It is the inverse of e. Letter “e” is a math constant that is commonly referred to as a natural exponential. Notably, just like Pi (π) that has a constant value of 3.14159, e also has a fixed value of approximately 2.718281828459. Where are In rules applied? e is used in many cases especially in mathematical scenarios such as decay equations, growth equations, and compound interest. Check the example below. In(x) is the time required to grow to x, right? But ex denotes the quantity of growth that has been achieved after a specific period, x. Notably, because e is applied in very many scenarios of math, physics, and economics, students take the logarithm featuring base e of a specific number to find a value. To put it differently, In was crafted as a shortcut for calculating log base e. It lets those reading a problem understand you are using the algorithm, taking base e, of a number. Let us demonstrate this using an example. In(x) = loge(x). Now, substitute the equation with a number: In(5) = loge(5) =1.609. ## The Main Log Rules When working with In in mathematics, there are four key natural logarithm rules that you need to understand. Here is a closer look at these rules: • ### Product Rule When working on In of multiplication of y and x, the answer is the total (sum) of the In of y and In of x. When put in an equation, this rule looks like this; ln(y)( x) = ln(y) + ln(x). Here is an example: ln(7)(5) = ln(7) + ln(5). • ### Quotient Rule When working on the In of the division of y and x, the answer is the difference of the In of y and In of x. To demonstrate this in an equation, here is how it will look like; ln(y/x) = ln(y) – ln(x). See the example below: ln(8/4) = ln(78) – ln(4). • The reciprocal rule Unlike in the quotient rule, the natural log of a reciprocal of a number, call it x, is the opposite of the In of x. When put in an equation, it appears like this; ln(1/x)=−ln(x). Here are some demonstrations using an example; ln(⅓)= -ln(3). • ### Power Rule When calculating the natural log of a number, call it x, raised to the power of y, you simply need to multiply y by In of x. In an equation, it will look like this; ln(xy) = y * ln(x). Take the example; ln(52) = 2 * ln(5). ## The Main Properties of Natural Log On top of the natural log and e rules that we have looked at above, it is important to also appreciate that there are a number of properties you need to understand when studying or adding natural logs. And you know what? You need to memorize the properties of ln to make related calculations easy. Specific scenario In property ln of a Negative Number The ln of any negative number is undefined ln of 0 ln(0) is undefined ln of 1 ln(1)=0 ln of Infinity ln(∞)= ∞ ln of e ln(e)=1 ln of e raised to the x power ln(ex) = x e raised to the ln power eln(x)=x Take a closer look at the above table. You will realize that the last three rows (e, f and g), In(e)=1. Note: this turns out right even when one is raised to the power of any other number. Why? Because the In and e in the rows serve as functions to each other. ## Solving Problems in College Natural Log Problems Now that we have looked at ln rules and ln properties, it is time to get down to solving real problems. Below, we take a closer look at natural log identities and related problems. Check how they are done and try to solve similar questions • Problem one: Can you solve ln (72/5). To begin with, note we are going to use the quotient rule. This will get us, ln(72) – ln(5). Second, we apply the power rule to get this, 2ln(7) -ln(5). Having calculated the equation up to this point, we can leave it that way. But you can also go ahead and use a calculator to get a specific answer, 2(1.946) – 1.609 = 2.283. • Problem two: Can you calculate ln (5x-6)=2. When you get an equation featuring multiple variables in the parenthesis, the first thing is making e your base, right? Then, everything else should be considered exponent of e. Move on and put In and e next to each other. Here is a demonstration. From the natural log laws, we know that eln(x)=x. Therefore, the equation will look like this, eln(5x-6)=e2. Because we also know eln(x)=x, eln(5x-6)= 5x-6, it implies that 5x-6= e2. Also from the e^ln rules, we know that e is a constant. Therefore, we can easily establish the value of e2. We can do this by using a calculator or e’s value. Have a look: 5x-6 =7.389 Then, we would add six to each side of the equation to get; 5x= 13.389 And, finally, divide both sides by number five; x= 2.678 ## The Differences between Natural Logs and Logarithms in College Math To get it right on ln and e rules, it is also important to understand that natural logs are different from algorithms. When you take any logarithm, it is the opposite of its power. Therefore, the main difference between logarithms and natural logs is the base you apply. Logarithms always use a base 10 but natural logs take a base of e. But you can also convert each to the other using the following equations: • log10(x) = ln(x) / ln(10) • ln(x) = log10(x) / log10(e) To demonstrate the differences even more effectively, we will list the rules of logarithms and rules of natural logs in a table. Have a look: In Rules Logarithm rules ln(xy)= ln(x)+ln(y) ln(x/y)=ln(x)−ln(y) ln(xa)= aln(x) ln(ex)= x eln(x) =x log(xy)=log(x)+log(y) log(x/y)=log(x)−log(y) log(xa)= alog(x) log(10x)= x 10log(x) = x ## Summary of Rules and Properties of ln and e The main thing that we have demonstrated in this post is that In or natural log, is the inverse of e. While you might see them difficult at first, they are not as complex after understanding the rules. We also demonstrated that the primary difference between logarithms and natural logs is the base. This can simply your ability to do math homework, especially when face with these issues and algebra rules. If not, there are other options. ## e^ln Rules Are Not That Hard If you are finding calculations related to natural logs complex, there is no need to stress yourself. Go ahead and use professional math help. We provide the best online assignment help, so we can solve any of your math problems. Then, practice more to understand the properties of In and e and associated problems. With time, you will understand and natural logs will be fun!<\|endoftext\|>	4.46875
863	When European settlers arrived in what would become Canada in the early 16th century, the number of Indigenous people ranged from an estimated low of 350,000 to as high as 2,000,000. By Confederation, more than 350 years later, the Indigenous population had not grown, as might be expected, but had shrunk dramatically. In 1867, there were between 100,000 and 125,000 First Nations people here, along with about 10,000 Métis in Manitoba and 2,000 Inuit across the Arctic (see Demography of Indigenous People). The reasons for their decline are tied to such factors as war, illness, and starvation, arising directly from European settlement and habits. As The Canadian Encyclopedia notes, “The Indigenous population… continued to decline until the early 20th century.” Even after that trend reversed, other problems continued, including discrimination, ignorance or misunderstanding of Indigenous cultures, and government laws and policies that often had disastrous effects. Those challenges and hardships cannot be forgotten, and National Aboriginal History Month is an opportune time to remember them. Yet, it is also important to be aware of the achievements of Indigenous peoples, and the manner in which they have enriched the lives of all Canadians. At our organization, Historica Canada, we highlight the triumphs and tragedies involving Indigenous peoples in Canada through programs such as Indigenous Arts & Stories, The Canadian Encyclopedia and our Heritage Minutes. Any list of events in either category is certain to be incomplete, but the process is ongoing. Here are some stories we’ve recently represented: a new Heritage Minute, released on 21 June 2016 (National Aboriginal Day), chronicles a tragic story arising from the long-standing forced enrolment of Indigenous youth in residential schools. That Minute tells the story of Chanie Wenjack, an Anishinaabe boy who ran away from his residential school and subsequently died from hunger and exposure. His death sparked the first inquest into the treatment of Indigenous children in Canadian residential schools. A second new Minute also released during National Aboriginal History Month depicts a treaty negotiation through the eyes of Cree people in Northern Ontario, who saw the process in much different terms than their counterparts across the table. Titled Naskumituwin (or “oral agreement” in Cree), the Minute tells of the story behind the 1921 signing of Treaty 9, which covers a vast tract of Cree and Ojibwa land in Ontario. A third new Minute, released in October 2016, celebrates Kenojuak Ashevak, the world-renowned artist who was at the forefront of the global popularization of Inuit art (see The Art of the North). Some other past Minutes tell stories of Indigenous traditions and defining events. They include: Indigenous achievements during the War of 1812 (many while in alliance with the British), particularly at the Battle of Queenston Heights; the heroic life and troubled death of Tommy Prince, one of Canada’s most decorated military figures; the hanging of the still-controversial Métis leader, Louis Riel, in 1885; and a Minute devoted to the significance of the Inukshuk, the Inuit symbol that serves as a statement in the wilderness to declare, as one character in the Minute says, “now the people will know we were here.” Those efforts barely even scratch the surface of Indigenous peoples’ history in Canada. At the same time, it’s worth noting the wide range of peoples that the term Indigenous encompasses. As of 2010, the most recent year for which statistics are available, the term included: 617 First Nations communities and more than 50 nations, 8 Métis settlements and 53 Inuit communities. Collectively, that includes more than 60 languages. In the 2011 National Household Survey, the most recent, more than 1.8 million people declared Indigenous ancestry. All of this goes to show that Indigenous peoples in Canada are distinct and diverse, with many different cultures, traditions and lifestyles. Their diversity gives them something in common with other Canadians, in a country that is increasingly defined by that quality. Yet, at the same time, they are increasingly proud of being distinct. And, more than ever, they are determined to stay that way.<\|endoftext\|>	3.84375
1,868	USING OUR SERVICES YOU AGREE TO OUR USE OF COOKIES # What Is The Prime Factorization Of 62 What is the prime factorization of 62? Answer: 2 * 31 The prime factorization of 62 has 2 prime factors. If you multiply all primes in the factorization together then 62=2 * 31. Prime factors can only have two factors(1 and itself) and only be divisible by those two factors. Any number where this rule applies can be called a prime factor. The biggest prime factor of 62 is 31. The smallest prime factor of 62 is 2. ## How To Write 62 As A Product Of Prime Factors How to write 62 as a product of prime factors or in exponential notation? First we need to know the prime factorization of 62 which is 2 * 31. Next we add all numbers that are repeating more than once as exponents of these numbers. Using exponential notation we can write 62=21311 For clarity all readers should know that 62=2 31=21311 this index form is the right way to express a number as a product of prime factors. ## Prime Factorization Of 62 With Upside Down Division Method Prime factorization of 62 using upside down division method. Upside down division gives visual clarity when writing it on paper. It works by dividing the starting number 62 with its smallest prime factor(a figure that is only divisible with itself and 1). Then we continue the division with the answer of the last division. We find the smallest prime factor for each answer and make a division. We are essentially using successive divisions. This continues until we get an answer that is itself a prime factor. Then we make a list of all the prime factors that were used in the divisions and we call it prime factorization of 62. 2\|62 We divide 62 with its smallest prime factor, which is 2 31 The division of 2/62=31. 31 is a prime factor. Prime factorization is complete The solved solution using upside down division is the prime factorization of 62=2 31. Remember that all divisions in this calculation have to be divisible, meaning they will leave no remainder. ## Mathematical Properties Of Integer 62 Calculator 62 is a composite figure. 62 is a composite number, because it has more divisors than 1 and itself. This is an even integer. 62 is an even number, because it can be divided by 2 without leaving a comma spot. This also means that 62 is not an odd digit. When we simplify Sin 62 degrees we get the value of sin(62)=-0.73918069664922. Simplify Cos 62 degrees. The value of cos(62)=0.67350716232359. Simplify Tan 62 degrees. Value of tan(62)=-1.0975097786623. When converting 62 in binary you get 111110. Converting decimal 62 in hexadecimal is 3e. The square root of 62=7.8740078740118. The cube root of 62=3.9578916096804. Square root of √62 simplified is 62. All radicals are now simplified and in their simplest form. Cube root of ∛62 simplified is 62. The simplified radicand no longer has any more cubed factors. ## Write Smaller Numbers Than 62 As A Product Of Prime Factors Learn how to calculate factorization of smaller figures like: ## Express Bigger Numbers Than 62 As A Product Of Prime Factors Learn how to calculate factorization of bigger amounts such as: ## Single Digit Properties For 62 Explained • Integer 6 properties: 6 is even and a composite, with the following divisors:1, 2, 3, 6. Also called perfect number since the sum of the divisors(excluding itself) is 6. The first perfect figure, the next ones are 28 and 496. Six is highly a composed, semiprimo, congruent, scarcely total, Ulam, Wedderburn-Etherington, multi-perfect, integer-free number. Complete Harshad, which is a quantity of Harshad in any expressed base. The factorial of 3 and a semi-perfect digit. The third triangular and the first hexagonal value. All perfect even amounts are triangular and hexagonal. Six is the smallest amount different from 1 whose square (36) is triangular(the next in the line that enjoys this property is 35). Strictly a non-palindrome. A numeral is divisible by 6 if and only if it is divisible by both 2 and 3. Part of the Pythagorean triple (6, 8, 10). Being the product of the first two primes (6=2×3), it is a primitive. In the positional numbering system based on 5 it is a repeated number. An oblong, of the form n(n+1). • Integer 2 properties: 2 is the first of the primes and the only one to be even(the others are all odd). The first issue of Smarandache-Wellin in any base. Goldbach's conjecture states that all even numbers greater than 2 are the quantity of 2 primes. It is a complete Harshad, which is a number of Harshad in any expressed base. The third of the Fibonacci sequence, after 1 and before 3. Part of the Tetranacci Succession. Two is an oblong figure of the form n(n+1). 2 is the basis of the binary numbering system, used internally by almost all computers. Two is a number of: Perrin, Ulam, Catalan and Wedderburn-Etherington. Refactorizable, which means that it is divisible by the count of its divisors. Not being the total of the divisors proper to any other arithmetical value, 2 is an untouchable quantity. The first number of highly cototent and scarcely totiente (the only one to be both) and it is also a very large decimal. Second term of the succession of Mian-Chowla. A strictly non-palindrome. With one exception, all known solutions to the Znam problem begin with 2. Numbers are divisible by two (ie equal) if and only if its last digit is even. The first even numeral after zero and the first issue of the succession of Lucas. The aggregate of any natural value and its reciprocal is always greater than or equal to 2. ## Finding Prime Factorization Of A Number The prime factorization of 62 contains 2 primes. The prime factorization of 62 is and equals 2 * 31. This answer was calculated using the upside down division method. We could have also used other methods such as a factor tree to arrive to the same answer. The method used is not important. What is important is to correctly solve the solution. ## List of divisibility rules for finding prime factors faster Knowing these divisibility rules will help you find primes more easily. Finding prime factors faster helps you solve prime factorization faster. Rule 1: If the last digit of a number is 0, 2, 4, 6 or 8 then it is an even integer. All even integers are divisible by 2. Rule 2: If the sum of digits of a number is divisible by 3 then the figure is also divisible by 3 and 3 is a prime factor(example: the digits of 102 are 1, 0 and 2 so 1+0+2=3 and 3 is divisible by 3, meaning that 102 is divisible by 3). The same logic works also for number 9. Rule 3: If the last two digits of a number are 00 then this number is divisible by 4(example: we know that 212=200+12 and 200 has two zeros in the end making it divisible with 4. We also know that 4 is divisible with 12). In order to use this rule to it's fullest it is best to know multiples of 4. Rule 4: If the last digit of a integer is 0 or 5 then it is divisible by 5. We all know that 25=10 which is why the zero is logical. Rule 5: All numbers that are divisible by both 2 and 3 are also divisible by 6. This makes much sense because 23=6. ## What Is Prime Factorization Of A Number? In mathematics breaking down a composite number(a positive integer that can be the sum of two smaller numbers multiplied together) into a multiplication of smaller figures is called factorization. When the same process is continued until all numbers have been broken down into their prime factor multiplications then this process is called prime factorization. Using prime factorization we can find all primes contained in a number.<\|endoftext\|>	4.53125
1,453	Courses Courses for Kids Free study material Free LIVE classes More # Area Perimeter of Triangle ## Introduction Last updated date: 19th Mar 2023 Total views: 28.8k Views today: 0.19k A triangle's perimeter is calculated as the whole length of its border. A triangle is a polygon with three sides that may be classed depending on the length of its sides and angles. Depending on the type of triangle, there are several formulae and techniques for calculating the perimeter. ## What is the Triangle's Perimeter? The perimeter of a triangle is calculated by the sum of its three sides. The term perimeter is derived from two Greek words: "peri" (circumference) and "metron" (measure). The perimeter of any 2D form is defined as the entire distance around it. Because the perimeter represents the length of a shape's boundaries, it is stated in linear units. Real-Life Triangle Perimeter Example: Assume we need to fence the triangle park. To get the measurements of the fence, sum the lengths of the park's three sides. The perimeter of a triangle is the length or distance of the triangle's border. ## Area of the triangle Several formulae may be used to determine the area of a triangle. The area of the triangle can be calculated by the Herons formula or through the trigonometric formula. However, the basic method for calculating the area of a triangle is: $\frac{1}{2}\times \text{base} \times \text{height}$ ## Triangle Perimeter Formula To find the perimeter of a triangle, just sum the lengths of the given sides. The simplest formula for calculating a triangle's perimeter is: Perimeter = a + b+ c Let us examine this formula using several sorts of triangles. ## Scalene Triangle Perimeter A scalene triangle has three sides that are of different lengths. The perimeter of a scalene triangle may be computed by adding the uneven sides together. The perimeter of a scalene triangle is calculated as Perimeter = a + b+ c, where "a", "b", and "c" are the three separate sides. ## Isosceles Triangle Perimeter An isosceles triangle has two sides that have the same length. The perimeter of an isosceles triangle may be computed by adding the equal and unequal sides together. The perimeter of an isosceles triangle is given by: Perimeter = 2a + b units, where “a” is the equal-length side and “b”, denotes the third side. ## Equilateral Triangle Perimeter All of three sides of an equilateral triangle are of equal length. The perimeter of an equilateral triangle is calculated as follows: Perimeter of an equilateral triangle = (3a) units, where “a” is the length of each side of the triangle. ## Right Triangle Perimeter A right-angled triangle or right triangle is a triangle in which one of the angles is 900. By adding the provided sides, the perimeter of a right triangle may be computed. The perimeter of a right triangle can be calculated by the formula: Perimeter = a + b+ c units, is right triangle perimeter. ## Isosceles Right Triangle Perimeter An isosceles right triangle is a right triangle having two equal sides and two equal angles. By adding the supplied sides, the perimeter of an isosceles right triangle may be computed. P = 2l + h is the formula for calculating the perimeter of an isosceles right triangle, where l is the length of the triangle's two equal legs or sides and h is the hypotenuse. ## How to Calculate the Perimeter of a Triangle? The perimeter of a triangle may be computed using the following steps: Step 1: Write down the dimensions of all the triangle's sides and double-check that they all have the same unit. Step 2: Add the sums of all the sides. Step 3: Provide the solution together with the unit. ## Sample questions 1. Equilateral triangle is the triangle with a. Two different side length b. All sides with the same length c. Two sides of the same length d. No equal sides Ans: All sides with the same length 2. Right angle triangle has a. One angle as right angle b. Two angles at right angle c. All the angles at right angle d. All the sides having the same length Ans: One angle as a right angle 3. To apply a fence around a triangular park we need to calculate a. Area of triangle b. Length of side of the triangle c. Perimeter of the triangle d. Angle of the triangle Ans: Perimeter of the triangle ## FAQs on Area Perimeter of Triangle 1. Find the length of the isosceles triangle if sides of equal length are of 17 units and the perimeter of the triangle is 48 units. Perimeter of the triangle = 48 units An isosceles triangle has two sides equal in length = 17 units Perimeter = 2a + b 48 =2 x 17 + b 48 = 34 + b b = 14 So, the length of a missing side is 14 units. 2. A book shaped in an equilateral triangle has a length of a side 12 inches. Find the perimeter of the triangle. Side of the triangle (a)= 12 inches Perimeter = 3a Perimeter = 3 x 12 = 36inches 3. The length of the base of a right-angled triangle is 4 units and the hypotenuse is 5 units. Find the perimeter of the triangle. To find the third side we will apply Pythagoras theorem: a2 + b2 = c2 Where a = 4 and c = 5 42 +b2 = 52 16+b2 = 25 b2 =25 -16 b2 = 9 b = 3 As, a = 4 b = 3, and c = 5 So, the perimeter of the triangle = a + b + c P = 4 + 3 +5 P = 12 units 4. The perimeter of the equilateral triangle is 279 units. Find the length of the side of the equilateral triangle. Perimeter = 279 units Perimeter = 3a 279 = 3 x a $\frac {279}{3} = a$ 93 = a 5. A triangle with sides of 20, 35, and 60 units needs to be fenced. What type of triangle is it and also find its perimeter. The triangle has three sides with different lengths. So, it is a scalene triangle. perimeter = a + b + c P = 20 + 35 + 60 P = 115 units<\|endoftext\|>	4.8125
621	June is here and the end of the school year in Nova Scotia is rapidly drawing near. In grade 7 math classes, it is time for students to break out their geometry sets and practice geometric constructions. The outcome for this unit (7G01) states, "Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors." The intention of this outcome is for students to be able to describe and demonstrate these constructions using a straight edge and compass. In developing the understanding of these constructions, students are exposed to a variety of methods and tools such as paper folding, Mira, and rulers. I've written about geometric constructions in the past, so I won't get into a discussion of the merits of Euclidean constructions. My concern is that this unit could turn into a series of steps to memorize for a short list of basic geometric constructions that lacks coherence and context. If you really need to construct a heptadecagon using only a straight edge and compass, I'd expect you to look up the steps, not have them memorized. Additionally, in my opinion, the exercises in the student textbook are routine and dull. I think that this is a unit that has a lot of potential for student engagement but could easily become tedious. Dan Meyer gave an inspiring ignite talk titled "Teaching the Boring Bits" at the 2013 CMC-North conference. In this talk, Dan encourages teachers to engage students by creating an intellectual need for new knowledge. Teachers should strive to provide students with a reason to want to know new mathematical skills and methods. A grade 7 teacher that I know thought that incorporating Islamic geometric designs into this unit would give a purpose and context for doing geometric constructions. Another factor in selecting this context was that she has a number students who are recent immigrants from the Middle East in her class. We brainstormed some ideas and developed several activities to infuse into this unit that might help give this outcome some coherence and allow students to be creative and artistic. The teacher started by using a template and pattern from Eric Broug's School of Islamic Geometric Design. Students used the template and followed the instructions to create and colour their designs which were then tessellated in a grid to make a group composition on the bulletin board. Later in the unit, students were challenged to construct eight-pointed stars using geometric constructions without a template (although a template could be used for students that needed additional supports). Creating this design using a straight edge and compass required students to perform the majority of the constructions required by this outcome. Students also had the opportunity to use their creativity to personalize their design and make it unique. A number of students were very interested in creating designs of increasing complexity. They were able to pursue this to apply their geometric skills to create some very impressive designs. Have you used any creative or artistic activities to teach students geometric constructions? If so, I'd appreciate hearing about it!<\|endoftext\|>	3.71875
486	In yesterday's lesson, the students created a checklist that they could use to evaluate a good leader. Today we will attempt to create a rating system that can be used for our whole class. First, I'll have groups share out their ideas, and I'll list them on the Smart Board. We'll look at commonalities among the groups for criteria. Then, I'll ask my students to decide if they want to make a checklist, assign a point system, create a rubric, or something else. Together we'll plan a system that we'll use to evaluate some leaders of the past and present. Once the form has been created, I'll print a copy for each group. I'll also display it on the board for reference. Next each group will be assigned a leader. I chose some people that the students would recognize and some that were obscure. I also chose some people who were famous or heroic but not necessarily leaders. Here is my list: Aung San Suu Kyi I did not let the students choose their leader, but I did let them choose their groups. Once they got into the groups, students came up with a plan for sifting through the information and looking for the criteria decided upon in the previous section. I assigned leaders at random...sort of. I tried to give leaders that would be interesting to each individual group. In other words, I assigned Jackie Robinson to a group of boys who were into sports, and Eleanor Roosevelt to several girls who leaders in our class. Now that students have gathered evidence, I'll have them reflect on their leaders by answering the following questions: 1. Is your person a good leader? Explain with evidence? 2. How does your person compare with Ernest Shackleton? Explain with evidence. I'll ask students to record this information on large posters and use markers so that it can be seen for tomorrow's gallery walk and presentations. I expect my students to support each and every claim with evidence. For example, if someone feels that Abraham Lincoln is brave, he or she will need to provide the proof. I want them to be as detailed as possible in their assessment. As a guide I'll ask them to discuss 3 to 5 of the criteria. It is also just fine if groups realize that their particular person was not really a leader. I just want them to support that opinion using evidence.<\|endoftext\|>	3.828125
600	Lee Rannals for redOrbit.com — Your Universe Online A new technique developed by an expert in craniofacial development and stem cell biology at King’s College London and his colleagues offers up a new type of tooth replacement. The scientists wrote in the Journal of Dental Research about how they developed a new method of replacing missing teeth with a bioengineered material enervated from a person’s own gum cells. Current methods for replacing teeth fail to reproduce a natural root structure, leading to loss of jaw bone due to eating and other jaw movements. This new way of bioengineering teeth with gum cells aims to put an end to this problem. “What is required is the identification of adult sources of human epithelial and mesenchymal cells that can be obtained in sufficient numbers to make biotooth formation a viable alternative to dental implants,” said Professor Paul Sharpe, lead researcher on the project. During the research, the team isolated human gum tissue from patients at the Dental Institute at King’s College London, and then grew more of it in the lab. After this, they combined the tissue with the cells of mice that form teeth. By translating this combination of cells into mice, the team was able to grow hybrid human and mouse teeth containing dentine and enamel. “Epithelial cells derived from adult human gum tissue are capable of responding to tooth inducing signals from embryonic tooth mesenchyme in an appropriate way to contribute to tooth crown and root formation and give rise to relevant differentiated cell types, following in vitro culture,” Sharpe said. “These easily accessible epithelial cells are thus a realistic source for consideration in human biotooth formation. The next major challenge is to identify a way to culture adult human mesenchymal cells to be tooth-inducing, as at the moment we can only make embryonic mesenchymal cells do this.” Scientists are using bioengineering to do some amazing things in health research. In 2011, scientists at the Children´s Center for Cancer and Blood Diseases and The Saban Research Institute of Children´s Hospital Los Angeles said they found a protein to fight leukemia. After bioengineering this protein into purified liquid form, they were able to see how it bonded to leukemia cells and caused their destruction within 24 hours. The protein, CD19-L, even killed leukemia cells that were highly resistant to both standard chemotherapy drugs and radiation. The team is now working to evaluate this new agent for clinical potential against leukemia and to confirm in preclinical studies that leukemic cell destruction can be achieved at non-toxic dose levels. “The CD19-ligand offers a previously unrecognized defense system against leukemia and opens a new range of therapeutic opportunities for the treatment of leukemia,” said Stuart Siegel, MD, director of the Center for Cancer and Blood Diseases at Childrens´ Hospital Los Angeles.<\|endoftext\|>	3.765625
1,230	In Science today we learnt about how to change the shape of objects made from different materials. We started our lesson by collecting three things we thought we could change the shape of and three things we couldn’t. We had a discussion about the properties of these materials. We then investigated which materials we could change by stretching, bending, squashing and twisting. We had great fun finding out! In our Science lesson today we learnt more about the properties of materials. We explored some different materials for ourselves to find out what properties they had. We focused on naming different materials but also describing them using scientific vocabulary. We focused on transparent, opaque, waterproof and flexible. In our topic we have been learning about the effects of global warming on polar bears. In PSHE today we explored where our food comes from. We were amazed to find how far some of our food travels around the world! We discussed how this contributes to global warming and ways in which we could avoid this. This week in our science lessons we have been experimenting and investigating. Today we planned and carried out an experiment to find out what makes ice melt more quickly. We shared our ideas and in groups put ice cubes in 3 different places in the classroom. We then observed them every 10 minutes to see how they had changed. Today in our Science lesson we carried out an experiment which we planned yesterday. Yesterday we received a letter from Professor Pole who had received our letters and was very impressed. Here is the letter he sent us: So in Science we planned and carried out an experiment to find the best material. We talked about what an umbrella needed to be and decided on these three things: flexible, waterproof and hard-wearing. We discussed materials that would be unsuitable and why: Jessica – Glass would be no good. It is waterproof but you wouldn’t be able to put it up and down and it might shatter. Amaya – Wood isn’t waterproof – it would rot. Seth – Plastic might be good because it is waterproof. My raincoat is made of plastic. Charlie – I think fabric might work because that’s what my umbrella is made from. Frankie – Metal wouldn’t bend and also it goes rusty. These were all good predictions! We did the experiment together. Have a look at our photographs for the results. We started our new year with a very exciting discovery because outside our classroom this morning there was a tent! We had no idea why or what it was doing there so we decided the only way to find out was to explore it for ourselves. We went outside and inside the tent Havana found a letter, Jessica found a backpack and a Millie found a bag. Everyone was very excited to read the letter. It said: We were then super-excited to try and find out who PP was and what film he was talking about. It was very chilly outside (Esmay told us it was minus two degrees!) so we took our findings inside to explore them further. We looked through the bag together and the more things we found the more we were convinced PP was definitely an explorer of some sort. He had some warm gloves, a hat, a woolly scarf, a compass, a clock, a torch, some snacks and lots of other things that suggested he explored in cold places. We then found a memory stick which had the film he spoke about in his letter. We watched it and discovered that PP was actually Professor Pole who is a nature explorer. He said it was minus 28 degrees in the North Pole today – BRR! He told us he explores all over the North Pole and is particularly interested in polar bears. He asked for our help with 3 things which we are interested to find out more about: 1: What animals live in the North Pole? 2. What are the landscape and climate like? 3. How do people survive in the North Pole? He has left his things with us while he explores further and left us a lovely bag of fiction and non-fiction books too. We have put them in our role-play area so we can practice being explorers too! We will keep you informed about how we get on with our research. What a wonderful trip! We went to City Kitchen today to learn about food and cook our own meals. When we arrived, we learnt some facts about how much sugar was in different cereals. Some of us were shocked to find out some breakfast cereals have more sugar in than a child should eat in their whole day! After using all that brain power to learn we decided we better feed our brains so we started making our lunch. Chrissy showed us how to make a dough from flour and water. We chose our own toppings and added plenty of cheese before putting them in the oven to bake. Sooner we could smell something delicious so we got ready to eat our pizzas. We all went and sat on a big table together and had some wedges and salad with our pizza. Everyone thought they were scrumptious! Charlie said it was ‘the best pizza in the world!’ We then went back into the kitchen and learnt some facts about how much sugar is in different sorts of drinks. We learnt that even drinks you think are going to be healthy, like flavoured water and fruit juice, have up to 8 teaspoons of sugar in! Next we made our own puddings. We had a whisking competition (which Jessica won) and had some frozen mango yoghurt, banana, berries, grapes, our whisked cream and some strawberry or chocolate sauce. It was called a Wednesday! Yum yum. In today’s Science lesson we started exploring materials. We had a collection of kitchen equipment to explore. We sorted the equipment into groups according to the material they were made from. We found some objects (such as rolling pins, spoons and forks) were made from different materials. Tomorrow we will explore the properties of materials further to discover why this is.<\|endoftext\|>	3.953125
1,900	Courses Courses for Kids Free study material Offline Centres More Store # Solve the differential equation $\left( {{e}^{x}}+1 \right)ydy+\left( y+1 \right)dx=0$. Last updated date: 15th Jul 2024 Total views: 448.5k Views today: 13.48k Verified 448.5k+ views Hint: Use a variable separable method to solve the given differential equation. Integrate with respect to ‘dx’ and ‘dy’ to both sides and simplify it to get the solution of the given differential equation. We have $\left( {{e}^{x}}+1 \right)ydy+\left( y+1 \right)dx=0.............\left( i \right)$ Dividing the whole equation by dx, we get, $\left( {{e}^{x}}+1 \right)y\dfrac{dy}{dx}+\left( y+1 \right)\dfrac{dx}{dx}=0$ Or $\left( {{e}^{x}}+1 \right)y\dfrac{dy}{dx}+\left( y+1 \right)=0..........(ii)$ Now, as we know, we have three types of differential equations i.e., separable, linear and homogeneous. Now, by observation, we get that if we divide equation (ii), by ‘y’ then we can separate variables ‘x’ and ‘y’ easily. Hence, the given differential equation belongs to a separable type. So, on dividing equation (ii), we get $\left( {{e}^{x}}+1 \right)\dfrac{y}{y}\dfrac{dy}{dx}+\left( \dfrac{y+1}{y} \right)=0$ Or $\left( {{e}^{x}}+1 \right)\dfrac{dy}{dx}+\dfrac{y+1}{y}=0$ Now transferring $\dfrac{y+1}{y}$ to other side, we get $\left( {{e}^{x}}+1 \right)\dfrac{dy}{dx}=-\dfrac{\left( y+1 \right)}{y}..........(iii)$ Now, we can transfer functions of variable ‘x’ to one side and functions of variable ‘y’ to another side to integrate the equation with respect to ‘dx’ and ‘dy’. So, equation (iii) can be written as $\left( \dfrac{y}{y+1} \right)dy=\dfrac{-1}{{{e}^{x}}+1}dx$ Now, we observe that variable are easily separated, so we can integrate them with respect to ‘x’ and ‘y’ hence, we get $\int{\dfrac{y}{y+1}dy=-\int{\dfrac{1}{{{e}^{x}}+1}dx............(iv)}}$ Let ${{I}_{1}}=\int{\dfrac{y}{y+1}dy}$ and ${{I}_{2}}=\int{\dfrac{1}{{{e}^{x}}+1}dx}$ Let us solve both the integration individually. So, we have ${{I}_{1}}$ as, ${{I}_{1}}=\int{\dfrac{y}{y+1}dy}$ Adding and subtracting ‘1’ in numerator, we get, ${{I}_{1}}=\int{\dfrac{\left( y+1 \right)-1}{\left( y+1 \right)}dy}$ Now, we can separate (y+1) as ${{I}_{1}}=\int{\dfrac{y+1}{y+1}dy}-\int{\dfrac{1}{y+1}dy}$ Or ${{I}_{1}}=\int{1dy}-\int{\dfrac{1}{y+1}dy}$ As we know, \begin{align} & \int{\dfrac{1}{x}dx}=\ln x \\ & \int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1}} \\ \end{align} So, ${{I}_{1}}$ can be simplified as, ${{I}_{1}}=\int{{{y}^{0}}dy-\ln \left( y+1 \right)+{{C}_{1}}}$ ${{I}_{1}}=y-\ln \left( y+1 \right)+{{C}_{1}}............\left( v \right)$ Now, we have ${{I}_{2}}$ as ${{I}_{2}}=\int{\dfrac{1}{{{e}^{x}}+1}dx}$ Multiplying by ${{e}^{-x}}$ in numerator and denominator we get, ${{I}_{2}}=\int{\dfrac{{{e}^{-x}}}{{{e}^{x}}.{{e}^{-x}}+{{e}^{-x}}}dx}$ As we have property of surds as, ${{m}^{a}}.{{m}^{b}}={{m}^{a+b}}$ So, we can write ${{I}_{2}}$ as ${{I}_{2}}=\int{\dfrac{{{e}^{-x}}}{1+{{e}^{-x}}}dx}..............\left( vi \right)$ Let us suppose $1+{{e}^{-x}}=t$. Differentiating both sides w.r.t. x, we get $-{{e}^{-x}}=\dfrac{dt}{dx}$ Where $\dfrac{d}{dx}{{e}^{x}}={{e}^{x}}$, so, we have ${{e}^{-x}}dx=-dt$ Substituting these values in equation (vi), we get ${{I}_{2}}=\int{\dfrac{-dt}{t}=-\int{\dfrac{dt}{t}}}$ Now, we know that $\int{\dfrac{1}{t}dt=\ln t}$ hence, ${{I}_{2}}=-\ln t+{{C}_{2}}$ Since, we have value of t as $1+{{e}^{-x}}$, hence ${{I}_{2}}$ in terms of x can be given as ${{I}_{2}}=-\ln \left( 1+{{e}^{-x}} \right)+{{C}_{2}}.............\left( vii \right)$ Hence, from equation (iv), (v) and (vii) we get, \begin{align} & y-\ln \left( y+1 \right)+{{C}_{1}}=-\left( -\ln \left( 1+{{e}^{-x}} \right)+{{C}_{2}} \right) \\ & y-\ln \left( y+1 \right)+{{C}_{1}}=\ln \left( 1+{{e}^{-x}} \right)-{{C}_{2}} \\ & y-\ln \left( y+1 \right)=\ln \left( 1+{{e}^{-x}} \right)+{{C}_{3}} \\ \end{align} Where $-{{C}_{2}}-{{C}_{1}}={{C}_{3}}$ Let us replace ${{C}_{3}}$ by ‘ln C’, so we get above equation as $y-\ln \left( y+1 \right)=\ln \left( 1+{{e}^{-x}} \right)+\ln C........\left( viii \right)$ Now, we know that $\text{ln }a+\ln b=\ln ab$, so, equation (viii) can be given as $y=\ln \left( y+1 \right)+\ln \left( C\left( 1+{{e}^{-x}} \right) \right)$ $y=\ln \left( C\left( y+1 \right)\left( 1+{{e}^{-x}} \right) \right)$ As we know that if ${{a}^{x}}=N$ then $x={{\log }_{a}}N$ or vice versa. Hence, above equation can be written as $C\left( y+1 \right)\left( 1+{{e}^{-x}} \right)={{e}^{y}}$ This is the required solution. Note: One can go wrong if trying to solve the given differential equation by a homogenous method, as the given equation is not a homogeneous differential equation so we cannot apply this method to a given variable separable differential equation. Observing the given differential equation as a variable separable equation is the key point of the question.<\|endoftext\|>	4.71875
2,976	Computer Memory Types And How They Affect Your Computer The memory your computer uses can be a big part of how the computer functions and how quickly it can perform. If you’re building a computer, however, it can be hard to know what to pick or why. That’s why we’ve put together this guide. There are several different technologies when it comes to memory. Here is an overview of these technologies and what they mean to your computer. Editors note: This article, originally published in 2007, was updated on in November 2016 with more current information on the latest memory technologies. ROM is basically a read-only memory, or memory that can be read but not written to. ROM is used in situations where the data being stored has to be held permanently. That’s because it’s a non-volatile memory — in other words the data is “hard-wired” into the chip. You can store that chip forever and the data will always be there, making that data very secure. The BIOS is stored on ROM because the user cannot disrupt the information. There are also a number of different types of ROM: Programmable ROM (PROM): This is basically a blank ROM chip that can be written to, but only once. It is much like a CD-R drive that burns the data into the CD. Some companies use special machinery to write PROMs for special purposes. The PROM was first invented way back in 1956. Erasable Programmable ROM (EPROM): This is just like PROM, except that you can erase the ROM by shining a special ultra-violet light into a sensor atop the ROM chip for a certain amount of time. Doing this wipes the data out, allowing it to be rewritten. EPROM was first invented in 1971. Electrically Erasable Programmable ROM (EEPROM): Also called flash BIOS. This ROM can be rewritten through the use of a special software program. Flash BIOS operates this way, allowing users to upgrade their BIOS. EEPROM was first invented in 1977. ROM is slower than RAM, which is why some try to shadow it to increase speed. Random Access Memory (RAM) is what most of us think of when we hear the word “memory” associated with computers. It is volatile memory, meaning all data is lost when power is turned off. RAM is used for temporary storage of program data, allowing performance to be optimized. Like ROM, there are different types of RAM. Here are the most common different types. Static RAM (SRAM) This RAM will maintain it’s data as long as power is provided to the memory chips. It does not need to be re-written periodically. In fact, the only time the data on the memory is refreshed or changed is when an actual write command is executed. SRAM is very fast, but is much more expensive than DRAM. SRAM is often used as cache memory due to its speed. There are a few types of SRAM: An older type of SRAM used in many PC’s for L2 cache. It is asynchronous, meaning that it works independently of the system clock. This means that the CPU found itself waiting for info from the L2 cache. Async SRAM began being used a lot in the 1990s. This type of SRAM is synchronous, meaning it is synchronized with the system clock. While this speeds it up, it makes it rather expensive at the same time. Sync SRAM became more popular in the late 1990s. Pipeline Burst SRAM: Commonly used. SRAM requests are pipelined, meaning larger packets of data re sent to the memory at once, and acted on very quickly. This breed of SRAM can operate at bus speeds higher than 66MHz, so is often used. Pipeline Burst SRAM was first implemented in 1996 by Intel. Dynamic RAM (DRAM) DRAM, unlike SRAM, must be continually re-written in order for it to maintain its data. This is done by placing the memory on a refresh circuit that re-writes the data several hundred time per second. DRAM is used for most system memory because it is cheap and small. There are several types of DRAM, complicating the memory scene even more: Fast Page Mode DRAM (FPM DRAM): FPM DRAM is only slightly faster than regular DRAM. Before there was EDO RAM, FPM RAM was the main type used in PC’s. It is pretty slow stuff, with an access time of 120 ns. It was eventually tweaked to 60 ns, but FPM was still too slow to work on the 66MHz system bus. For this reason, FPM RAM was replaced by EDO RAM. FPM RAM is not much used today due to its slow speed, but is almost universally supported. Extended Data Out DRAM (EDO DRAM): EDO memory incorporates yet another tweak in the method of access. It allows one access to begin while another is being completed. While this might sound ingenious, the performance increase over FPM DRAM is only around 30%. EDO DRAM must be properly supported by the chipset. EDO RAM comes on a SIMM. EDO RAM cannot operate on a bus speed faster than 66MHz, so, with the increasing use of higher bus speeds, EDO RAM has taken the path of FPM RAM. Burst EDO DRAM (BEDO DRAM): Original EDO RAM was too slow for the newer systems coming out at the time. Therefore, a new method of memory access had to be developed to speed up the memory. Bursting was the method devised. This means that larger blocks of data were sent to the memory at a time, and each “block” of data not only carried the memory address of the immediate page, but info on the next several pages. Therefore, the next few accesses would not experience any delays due to the preceding memory requests. This technology increases EDO RAM speed up to around 10 ns, but it did not give it the ability to operate stably at bus speeds over 66MHz. BEDO RAM was an effort to make EDO RAM compete with SDRAM. Synchronous DRAM (SDRAM): SDRAM became the new standard after EDO bit the dust. Its speed is synchronous, meaning that it is directly dependent on the clock speed of the entire system. Standard SDRAM can handle higher bus speeds. In theory, it could operate at up to 100MHz, although it was found that many other variable factors went into whether or not it could stabily do so. The actual speed capacity of the module depended on the actual memory chips as well as design factors in the memory PCB itself. To get around the variability, Intel created the PC100 standard. The PC100 standard ensures compatibility of SDRAM subsystems with Intel’s 100MHz FSB processors. The new design, production, and test requirements created challenges for semiconductor companies and memory module suppliers. Each PC100 SDRAM module required key attributes to guarantee full compliance, such as the use of 8ns DRAM components (chips) that are capable of operating at 125MHz. This provided a margin of safety in ensuring that that the memory module could run at PC100 speeds. Additionally, SDRAM chips must be used in conjunction with a correctly programmed EEPROM on a properly designed printed circuit board. The shorter the distance the signal needs to travel, the faster it runs. For this reason, there were additional layers of internal circuitry on PC100 modules. As PC speeds increased, the same problem was encountered for the 133 MHz bus, so the PC133 standard was developed. SDRAM first appeared in the early 1970s and was used until the mid 1990s. RAMBus DRAM (RDRAM): Developed by Rambus, Inc. and endorsed by Intel as the chosen successor to SDRAM. RDRAM narrows the memory bus to 16-bit and runs at up to 800 MHz. Since this narrow bus takes up less space on the board, systems can get more speed by running multiple channels in parallel. Despite the speed, RDRAM has had a tough time taking off in the market because of compatibility and timing issues. Heat is also an issue, but RDRAM has heatsinks to dissipate this. Cost is a major issue with RDRAM, with manufacturers needing to make major facility changes to make it and the product cost to consumers being too high for people to swallow. The first motherboards with RDRAM support came out in 1999. This type of memory is the natural evolution from SDRAM and most manufacturers prefer this to Rambus because not much needs to be changed to make it. Also, memory makers are free to manufacture it because it is an open standard, whereas they would have to pay license fees to Rambus, Inc. in order make RDRAM. DDR stands for Double Data Rate. DDR shuffles data over the bus over both the rise and fall of the clock cycle, effectively doubling the speed over that of standard SDRAM. Due to its advantages over RDRAM, DDR-SDRAM support was implemented by almost all major chipset manufacturers, and quickly became the new memory standard for the majority of PC’s. Speeds ranged from 100mhz DDR (with operating speed of 200MHz), or pc1600 DDR-SDRAM, all the way to current rates of 200mhz DDR (with operating speed of 400MHz), or pc3200 DDR-SDRAM. Some memory manufactures produce even faster DDR-SDRAM memory modules which readily appeal to the overclocker crowd. DDR was developed between 1996 and 2000. DDR-SDRAM 2 (DDR2): DDR2 features several advantages over conventional DDR-SDRAM (DDR), with the main one being that in each memory cycle DDR2 now transmits for 4 bits of information from logical (internal) memory to the I/O buffers. standard DDR-SDRAM only transmits 2 bits of information each memory cycle. Because of this, normal DDR-SDRAM requires the internal memory and I/O buffers to both operate at 200MHz to reach a total external operating speed of 400MHz. Due to DDR2’s ability to transmit twice as many bits per cycle from logical (internal) memory to the I/O buffers (this technology is formally known as 4 bit prefetch), the internal memory speed can actually run at 100MHz instead of 200MHz, and the total external operating speed will still be 400MHz. Mainly what all this comes down to is that DDR-SDRAM 2 will be able to operate at higher total operating frequencies thanks to its 4 bit prefetch technology (e.g. a 200mhz internal memory speed would yield a total external operating speed of 800mhz!) than DDR-SDRAM. DDR2 was first implemented in 2003. DDR-SDRAM 3 (DDR3): One of the main advantages of DDR3 over the likes of DDR2 and DDR is its focus on low power consumption. In other words, the same amount of RAM consumes a lot less power, so you can increase the amount of RAM you’re using for the same amount of power. How much does it reduce power consumption? By a hefty 40 percent, sitting at 1.5V compared to DDR2’s 1.8V. Not only that, but the transfer rate of the RAM is quite a bit faster, sitting between 800mHz – 1600mHz. The buffer rate is also significantly higher — DDR3’s preferred buffer rate is 8 bit, while DDR2’s is 4 bit. That basically means that the RAM can transmit twice as many bits per cycle as DDR2, and it transmits 8 bits of data from the memory to the I/O buffers. DDR3 isn’t the most recent form of RAM, but it is used on many computers. DDR3 was launched in 2007. DDR-SDRAM 4 (DDR4): Next up is DDR4, which takes the power savings to the next level — the operating voltage of DDR4 RAM is 1.2V. Not only that, but DDR4 RAM offers a higher transfer rate too, sitting at up to 3200mHz. On top of that, DDR4 adds four Bank Groups, each of which can singlehandedly take on an operation, meaning that the RAM can handle four sets of data per cycle. That makes it far more efficient than DDR3. DDR4 takes things a step further, too, bringing DBI, or Data Bus Inversion. What does that mean? If DBI is enabled, it basically counts the number of “0” bits in a single lane. If there are 4 or more, the byte if data is inverted and a ninth bit is added to the end, ensuring that five or more bits are “1.” What that does is it reduces data transmission delay, ensuring that as little power as possible is used. DDR5 RAM is currently the standard on most computers, however DDR5 is set to be finalized as a standard by the end of 2016. DDR4 was launched in 2014. Non-volatile RAM (NVRAM): Non-volatile RAM is a type of memory that, unlike other types of memory, doesn’t lose its data when it loses power. The best known form of NVRAM is actually flash storage, used in solid-state drives and USB drives. It doesn’t, however, come without its drawbacks — for example, it has a finite number of write cycles, and after that number the memory will start to deteriorate. Not only that, but it has some performance limitations that prevent it from being able to access data as fast as some other types of RAM. Suffice to say, there are a lot of different memory types. With this guide, we hope we made it clear what the different types of RAM, what they do and how they affect your computer. Got questions? Be sure to leave us a comment below or join us over in the PCMech Forums!<\|endoftext\|>	3.703125
711	{[ promptMessage ]} Bookmark it {[ promptMessage ]} Chapter 3 Section 05 # Chapter 3 Section 05 - Chapter 3 Section 05 Higher Order... This preview shows pages 1–8. Sign up to view the full content. 1 Chapter 3 Section 05 Higher Order Derivatives page 162 This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 2 Objectives 1. Find 2 nd and 3 rd derivatives. 2. Interpret 2 nd and 3 rd derivatives of functions. 3. Find a formula for the nth derivative of a function. 4. Work distance, velocity, acceleration applications. 3 Higher derivatives Since the derivative of f is a function, it may also have a derivative. If so, it is called the 2 nd derivative of f. )] ( [ ) ( 4 )] ( [ ' ' ' ) ( ' ' ' 3 )] ( [ ' ' ) ( ' ' 2 4 4 4 ) 4 ( ) 4 ( 3 3 3 2 2 2 x f D dx y d y x f derivative th x f D dx y d y x f derivative rd x f D dx y d y x f derivative nd = = = = = = = = = This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 4 Example: f(t) = t 4 – 7t 2 + 2 f’(t) = 4t 3 – 14t f’’(t) = 12t 2 – 14 f’’’(t) = 24t f (4) = 24 f (5) = 0 f (6) = 0 Etc. f (n) = 0 For any polynomial, f (n) = 0 for n ≥ degree + 1 5 4 2 2 2 2 2 2 2 ) 1 ( ) 2 2 )( 2 ( ) 2 2 ( ) 1 ( ' ' ) 1 ( 2 ' ) 1 ( ) 1 ( 2 ) 1 ( ' 1 of derivative second the Find + + + - + + = + + = + - + = + = x x x x x x y x x x y x x x x y x x y This preview has intentionally blurred sections. Sign up to view the full version. View Full Document 6 Page 165, #34. Finding a formula for f (n) (x). f(x) = x -2 f’(x) = -2x -3 f’’(x) = (-3)(-2)(x) -4 f’’’(x) = (-4)(-3)(-2)(x) -5 f (n) (x) = f (n) (x) =(-1) n (n+1)!(x) -(n+2) 7 We know that velocity is the rate of change of position. That is, given position, s, velocity, v This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. {[ snackBarMessage ]}<\|endoftext\|>	4.4375
1,457	# The Supposition Method ### What is the Supposition Method in Maths? The Supposition Method, also known as the Assumption Method, is a very useful Maths Heuristic that you can apply to solve Guess and Check Maths problem sums in Primary 4, Primary 5 and Primary 6. Much faster and systematic than the Guess and Check method, this method is often used by more sophisticated Math students. Before we learn how to use the Supposition method, let’s see how to identify questions that we can solve with this method first. ### Examples of Supposition Questions Here are some examples of Supposition questions for Primary 3 and Primary 4 students. There are 15 mosquitoes and pest busters somewhere in Singapore. There are 54 legs altogether. How many mosquitoes are there? Jay is practising 20 Math questions furiously on Practicle to gain some experience points. He receives 5 marks for every correct answer and for every wrong answer, he gets 2 marks deducted. If Jay earned 35 experience points for his Math practice in the end, how many questions did he get wrong? If you want to see how to solve Supposition questions for Primary 5 and Primary 6 using the Supposition or Assumption Method, don’t forget to check out our video on that. As you can see from the examples of supposition questions above, such questions usually give us a total that’s made up of a few kinds of items. Here we have mosquitoes and pest busters and these 2 items make up a total of 15. Besides that, the items will share something in common and we’ll know the number of that something for each item. The number of legs is the common item between the mosquitoes and pest busters. In addition, we also know the no. of legs for each item – 6 legs for a mosquito and 2 legs for the pest buster. Lastly, in most cases, we will be asked to find the exact number of one of the items. In this question, we’ll need to find the no. of mosquitoes. Now that we know how to look for a Supposition question that we can solve with the Supposition Method, we’re ready to learn how to use the Supposition method to solve it! ### How do we do the Supposition Method? The word “supposition” means “to assume”. Therefore, we’ll start to solve the question with an assumption – “Suppose this is the extreme situation…then what do we have?” After determining the situation, we’ll work step-by-step to dig up more information. Finally, we’ll arrive at the solution to the problem. Now, let’s see how this really works with an example by using the first question. . ### Supposition Method Example Explained Let’s look at this question on Supposition. There are 15 mosquitoes and pest busters somewhere in Singapore. There are 54 legs altogether. How many mosquitoes are there? How to do Supposition? Step 1: Make an assumption As mentioned, we’ll start by making an assumption. This is where we’ll assume an extreme case where all the items are of the same type. Here, we can either assume that all the items were mosquitoes or that all the items were pest busters. Well, we could choose to assume that all the items were pest busters since we hate mosquitoes, or you could assume that all the items were mosquitoes if you are that rare mosquito supporter, but which one would be the better choice? Let’s think about the number of legs each item has. We have 6 legs versus 2 legs. Since we are going to work with that number, let’s choose the smaller number for easier calculation. This also helps to reduce the chances of us making careless mistakes. So here we have it, our first step where we write our assumption: Assuming that all the items were pest busters, Now that we have made an assumption, what do we use it for? Step 2: Multiply to find the total in assumption The next step is to multiply it with the number of legs to find the total number of legs there will be if all the 15 items were pest busters. No. of legs of 1 pest buster x No. of pest busters in our assumption = 2 x 15 = 30 Looks like we have 30 legs. However, when we look at the question, we are supposed to have 54 legs in total. Oh no, looks like we have much fewer legs in our assumption. Let’s find out how many legs we are missing in our next step. Step 3: Find the Difference (The Gap between our Assumption and the Question) 54 – 30 = 24 When we subtract the total number of legs that we are supposed to have from the total number of legs we have in our assumption, you’ll notice that we are short of 24 legs. What shall we do to get more legs then? Remember that there were supposed to be mosquitoes and pest busters in the question? Since we have been working with only pest busters all along, it’s time to get the mosquitoes in! Step 4: Find the Difference (The Effect of Making a Replacement) Let’s think about 1 simple case first. When we add 1 mosquito into what we have, we’ll have to get rid of 1 pest buster, correct? This is because there will always be only 15 items at any one time. When we do that, we are actually replacing 1 pest buster with 1 mosquito. Then what happens to the total number of legs? Let’s find the difference in the number of legs between 1 mosquito and 1 pest buster. A mosquito has 6 legs while a pest buster has 2. So 6 – 2 gives us 4. In other words, we know that every time we replace a pest buster with a mosquito, the total number of legs increases by 4. So far so good? Now that’s super helpful! Step 5: Divide to find the number of replacements Remember that we need an extra of 24 legs? Since we know that replacing a pest buster with a mosquito increases the total number of legs by 4, let’s find out how many replacements do we need to make up for these 24 legs. We’ll do that by dividing the difference in the total no. of legs between our assumption and reality with the difference in the total no. of legs caused by replacing 1 pest buster with a mosquito. 24 divided by 4 gives us 6. Now we know that we’ll need to replace the 6 pest busters we have with 6 mosquitoes in order to make up for the extra 24 legs. That’s how we know we have 6 mosquitoes out of the 15 items. ### Conclusion Hopefully this post has helped you understand how to apply the Supposition method to solve Guess and Check questions more efficiently. Do you prefer the Guess and Check method or the Supposition method? Let us know in the comments below.<\|endoftext\|>	4.84375
516	Period: Early Jurassic Order, Suborder, Family: Saurischia, Theropoda, Podokesauridae Location: Africa (Zimbabwe), North America (United States) Length: 10 feet (3 meters) The agile and fleet-footed carnivore Syntarsus was a small predator. It strongly resembled Coelophysis from the Late Triassic of Arizona and New Mexico. The bones of the head and jaws were large compared to Coelophysis, and the neck was somewhat shorter. Syntarsus was sturdier and slightly larger than Coelophysis. Its jaws had many small, bladelike, serrated teeth for slicing the flesh of its prey. Syntarsus probably ate other small reptiles and fish, and it lived along stream courses in herds. Like Coelophysis, a large number of Syntarsus remains were found in one area, suggesting that they were social animals. The animal is named for its fused ankle, or tarsus, joint. Its name means "fused tarsus." This ankle gave it greater speed and endurance when running. It would have needed to run quickly to escape predators. The front legs were quite small and weak, while the rear legs were stout. The tail was large and probably stiff. The limbs' proportions suggest that Syntarsus may have moved by "saltation," or hopping, similar to rabbits or kangaroos. This hopping produces abrupt and unpredictable movements in order to escape predators. The many skeletons from Zimbabwe have two body forms: males were smaller and more slender, while the females were robust and larger. There were many more females than males. The bones of Syntarsus had so many bone cells and blood vessels that they resemble the bones of birds and mammals. This may mean that Syntarsus was a warm-blooded dinosaur. It is difficult to pinpoint ancestral relationships, but a case may be made that Coelophysis is a close ancestor of Syntarsus. Changes in Syntarsus include fewer teeth; larger openings in the skull for larger and stronger jaw muscles; growth of small plates of bone between the teeth for more biting strength; smaller front limbs and weaker hands; stronger pelvis and bottom of the spine; and fusion of the ankle joints. Syntarsus became more powerful than Coelophysis. Similar trends happened to all the predatory dinosaurs that appeared in the Jurassic. This resulted in the development of the large and ferocious predators such as Allosaurus and its relatives millions of years later.<\|endoftext\|>	4
2,286	Teacher resources and professional development across the curriculum Teacher professional development and classroom resources across the curriculum Topic Overview: Part 1: Factoring Quadratic Expressions Part 2: Understanding Basic Recursion Factoring is the process of rewriting a number or expression as a product of two or more numbers or expressions. It can be used to break a polynomial into smaller parts. By writing the factors of a polynomial, it is often easier to solve equations. The distributive property plays a big role when multiplying factors to get a product. Explanation A factor (of a polynomial) is a polynomial that, when multiplied by another polynomial, yields the original polynomial. For instance, x and (x - 3) are factors of x2 - 3x because x(x - 3) = x2 - 3x. Similarly, (x - 5) and (x + 2) are factors of x2 - 3x - 10, because (x - 5)(x + 2) = x2 - 3x - 10. When used as a verb, factor means to divide a number or expression into a product of other numbers or expressions. As an example, the number 30 can be factored as 1 x 30, 2 x 15, 3 x 10, 5 x 6, or 2 x 3 x 5. The last set of factors is called the prime factorization of 30 because the factors are prime numbers. On the other hand, there is typically only one way to factor a polynomial, especially when the leading coefficient of x2 is 1. For instance, the expression x2 + 2x - 24 can only be factored as (x + 6)(x - 4). Many polynomials, such as x2 + 2x - 7, cannot be factored at all, and are therefore known as prime polynomials. Some polynomials, however, contain coefficients with common factors, and they may be factored in more than one way. For instance, 2x2 + 4x - 48 can be factored in several different ways: 2(x + 6)(x - 4) or (2x + 12)(x - 4) or (x + 6)(2x - 8). Factoring is one method of finding the solutions of a polynomial equation. Using factoring, the quadratic equation x2 + 2x - 15 = 0 can be rewritten as (x + 5)(x - 3) = 0. This shows that either x + 5 = 0 or x - 3 = 0, because one or both of the factors must equal zero if their product is to equal zero. Therefore, either x = -5 or x = 3. Other examples: • The height (h) of a ball thrown into the air with an initial vertical velocity of 24 feet per second from a height of 6 feet above the ground is represented by the equation h = 16t2 + 24t + 6 where t is the time, in seconds, that the ball has been in the air. After how many seconds is the ball at a height of 14 feet? • The parabola y = x2 + 18x - 40 crosses the x-axis at (2, 0) and (-20, 0); the function can be factored as y = (x + 20)(x - 2). • A garden that is 20 feet by 30 feet is bordered by a cement walkway. If the width of the walkway is w feet, the area of the garden and walkway combined is (20 + 2w)(30+2w) = 600 + 100w+ 4w2 or (2w+20)(2w+30) = 4w2 + 100w + 600 • A rectangular box has a volume of 280 in3. Its dimensions are 4 in * (x + 2) in * (x + 5) in. The formula V = lwh can be used to find the value of x, as follows: 4(x + 2)(x + 5) = 280 4x2 + 28x + 40 = 280 4x2 + 28x - 240 = 0 x2 + 7x - 60 = 0 (x + 12)(x - 5) = 0 x = -12 or x = 5 Because x = -12 does not make sense in the context of the problem, it cannot be an answer. Consequently, x = 5, and the dimensions are 4 in, 7 in, and 10 in. • The equation -0.00239d2 + 1.199d = 0 can be used to model a home run that Mickey Mantle hit on May 22, 1963, where d represents the distance of the home run in feet. By factoring the expression as 0.00239d (d - 501.6736) = 0, students can see that Mantle's hit measured about 502 feet. Mathematical Definition A factor (of a number) is an integer that, when multiplied by another integer, results in the original number. For instance, 3 is a factor of 18, because 3 x 6 = 18, and a and b are factors of ab. A factor (of a polynomial) is a polynomial that, when multiplied by another polynomial, results in the original polynomial. For instance, (x + 2) and (x + 4) are factors of x2 + 6x + 8, because (x + 2)(x + 4) = x2 + 6x + 8. Similarly, 4 and (x + 5) are factors of 4x + 20 because 4(x + 5) = 4x + 20. To factor a number (or a polynomial) is to name the number by its factors; that is, to write the number (or polynomial) as a product of two or more integers (or polynomials). When used as a verb, factoring refers to the process opposite of multiplying. In a sense, it is similar to dividing - to factor a number or polynomial is to divide it into other numbers or polynomials. Following is a definition of factor from a mathematics dictionary: Factor: As a verb, to resolve into factors. One factors 6 when he writes it in the form 2 x 3. [As a noun,] a factor of an object (perhaps of some specified type) divides the given object. factor of an integer: An integer whose product with some integer is the given integer. For example, 3 and 4 are factors of 12, since 3 x 4 = 12; the positive factors of 12 are 1, 2, 3, 4, 6, 12, and the negative factors are -1, -2, -3, -4, -6, -12. factor of a polynomial: One of two or more polynomials whose product is the given polynomial. Sometimes one of the polynomials is not allowed to be the constant 1, but usually in elementary algebra a polynomial with rational coefficients is considered factorable if and only if it has two or more nonconstant polynomial factors whose coefficients are rational (sometimes it is required that the coefficients be integers). For example, (x2 - y2) has the factors (x - y) and (x + y) in the ordinary (elementary) sense; (x2 - 2y2) has the factors and in the field of real numbers; (x2 + y2) has the factors (x - iy) and (x + iy) in the complex field. (Source: James, Robert C. and Glenn James. Mathematics Dictionary (5th edition). New York: Chapman & Hall, 1992) Role in the Curriculum Exposure to factoring is important for Algebra 1 students because it illuminates many aspects of the nature of mathematics and serves as a bridge to the study of advanced mathematical topics. The key word here is exposure. In the past, students would be entrenched in the study of factoring for an entire unit and spend a month learning myriad methods; today, the topic receives less emphasis. Students should understand the various representations for polynomials, one of which is its factored form, rather than mastering every factoring technique. Read what teacher educator Diane Briars has to say about the role of factoring in the curriculum: Read transcript from teacher educator Diane Briars In terms of solving quadratics, we need to recognize that factoring is limited as a general method... Read More The NCTM Principles and Standards for School Mathematics (PSSM) does not explicitly mention factoring polynomials. However, the Algebra Standard lists the following expectation: "Students should understand relations and functions and select, convert flexibly among, and use various representations for them; ... [and] understand the meaning of equivalent forms of expressions, equations, inequalities, and relations." (PSSM 2000, p. 395) Recognizing equivalent forms of expressions and being able to convert flexibly among them means that a student should be able to write a polynomial in factored form. That is, a student should understand that x2 + 7x + 10 = (x + 2)(x + 5). Further, students should recognize that both expressions represent a quadratic function that crosses the x-intercept at (2, 0) and (-5, 0). According to Workshop 5 video teacher Tom Reardon, factoring trinomials gives his algebra students their first exposure to quadratic functions and lays the foundation for in-depth study of quadratics later in the year. Students should be able to factor expressions by graphing the equivalent function and understanding the relationship between the x-intercepts and the factors. The strength of the graphical technique is that it is generalized to all polynomials and not just quadratic expressions. Computer Algebra Systems (CAS) can also assist in factoring complicated expressions. (CAS manipulate a formula symbolically using the computer. Factoring, finding roots, and simplifying polynomials are some of the typical uses of CAS.) Students should be able to factor some expressions mentally, such as x2 + 2x + 1, while more complex expressions, such as 3.2x2 + 7x - ½, should be solved graphically or with CAS. They should recognize when one technique is more efficient or beneficial than the others. Read what teacher Tom Reardon has to say about the role of factoring in the algebra curriculum: Read transcript from teacher Tom Reardon It's a technique that's going to be used to solve quadratic equations later, or maybe help work with rational expressions... Read More<\|endoftext\|>	4.875
1,506	Good Afternoon! Good Afternoon! Good Afternoon! - - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - - Presentation Transcript 1. Good Afternoon! Our lesson today will cover The Addition Properties AND Adding multi-digit numbers Let's warm up.... Write < , > , or = 1.) 12-4 11-5 1) : > 2.) 6+7 8+5 2) : = 3.) 9+6 21-4 3) : < 4) : < 4.) 18-11 5+3 CONFIDENTIAL 2. Let's review what we covered in the last lesson. We can use PLACE VALUE to ROUND. Let's round 2,345to the nearest thousands place. First, we need to underline the thousands place. 2,345 CONFIDENTIAL 3. 2,345 Second, we need to look at the digit just to the right of the underlined digit. * If that digit is 5 ormore, we increase the underlined digit by 1. * If that digit is 4or less, we do not change the underlined digit. CONFIDENTIAL 4. 2,345 Since this digit is 4 or less, we leave the 2alone. Finally, we changeall the digits to the right of the underlined digit to 0. 2,345 rounds to 2000 CONFIDENTIAL 5. SET of data 52, 45, 48, , 69, 74 The middle number is called the MEDIAN CONFIDENTIAL 6. ADDITION OF MULTI-DIGIT NUMBERS There are several properties of addition COMMUTATIVE PROPERTY: We can change the order and the sum stays the same. EXAMPLE: 4 + 6 = 10 6 + 4 = 10 ASSOCIATIVE PROPERTY: We can change the grouping and the sum stays the same. EXAMPLE: 2 + (3+ 8) = 10 (2 + 3)+ 8= 10 IDENTITY PROPERTY: We can add zero and the sum stays the same. EXAMPLE: 7 + 0 = 7 CONFIDENTIAL 7. Complete each problem using the COMMUTATIVE PROPERTY. "Change the Order" 36 + 98 = 98 + 36 + + 0 + 76 = 76 + 0 CONFIDENTIAL 8. Complete each problem using the ASSOCIATIVE PROPERTY. "Re-group" 20 + (4 + 6) = ( (20 + 4) + 6 + ) + (8 + 7) + 5 = ) ( + + 8 + (7 + 5) CONFIDENTIAL 9. Complete each problem using the IDENTITY PROPERTY. "Keep the same number" 15 15 + 0 = + 34 = 34 0 CONFIDENTIAL 10. We add numbers together by first LINING UP THE DIGITS 4,321 2,634 + The ones are lined up The thousands are lined up. The tens are lined up. The hundreds are lined up. CONFIDENTIAL 11. Then we need to determine if we must REGROUP 7+5=12 so we must REGROUP 12 ones as 1 ten and 2 ones 1 197 + 35 2 CONFIDENTIAL 12. We continue to REGROUP 1 1 197 + 35 2 3 1+9+3=13 so we must REGROUP 13 tens as 1 hundred and 3tens CONFIDENTIAL 13. Finally, we add the hundreds. This is how we add multi-digit numbers by REGROUPING. 1 1 197 + 35 1+1=2 2 2 3 CONFIDENTIAL 14. Here are some problems for you to try.... 467 + 84 6,725 + 5,932 551 12,657 4,832 + 168 5,000 CONFIDENTIAL 15. 1.) Jacob read 256 pages the first week, 367 pages the second week, and 184pages the third week. How many pages did he read over the 3 weeks? 1) : 256 + 367 + 184 =807 CONFIDENTIAL 16. 2.) 1,2,3,4,5,6 + 2) : 642 +531 (Sample answer) Use the numbers above one time only to make thelargest sumpossible. CONFIDENTIAL 17. 3.) What is wrong with this problem? 6,789 + 2,935 3) : The correct answer is 9,724. 9, 2 6 4 CONFIDENTIAL 18. Assessment 1.) Use the COMMUTATIVE Property of Addition to complete this problem 1) : 84 + 37 37 + 84 = + 2.) Do you think I can use the COMMUTATIVE Property of Addition with three numbers? + 4 + 5 + 8 = + 2) : Yes. 5+8+4 or 8+4+5 CONFIDENTIAL 19. 3.) Use the ASSOCIATIVE Property of Addition to complete this problem. Be sure to include the parentheses! 25 + (5 + 9) = 3) : (25 + 5) + 9 4.) Do you think it is easier to add the left side of the equation or the right side? Why? CONFIDENTIAL 20. 5.) Use the IDENTITY Property of Addition to complete the problem. 4,732 + 5) : 0 and 4,732 = CONFIDENTIAL 21. Find the sum: 13,563 45,258 6) 7) + 4,709 + 787 18,272 46,045 8) 7,064 + 9,673 16,737 CONFIDENTIAL 22. Very Good! Let's Review There are several properties of addition COMMUTATIVE PROPERTY: We can change the order and the sum stays the same. EXAMPLE: 4 + 6 = 10 6 + 4 = 10 ASSOCIATIVE PROPERTY: We can change the groupingand the sum stays the same. EXAMPLE: 2 + (3+ 8) = 10 (2 + 3)+ 8= 10 IDENTITY PROPERTY: We can add zero and the sum stays the same. EXAMPLE: 7 + 0 = 7 CONFIDENTIAL 23. We add numbers together by first LINING UP THE DIGITS Then we need to determine if we must REGROUP 7+5=12 so we must REGROUP 12 ones as 1 ten and 2 ones 1 197 + 35 This is how we regroup. 2 CONFIDENTIAL 24. You did a wonderful job today! See you in the next session. CONFIDENTIAL<\|endoftext\|>	4.65625
485	There’s a school of thought which believes that playing computer games, far from causing the eyesight to deteriorate, can actually improve it to a large degree – that is if you have poor eyesight to begin with. Research Carried Out By A Canadian University An academic at McMaster University in Canada had come across earlier research which suggested that a certain amount of gaming made a significant improvement in the eyesight of young people who had been almost blind as babies. Patients Had Congenital Cataracts The patients in the study had been born with cataracts in their eyes which were subsequently removed. The operation did not result in them having good vision as the babies were left being able to distinguish light but without any detail. Over four weeks this research sample of young men and women played video games for forty hours – at ten hours a week. A reported improvement in vision was then noted, shown by the observation that the patients could read two more lines on an eye chart, something they had been unable to do prior to the gaming experiment. So why did playing these screen-based games have such a positive effect? It is thought to be because the brain responds by stimulating sluggish cells and encouraging fresh links. Good News For Young Gamers With Visual Impairment The research is likely to be popular with the young who have had very poor vision as babies. By indulging in visual, sports-based games on a Playstation or Wii they could actually be helping the quality of their eyesight to improve. After spending forty hours of time gaming over four weeks , the patient sample could discern the difference between faces and the location of movement. This was in addition to being able to view more lines on an eye chart. The Professor who set up this research study presented her discoveries to the American Association for the Advancement of Science. She stated that: “Videogames have got a lot going for them in terms of them being an optimal visual therapy.” Her observations concluded that during the first ten years of life the brain learns to decipher visual material. However it needs to receive a critical amount to do this and if this is not possible (due to an issue like cataracts) then it will find it difficult. By interacting with vigorous games the brain is encouraged to make a sustained effort and forms fresh connections which have a positive visual result.<\|endoftext\|>	3.859375
798	# Question: In the given quadrilateral beginalignABCDendalign , find the value of beginalignxendalign in radians. Gauthmathier1391 In the given quadrilateral \begin{align}ABCD\end{align}, find the value of \begin{align}x\end{align} in radians. Good Question (71) Report ## Gauth Tutor Solution Yoko Electrical engineer Tutor for 2 years \begin{align}\frac{5\pi}{6}\end{align} Explanation To convert from degrees to radians, multiply by \begin{align}\frac{\pi}{180^\circ}\end{align}. \begin{align}\eqalign{ && 60^\circ \\ && 60^\circ \times \frac{\pi}{180^\circ} \\ && \frac{60^\circ \pi}{180^\circ} \\ && \frac{\pi}{3} }\end{align} \begin{align}\eqalign{ && 360^\circ \\ && 360^\circ \times \frac{\pi}{180^\circ} \\ && \frac{360^\circ \pi}{180^\circ} \\ && 2 \pi }\end{align} Let: \begin{align}\angle A = \frac{\pi}{6}\end{align} \begin{align}\angle B = \frac{\pi}{3}\end{align} \begin{align}\angle C = x\end{align} \begin{align}\angle D = \frac{2 \pi}{3}\end{align} The sum the angles of a quadrilateral is \begin{align}360^\circ\end{align}or \begin{align}2 \pi\end{align} radians. \begin{align}\eqalign{ \angle A + \angle B + \angle C + \angle D &=2 \pi \\ \frac{\pi}{6} + \frac{\pi}{3} + x + \frac{2 \pi}{3} &=2 \pi \\ \frac{\pi}{6} + \frac{2 \pi}{6} + \frac{6x}{6} + \frac{4 \pi}{6} &=2 \pi \\ \frac{\pi + 2 \pi + 6x + 4 \pi}{6} &=2 \pi \\ \frac{7\pi + 6x}{6} &=2 \pi \\ \frac{7\pi + 6x}{6} \ {\color{red}\times \ 6} &=2 \pi \ {\color{red}\times \ 6} \\ 7 \pi + 6x &=12 \pi \\ 7 \pi \ {\color{red}- \ 7 \pi} + 6x &=12 \pi \ {\color{red}- \ 7 \pi} \\ 6x &=5 \pi \\ \frac{6x}{\color{red}6} &=\frac{5 \pi}{\color{red}6} \\ x &=\frac{5 \pi}{6} }\end{align} The value of \begin{align}x\end{align} is \begin{align}\frac{5 \pi}{6}\end{align} radians. 5 Thanks (74) Feedback from students Detailed steps (85) Clear explanation (83) Excellent Handwriting (80) Help me a lot (76) Easy to understand (70) Write neatly (32) • 12 Free tickets every month • Always best price for tickets purchase • High accurate tutors, shorter answering time<\|endoftext\|>	4.40625
414	# Graphing Basic Exponential Functions How can we use the graph of $y=2^x$ to sketch the graph of $y=2^{x-1}$? - Note that for any $x,y$ graph: $y = f(x-1)$ is just the graph $y = f(x)$ shifted over one unit to the right. - Why does it shift over one unit to the right instead of left? – jaykirby Jul 3 '13 at 11:19 I'll explain it like this: where you once had $f(0)$, you now have $f(0-1)=f(-1)$. That is, $f(-1)$ was shifted to the right to cover $x=0$. – Omnomnomnom Jul 3 '13 at 11:21 Could you try explaining it to me without the f's..I get confused easily by that.. – jaykirby Jul 3 '13 at 11:24 @omnomnomnon: What happens to the y value, it is always going to be one less isn't it? – jaykirby Jul 3 '13 at 11:44 So to explain it without all the $f$'s, let's look at $2^x$ and $2^{x-1}$ over the numbers $\{1,2,3\}$. We have: $$2^1=2;2^{1-1}=1\\ 2^2=4; 2^{2-1}=2\\ 2^3=8; 2^{3-1}=4$$ At each $x$, the second graph takes the value that was the $y$ at one to its left. The result, if you graphed this, is that $2^{x-1}$ is shifted to the right by one. Does that make sense? – Omnomnomnom Jul 3 '13 at 14:01<\|endoftext\|>	4.40625
198	The sun is the source of energy and heat for our planet. On Friday, we taught your little one that the sun’s energy travels through space, into the earth’s atmosphere, and finally onto earth’s surface. The sun’s radiation warms the earth’s atmosphere and surface and this process is called heat transfer. To help your little one understand this, we studied how heat transfer affects matter. To a child, matter is anything that they can touch. Because heat transfer can be an abstract concept for the young learner, we did a variety of activities that enabled your child to interact with it in a hands-on way. It is for this reason that we created mud pies! By making mud pies and allowing them to dry over night, your little one was able to experience how the sun works with their eyes and their hands. This enabled them to experience the process of heat transfer in a way that they could understand.<\|endoftext\|>	3.953125
600	Phonemic awareness is the awareness that words are composed of sequences or strings of individual sounds called phonemes. Phonemes are the smallest parts of sound in a spoken word. The word, “at,” has two sounds or phonemes, / ă / / t /. The word, “dog,” has three phonemes, / d / / ŏ / / g /. The word, “box,” only has three letters, but it has four phonemes, / b / / ŏ / / k / / s /. As evidenced by the word, “box,” phonemes are completely separate entities from the symbols that we call letters of the alphabet, or graphemes. A grapheme is the smallest part of written language that represents a phoneme in the spelling of a word. A grapheme may be just one letter, such as “t” or “d” or several letters, such as “aw” or “eigh”. Graphemes represent the phonemes in written language. Students must have the ability to identify and visually image the number, order, and identity of sounds and letters within words. These abilities underlie accurate word attack and spelling. Children who have these skills are likely to have an easier time learning to read and spell than children who have few or none of these skills. Weakness in these functions causes individuals to add, omit, substitute, and reverse sounds and letters within words while reading and spelling. The five key skills that serve as the foundation of phonemic awareness are: Replication – the ability to repeat a sound that they hear Blending – the ability to join a string of phonemes together to create a word Segmenting – the ability to break a word into its individual phonemes Substitution – the ability to replace a phoneme with a new phoneme, creating a new word Rhyming – the ability to find words with the same rhyme The onset is the beginning of the word, usually consisting of consonants. The rhyme (rime) is the rest of the word without the initial consonant structure (single consonant, consonant digraph, or consonant blend). For example, in the word, “rest,” the consonant “r” is the onset. The remaining letters in the word, “est,” is the rhyme. Words that end with the phonemes / ĕ / / s / / t / rhyme with “rest” (e.g., test, guest, blessed). (Excerpted from, “School Success for Kids with Dyslexia and Other Reading Difficulties” by Walter Dunson, Ph.D.) Image courtesy of Pixabay<\|endoftext\|>	4.03125
1,473	Solve the inequality 𝑥 plus nine times 𝑥 minus two is less than or equal to 22𝑥 minus 74. We’re going to solve this inequality in three steps. Firstly, we’re going to rearrange the inequality that we have, putting it in the form 𝑓 of 𝑥 is less than or equal to zero. Then we’re going to sketch a graph of 𝑓 of 𝑥. And we’ll see the once we’ve graphed 𝑓 of 𝑥, we will be able to conclude. So naturally we start with our first step. We write down the inequality that we have. We expand the brackets on the left-hand side, so 𝑥 plus nine times 𝑥 minus two becomes 𝑥 squared plus seven 𝑥 minus 18. We add 74 to both sides, so we get 𝑥 squared plus seven 𝑥 plus 56 on the left-hand side and just 22𝑥 on the right-hand side. And finally we subtract 22𝑥 from both sides to get 𝑥 squared minus 15𝑥 plus 56 is less than or equal to zero. We have written the inequality in the required form for step one, so we’re now ready to move on to step two. Okay, so now we have to graph 𝑓 of 𝑥. What is 𝑓 of 𝑥? Well it’s the left-hand side of the inequality that we got after completing step one. We can see that 𝑓 of 𝑥 is a quadratic function. And of course to graph a quadratic function, it’s very helpful to know the factors because they tell you the 𝑥- intercept of the graph. So let’s attempt to factor 𝑓 of 𝑥. We’re looking for two numbers whose sum is negative 15 and whose product is 56. Two numbers which satisfy those requirements are negative seven and negative eight. And so 𝑓 of 𝑥 when factored is 𝑥 minus seven times 𝑥 minus eight. So let’s try to sketch a graph. We can see that 𝑓 of 𝑥 has zeros at 𝑥 equals seven and 𝑥 equals eight, so we know its graph must pass through the points seven, zero and eight, zero, which are marked. We also know that as 𝑓 of 𝑥 is a quadratic function, its graph will be 𝑎 parabola. So the only question is whether it’s an upward- or downward-facing parabola. How can we tell? Well for the graph of 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏 𝑥 plus 𝑐, if the coefficient of 𝑥 squared, 𝑎, is greater than zero, then we have an upward-facing parabola; and if 𝑎 is less than zero, then we have a down-facing parabola. So which case do we have here? The coefficient of 𝑥 squared is just one, which is greater than zero, and so we have an upward-facing parabola. As a result, the graph of 𝑓 of 𝑥 looks something like this. Now we have a sketch of the graph of 𝑓 of 𝑥. Of course this sketch isn’t particularly accurate, but it’s good enough for what we need it for. We can now conclude. The inequality we’re trying to solve now is 𝑓 of 𝑥 is less than or equal to zero. 𝑓 of 𝑥 is less than or equals to zero when its graph is below the 𝑥-axis, which happens between 𝑥 equals seven and 𝑥 equals eight. So that’s the solution to the inequality informally, but we need to make it mathematical. First of all, we need to clarify what we mean by between, our 𝑥 equals seven and 𝑥 equals eight included in the solution set of this inequality. Looking at the graph we can remember that 𝑓 of seven is zero and so is definitely less than or equal to zero; similarly, 𝑓 of eight is zero and so it satisfies 𝑓 of 𝑥 is less than or equal to zero. So 𝑥 equal seven and 𝑥 equals eight are included in the solution sets. Here’s one way to express this mathematically; we say that seven is less than or equal to 𝑥 which is less than or equal to eight. And the fact that we are using less than or equal to signs instead of just less than signs tells us that seven and eight are allowed values of 𝑥. We can also express this fact using set notation and interval notation. So we say that 𝑥 is in the interval from seven to eight. And the fact that we’re using square brackets here instead of round parentheses tells us that the endpoints seven and eight are included in this interval. Let’s just recap what we’ve done. We took this inequality and rearranged it to the form 𝑓 of 𝑥 is less than or equal to zero; that was the first step. Had the inequality sign been just a less than sign, of course we would be rearranging to just 𝑓 of 𝑥 is less than zero. The important thing is that on the right-hand side of the inequality sign whatever it maybe, we have zero. We then took this 𝑓 of 𝑥 and we sketched its graph, and we factored 𝑓 of 𝑥 to allow us to do this. Finally, we concluded from the graph what the solution was. There were two things that we needed from the graph in order to be able to conclude: we needed the 𝑥-intercepts, seven and eight, and we also needed to know the orientation of the parabola which passes through these two points. Had we drawn a downward-facing parabola, we’ve have got the wrong answer. To check that we’ve got the answer right, you might like to check that values inside the interval that we’ve got do indeed satisfy the original inequality that we had: 𝑥 plus nine times 𝑥 minus two is less than or equal to 22𝑥 minus 74. And further, you might like to check that values outside this interval do not satisfy the inequality.<\|endoftext\|>	4.28125
875	NEW ORLEANS — How is a little Pacific island like the planet Mars? Let James Garvin count the ways. In December 2014, an underwater volcano amid the islands of Tonga in the South Pacific erupted. When the eruption ended and ashes settled a month later, a new island had emerged, rising 400 feet above the ocean’s surface. Scientists unofficially named the island Hunga Tonga-Hunga Ha’apai, a concatenation of two older, uninhabited islands it nestles between. Since then, scientists have been tracking how the new land mass has eroded and shifted. What they have found could make the island a Rosetta Stone to understanding volcanic features on Mars that also appear to have erupted underwater, providing clues about when the red planet was wet several billion years ago. “We see things that remind us of this kind of volcano at similar scales on Mars,” said Dr. Garvin, the chief scientist at NASA’s Goddard Space Flight Center in Greenbelt, Md. “And literally, there are thousands of them, in multiple regions.” He and colleagues presented the findings on Monday at a news conference at a meeting of the American Geophysical Union here. Networks of river channels chiseled into Mars persuasively argue that liquid water once flowed across the red planet, but the current thinking of many planetary scientists is that Mars remained frozen through much of its history, punctuated with episodes of melting and flowing water. Life on Mars On the side of Mauna Loa volcano in Hawaii, six individuals are living in Mars-like conditions as part of a NASA-funded behavioral research study. We chronicle their mission in 360 video. Some Martian volcanoes that look as if they erupted underwater could offer clues. By analyzing these leftover structures, scientists may be able to tease out information like how deep the water was when the volcanoes erupted and how long water persisted. But to fully understand the terrain on Mars, researchers need a model to compare it against, and that’s where the new island comes in. While Tonga is in the middle of the ocean, Hunga Tonga-Hunga Ha’apai and its neighbors sit on the rim of a large volcano that rises about a mile above the deep ocean floor. Thus the water around the island is shallow, perhaps similar to what existed around the Martian volcanoes. Since the eruption, satellites have repeatedly viewed the new Tongan island, not much more than a square mile in size, allowing scientists to generate detailed maps of the shifting topography. Scientists have also made visits to map the surrounding seafloor and walk around for up-close examination. That’s an advantage that Earth scientists have over Mars researchers — they can directly compare what satellites see from orbit with samples they pick up. Islands formed by explosive underwater eruptions are usually short-lived, the ash washed away by crashing waves. In the initial months of its existence, Hunga Tonga-Hunga Ha’apai shifted in shape quickly. Initially oval, the island’s southern shore eroded rapidly, allowing the Pacific Ocean to break through into the lake at the center of the island. Steep walls around the lake appeared in danger of collapse, and it looked as if the island might have been about to vanish. But then a sandbar formed, sealing off the lake again, and the landscape stabilized. When conditions are right, chemical reactions with warm water cement volcanic ash into resilient rock, and the scientists speculate that similar reactions may have occurred on Hunga Tonga-Hunga Ha’apai. It is only the third such island in the last 150 years to survive more than a few months. They estimate that the island could now last for decades. The evolving island can be compared with erosion patterns around Martian volcanoes. If some of the volcanic shapes on Mars matched an intermediate state of Hunga Tonga-Hunga Ha’apai, that could suggest that the water disappeared and the erosion stopped. “That will give us a window into some of those murkier times of Mars, when we think there were standing bodies of water,” Dr. Garvin said.<\|endoftext\|>	3.84375
1,332	Computational thinking has become a battle cry for coding in K–12 education. It is echoed in statewide efforts to develop standards, in changes to teacher certification and graduation requirements, and in new curriculum designs.1 The annual Hour of Code has introduced millions of kids to coding inspired by Apple cofounder Steve Jobs who said, "everyone should learn how to program a computer because it teaches you how to think." Computational thinking has garnered much attention but people seldom recognize that the goal is to bring programming back into the classroom. In the 1980s many schools featured Basic, Logo, or Pascal programming computer labs. Students typically received weekly introductory programming instruction.6 These exercises were often of limited complexity, disconnected from classroom work, and lacking in relevance. They did not deliver on promises. By the mid-1990s most schools had turned away from programming. Pre-assembled multimedia packages burned onto glossy CD-ROMs took over. Toiling over syntax typos and debugging problems were no longer classroom activities. Computer science is making a comeback in schools. We should not repeat earlier mistakes, but leverage what we have learned.5 Why are students interested in programming? Under what circumstances do they do it, and how?2 Computational thinking and programming are social, creative practices. They offer a context for making applications of significance for others, communities in which design sharing and collaboration with others are paramount. Computational thinking should be reframed as computational participation. This idea expands on Jeannette Wing's original definition of computational thinking.7 Computational participation involves solving problems, designing systems, and understanding human behavior in the context of computing. It allows for participation in digital activities. Many kids use code outside of school to create and share. Youth-generated websites have appeared to make and share programmable media online. These sites include video games, interactive art projects, and digital stories. They are inherently do-it-yourself (DIY), encouraging youth programming as an effective way to create and share online, and connect with each other, unlike learned disciplines such as algebra or chemistry. Through individual endeavor mixed with group feedback and collaboration the DIY ethos opens up three new pathways for engaging youth. From building code to creating shareable applications. Programming that prizes coding accuracy and efficiency as signifiers of success is boring. To learn programming for the sake of programming goes nowhere for children unless they can put those skills to use in a meaningful way. Today children program to create applications like video games or interactive stories as part of a larger learning community.3 They are attracted to the possibility of creating something real and tangible that can be shared with others. Programming is not an abstract discipline, but a way to "make" and "be" in the digital world. From tools to communities. Coding was once a solitary, tool-based activity. Now it is becoming a shared social practice. Participation spurred by open software environments and mutual enthusiasm shifts attention from programming tools to designing and supporting communities of learners. The past decade has brought many admirable introductory programming languages to make coding more intuitive and personal. Developers and educators realize that tools alone are not enough. Audiences are needed, and a critical mass of like-minded creators. Scratch, Alice, and similar tools have online communities of millions of young users. Children can work and share programs on a single website. This tacitly highlights the community of practice that has become a key for learning to code. From "from scratch" creation to "remixing." These new, networked communities focus on remixing. Students once created programs from scratch to demonstrate competency. Now they pursue seamless integration via remixing as the new social norm, in the spirit of the open source movement. Sharing one's code encourages others to sample creations, adjust them, and add to them. Such openness heightens potential for innovation across the board. Young users embrace sampling and sharing more freely, challenging the traditional top-down paradigm. How do we facilitate broader and deeper participation in the design of the programming activities, tools, and practices? These three shifts signal a social turn in K–12 computing. They move from a predominantly individualistic view to greater focus on underlying social and cultural dimensions of programming. We should rethink what and how students learn to become full participants in networked communities. It is not possible to addresses all of the difficulties of implementing computational participation by placing students in groups, having them program applications, and encouraging them to remix code. Computational participation will present new challenges in bringing programming back into schools. How do we facilitate broader and deeper participation in the design of the programming activities, tools, and practices? Computational thinking is a social practice. We must broaden access to communities of programming.4 Children are not "digital natives" who naturally migrate online. Establishing membership in the programming community is not easy. Groups with powerful learning cultures are often exclusive cultures. Students need strategies to cope with the vulnerability of sharing one's work for others to comment on and remix. In addition, students need a more expansive menu of computing activities, tools, and materials. Designing authentic applications is an important step in the right direction, but games, stories, and robotics are not the only applications for achieving this goal. We need different materials to expand students' perspectives and perceptions of computing. Broadening computational participation gets students into the clubhouse. The next challenge is to help them develop fluency that permits them to engage deeply, making their participation meaningful and enriching. These levels of computational participation are still rare. To learn to code students must learn the technicalities of programming language and common algorithms, and the social practices of programming communities. Computational participation provides new direction for programming in K–12 education. It moves us beyond tools and code to community and context. It equips designers, educators, and researchers to broaden and deepen computational thinking on a larger scale than previously. Users of digital technologies for functional, political, and personal purposes need a basic understanding of computing. Students must understand interfaces, technologies, and systems that they encounter daily. This will empower them and provides them with the tools to examine and question design decisions they encounter. Computing for communicating and interacting with others builds relationships. Education activist Paulo Freire once said that "reading the word is reading the world." He was right. Today, reading code is about reading the world. It is needed to understand, change, and remake the digital world in which we live. The Digital Library is published by the Association for Computing Machinery. Copyright © 2016 ACM, Inc. No entries found<\|endoftext\|>	4
1,314	# Unlocking the Secrets of .166 as a Fraction: A Comprehensive Guide and Simplified Explanation ## Understanding .166 as a Fraction: An Introduction to Decimal to Fraction Conversion Converting decimals to fractions can be a challenging task for many individuals. However, it is an essential skill to have, especially when dealing with mathematical calculations or measurements. In this article, we will focus on understanding and converting the decimal .166 to a fraction. To convert a decimal to a fraction, it is important to understand the place value of each digit after the decimal point. In the example of .166, the digit 1 is in the tenths place, the digit 6 is in the hundredths place, and the digit 6 is in the thousandths place. When converting a decimal to a fraction, we utilize the fact that the decimal place value is related to the power of 10. For example, the tenths place is equivalent to 10-1, the hundredths place is equivalent to 10-2, and the thousandths place is equivalent to 10-3. To convert .166 to a fraction, we can write it as 166/1000. We can simplify this fraction further by finding the greatest common divisor for both the numerator and the denominator, which is 2 in this case. Further simplification gives us the fraction 83/500. ## The Simplified Fraction Representation of .166 Converting decimal numbers into simplified fraction representations can be a useful skill to have, especially when dealing with measurements or calculations that require precise fractions. In this article, we will focus on the simplified fraction representation of the decimal number .166. To convert .166 into a simplified fraction, we need to understand that the decimal point separates the whole number part from the fractional part. In this case, the whole number part is 0, and the fractional part is 166. To simplify the fraction, we need to consider the lowest common denominator (LCD). Step 1: Identifying the LCD: The fractional part of .166 is 166. Since 166 is not divisible by any other number besides 1 and itself, the LCD is 1. Step 2: Writing the simplified fraction: To convert .166 into a fraction, we simply write the fractional part (166) over the LCD (1). Therefore, the simplified fraction representation of .166 is 166/1. You may also be interested in: 365°F to °C Conversion Made Easy: A Comprehensive Guide for Hassle-Free Temperature Conversions It’s important to note that even though the fraction 166/1 is already in its simplest form, it may not be the most conventional way to express .166 as a fraction. In such cases, it is common to multiply the numerator and denominator by a suitable number to further simplify the fraction. However, for the decimal .166, the fraction 166/1 is already the simplified representation. ## Explaining the Steps to Convert .166 to a Fraction ### Converting .166 to a Fraction To convert a decimal number like .166 to a fraction, you need to follow a few simple steps. The process involves understanding the place value of each digit in the decimal number and then expressing it as a fraction in its simplest form. Step 1: Write down the decimal number as the numerator of the fraction. In this case, the numerator is .166. Step 2: Determine the denominator of the fraction. The denominator is based on the number of decimal places in the decimal number. Since .166 has three decimal places, the denominator will be 1000 (10 raised to the power of 3). Step 3: Simplify the fraction, if necessary. In this case, .166 can be simplified by dividing both the numerator and denominator by their greatest common divisor. However, in many cases, decimals cannot be simplified further. Step 4: Write the simplified fraction. In this example, .166 as a fraction is 166/1000. To further simplify it, divide both the numerator and denominator by 166 to get the simplest form of the fraction. Converting decimals to fractions is an essential skill in mathematics and can be useful in various real-world applications. By following these steps, you can easily convert a decimal like .166 into its equivalent fraction representation. ## Practical Use Case of .166 as a Fraction in Everyday Situations ### 1. Cooking Measurements In cooking, precise measurements are crucial for achieving the desired results in a recipe. When a recipe involves small amounts of an ingredient, such as spices or flavorings, it is often expressed in fractions. Using .166 as a fraction can be helpful when doubling or halving a recipe, as it allows for more accurate measurements. For example, if a recipe calls for 1/3 cup of an ingredient and you need to halve it, you would use approximately .166 cups. You may also be interested in: Converting 3.6 km to Miles: How to Easily Calculate the Distance in Your Preferred Units ### 2. Currency Conversion When traveling internationally, it’s important to be able to convert currencies to understand the value of items or services. In some cases, the exchange rate may result in fractional amounts. Using .166 as a fraction can help in understanding the approximate value of a currency conversion. For instance, if the exchange rate is 0.5 for a currency, multiplying it by .166 would give you an approximate value of 0.083, which can be useful for estimating expenses. You may also be interested in: Simplify Fractions Made Easy: Simplifying 6/100 and Mastering Fraction Reducing Techniques ### 3. Percentage Calculations Percentages are commonly used in various everyday situations, such as calculating discounts, taxes, or interest rates. Sometimes, percentages may result in fractions. For example, if a product is on sale for 25% off, you can use .166 as a fraction to determine the discount amount. Multiplying the original price by .166 would give you an approximate value of the discount percentage. In conclusion, .166 as a fraction has practical applications in everyday situations such as cooking measurements, currency conversions, and percentage calculations. It can be used to achieve more accurate measurements in recipes, estimate the value of currency exchanges, and calculate percentage-based amounts. Understanding how to use .166 as a fraction can help in various aspects of daily life where precise calculations are necessary.<\|endoftext\|>	4.96875
773	An adjective is a word that qualifies a noun. Without belabouring the subject matter, let’s look at some complex aspects of the adjectives especially ones tested in examinations. Adjective of Quantity Adjective of Quantity shows the amount of nouns without the exact number. These adjectives help to show the amount or the estimated amount of the noun unlike the adjective of number that is very specific. Examples are: little, a little, few, a few, many, much, some, enough, plenty, whole, all etc. Our focus in this section is on the first six adjectives in the examples. Little and A Little Little and a little are quantifiers used for uncountable nouns like water, sand, air, information, advice, bread, milk etc. Little denotes a very small amount of the noun. It means the amount is so small that it is not enough. - Bathing will be impossible as we have little - I can’t buy that form now because I have little money in my account. - Mummy couldn’t prepare the breakfast; we had little - Little honey was scraped from the surface. A little is used to refer to an amount of the noun though small but considerably enough. - A little water was available to clean the baby. - I stayed back because I still had a little - We enjoyed a little peace compared to our counterpart. - The pay is a little bit better. Few and A Few Few and a few are adjectives of quantity used for countable nouns like book, game, child, star, father etc. Few is used to quantify the plural countable noun. It refers to a very small number of something. It denotes inadequacy just like little. - The visitors were made to stand because we had few - The principal couldn’t address us; we were few. - Your few dogs cannot withstand our security pressure. - He was taken for a ride because he was a man of few A Few is used for plural countable nouns and it refers to a small number but not in the negative. It conveys a sense of considerable adequacy. - A few of the students helped to rearrange the hall. - I was able to revise a few topics that helped me in the exam. - With a few songs, the choir rocked the stage. - A few of the people were stranded and the programme couldn’t start. Summarily, a little and little are used for the uncountable noun. While little denotes inadequacy, a little show adequacy. Same goes for a few and few safe the fact that they are used for the plural countable noun. To aid your memory, let the ‘a’ in front of a little and a few stand for ‘adequacy’. So, anytime you need to express an amount or a number of something that is small yet considerably enough or adequate, put ‘a’ before you little or few. However, if the amount or number is not enough, then, remove ‘a’. I hope this helps you. Many and Much Many is used with the plural countable noun and Much is used with the uncountable noun. The comparative and superlative adjectives of both are more and most respectively. Many of the musicians don’t write their songs. Ronaldo will surely score many goals this season. Many Facebook users are below forty years. How much salt will you need? He does have much money. The family had so much fun with the music stars. We saw many animals in the park.<\|endoftext\|>	3.859375
10,169	The Ptolemaic Kingdom in 300 BC (in blue). \|Languages\|\|Greek, Egyptian, Berber\| \|Religion\|\|Cult of Serapis\| \|•\|\|305–283 BC\|\|Ptolemy I Soter (first)\| \|•\|\|51–30 BC\|\|Cleopatra VII (last)\| \|Historical era\|\|Classical antiquity\| \|Today part of\|\| Cyprus Part of a series on the \|History of Egypt\| The Ptolemaic Kingdom (//; Ancient Greek: Πτολεμαϊκὴ βασιλεία, Ptolemaïkḕ Basileía) was a Hellenistic kingdom based in Egypt. It was ruled by the Ptolemaic dynasty which started with Ptolemy I Soter's accession after the death of Alexander the Great in 323 BC and which ended with the death of Cleopatra VII and the Roman conquest in 30 BC. The Ptolemaic Kingdom was founded in 305 BC by Ptolemy I Soter, who declared himself Pharaoh of Egypt and created a powerful Hellenistic dynasty that ruled an area stretching from southern Syria to Cyrene and south to Nubia. Alexandria became the capital city and a major center of Greek culture and trade. To gain recognition by the native Egyptian populace, they named themselves the successors to the Pharaohs. The later Ptolemies took on Egyptian traditions by marrying their siblings, had themselves portrayed on public monuments in Egyptian style and dress, and participated in Egyptian religious life. The Ptolemies had to fight native rebellions and were involved in foreign and civil wars that led to the decline of the kingdom and its final annexation by Rome. Hellenistic culture continued to thrive in Egypt throughout the Roman and Byzantine periods until the Muslim conquest. - 1 History - 2 Culture - 3 Cities - 4 Demographics - 5 Agriculture - 6 List of Ptolemaic rulers - 7 See also - 8 References - 9 Further reading - 10 External links The era of Ptolemaic reign in Egypt is one of the most well documented time periods of the Hellenistic Era; a wealth of papyri written in Greek and Egyptian of the time have been discovered in Egypt. In 332 BC, Alexander the Great, King of Macedon invaded the Achaemenid satrapy of Egypt. He visited Memphis, and traveled to the oracle of Amun at the Oasis of Siwa. The oracle declared him to be the son of Amun. He conciliated the Egyptians by the respect he showed for their religion, but he appointed Macedonians to virtually all the senior posts in the country, and founded a new Greek city, Alexandria, to be the new capital. The wealth of Egypt could now be harnessed for Alexander's conquest of the rest of the Persian Empire. Early in 331 BC he was ready to depart, and led his forces away to Phoenicia. He left Cleomenes as the ruling nomarch to control Egypt in his absence. Alexander never returned to Egypt. Following Alexander's death in Babylon in 323 BC, a succession crisis erupted among his generals. Initially, Perdiccas ruled the empire as regent for Alexander's half-brother Arrhidaeus, who became Philip III of Macedon, and then as regent for both Philip III and Alexander's infant son Alexander IV of Macedon, who had not been born at the time of his father's death. Perdiccas appointed Ptolemy, one of Alexander's closest companions, to be satrap of Egypt. Ptolemy ruled Egypt from 323 BC, nominally in the name of the joint kings Philip III and Alexander IV. However, as Alexander the Great's empire disintegrated, Ptolemy soon established himself as ruler in his own right. Ptolemy successfully defended Egypt against an invasion by Perdiccas in 321 BC, and consolidated his position in Egypt and the surrounding areas during the Wars of the Diadochi (322–301 BC). In 305 BC, Ptolemy took the title of King. As Ptolemy I Soter ("Saviour"), he founded the Ptolemaic dynasty that was to rule Egypt for nearly 300 years. All the male rulers of the dynasty took the name "Ptolemy", while princesses and queens preferred the names Cleopatra, Arsinoe and Berenice. Because the Ptolemaic kings adopted the Egyptian custom of marrying their sisters, many of the kings ruled jointly with their spouses, who were also of the royal house. This custom made Ptolemaic politics confusingly incestuous, and the later Ptolemies were increasingly feeble. The only Ptolemaic Queens to officially rule on their own were Berenice III and Berenice IV. Cleopatra V did co-rule, but it was with another female, Berenice IV. Cleopatra VII officially co-ruled with Ptolemy XIII Theos Philopator, Ptolemy XIV, and Ptolemy XV, but effectively, she ruled Egypt alone. The early Ptolemies did not disturb the religion or the customs of the Egyptians, and indeed built magnificent new temples for the Egyptian gods and soon adopted the outward display of the Pharaohs of old. During the reign of Ptolemies II and III thousands of Macedonian veterans were rewarded with grants of farm lands, and Macedonians were planted in colonies and garrisons or settled themselves in the villages throughout the country. Upper Egypt, farthest from the centre of government, was less immediately affected, even though Ptolemy I established the Greek colony of Ptolemais Hermiou to be its capital. But within a century Greek influence had spread through the country and intermarriage had produced a large Greco-Egyptian educated class. Nevertheless, the Greeks always remained a privileged minority in Ptolemaic Egypt. They lived under Greek law, received a Greek education, were tried in Greek courts, and were citizens of Greek cities. The first part of Ptolemy I's reign was dominated by the Wars of the Diadochi between the various successor states to the empire of Alexander. His first object was to hold his position in Egypt securely, and secondly to increase his domain. Within a few years he had gained control of Libya, Coele-Syria (including Judea), and Cyprus. When Antigonus, ruler of Syria, tried to reunite Alexander's empire, Ptolemy joined the coalition against him. In 312 BC, allied with Seleucus, the ruler of Babylonia, he defeated Demetrius, the son of Antigonus, in the battle of Gaza. In 311 BC, a peace was concluded between the combatants, but in 309 BC war broke out again, and Ptolemy occupied Corinth and other parts of Greece, although he lost Cyprus after a sea-battle in 306 BC. Antigonus then tried to invade Egypt but Ptolemy held the frontier against him. When the coalition was renewed against Antigonus in 302 BC, Ptolemy joined it, but neither he nor his army were present when Antigonus was defeated and killed at Ipsus. He had instead taken the opportunity to secure Coele-Syria and Palestine, in breach of the agreement assigning it to Seleucus, thereby setting the scene for the future Syrian Wars. Thereafter Ptolemy tried to stay out of land wars, but he retook Cyprus in 295 BC. Feeling the kingdom was now secure, Ptolemy shared rule with his son Ptolemy II by Queen Berenice in 285 BC. He then may have devoted his retirement to writing a history of the campaigns of Alexander—which unfortunately was lost but was a principal source for the later work of Arrian. Ptolemy I died in 283 BC at the age of 84. He left a stable and well-governed kingdom to his son. Ptolemy II Philadelphus, who succeeded his father as King of Egypt in 283 BC, was a peaceful and cultured king, and no great warrior. He did not need to be, because his father had left Egypt strong and prosperous. Three years of campaigning at the start of his reign (called the First Syrian War) left Ptolemy the master of the eastern Mediterranean, controlling the Aegean islands and the coastal districts of Cilicia, Pamphylia, Lycia and Caria. However, some of these territories were lost near the end of his reign as a result of the Second Syrian War. In the 270s BC, Ptolemy II defeated the Kingdom of Kush in war, gaining the Ptolemies free access to Kushite territory and control of important gold-mining areas south of Egypt known as Dodekasoinos. As a result, the Ptolemies established hunting stations and ports as far south as Port Sudan, from where raiding parties containing hundreds of men searched for war elephants. Hellenistic culture would acquire an important influence on Kush at this time. Ptolemy's first wife, Arsinoe I, daughter of Lysimachus, was the mother of his legitimate children. After her repudiation he followed Egyptian custom and married his sister, Arsinoë II, beginning a practice that, while pleasing to the Egyptian population, had serious consequences in later reigns. The material and literary splendour of the Alexandrian court was at its height under Ptolemy II. Callimachus, keeper of the Library of Alexandria, Theocritus and a host of other poets, glorified the Ptolemaic family. Ptolemy himself was eager to increase the library and to patronise scientific research. He spent lavishly on making Alexandria the economic, artistic and intellectual capital of the Hellenistic world. It is to the academies and libraries of Alexandria that we owe the preservation of so much Greek literary heritage. Ptolemy III Euergetes ("the benefactor") succeeded his father in 246 BC. He abandoned his predecessors' policy of keeping out of the wars of the other Macedonian successor kingdoms, and plunged into the Third Syrian War with the Seleucids of Syria, when his sister, Queen Berenice, and her son were murdered in a dynastic dispute. Ptolemy marched triumphantly into the heart of the Seleucid realm, as far as Babylonia, while his fleets in the Aegean made fresh conquests as far north as Thrace. This victory marked the zenith of the Ptolemaic power. Seleucus II Callinicus kept his throne, but Egyptian fleets controlled most of the coasts of Asia Minor and Greece. After this triumph Ptolemy no longer engaged actively in war, although he supported the enemies of Macedon in Greek politics. His domestic policy differed from his father's in that he patronised the native Egyptian religion more liberally: he has left larger traces among the Egyptian monuments. In this his reign marks the gradual "Egyptianisation" of the Ptolemies. The decline of the Ptolemies In 221 BC, Ptolemy III died and was succeeded by his son Ptolemy IV Philopator, a weak and corrupt king under whom the decline of the Ptolemaic kingdom began. His reign was inaugurated by the murder of his mother, and he was always under the influence of royal favourites, male and female, who controlled the government. Nevertheless, his ministers were able to make serious preparations to meet the attacks of Antiochus III the Great on Coele-Syria, and the great Egyptian victory of Raphia in 217 BC secured the kingdom. A sign of the domestic weakness of his reign was the rebellions by native Egyptians that took away over half the country for over 20 years. Philopator was devoted to orgiastic religions and to literature. He married his sister Arsinoë, but was ruled by his mistress Agathoclea. Ptolemy V Epiphanes, son of Philopator and Arsinoë, was a child when he came to the throne, and a series of regents ran the kingdom. Antiochus III of The Seleucid Empire and Philip V of Macedon made a compact to seize the Ptolemaic possessions. Philip seized several islands and places in Caria and Thrace, while the battle of Panium in 198 BC transferred Coele-Syria from Ptolemaic to Seleucid control. After this defeat Egypt formed an alliance with the rising power in the Mediterranean, Rome. Once he reached adulthood Epiphanes became a tyrant, before his early death in 180 BC. He was succeeded by his infant son Ptolemy VI Philometor. Now when the kingdom was established before Antiochus, he thought to reign over Egypt that he might have the dominion of two realms. Wherefore he entered into Egypt with a great multitude, with chariots, and elephants, and horsemen, and a great navy, and made war against Ptolemy king of Egypt: but Ptolemy was afraid of him, and fled; and many were wounded to death. Thus they got the strong cities in the land of Egypt and he took the spoils thereof. Philometor's younger brother (later Ptolemy VIII Euergetes II) was installed as a puppet king. When Antiochus withdrew, the brothers agreed to reign jointly with their sister Cleopatra II. They soon fell out, however, and quarrels between the two brothers allowed Rome to interfere and to steadily increase its influence in Egypt. Eventually Philometor regained the throne. In 145 BC he was killed in the Battle of Antioch. The later Ptolemies Philometor was succeeded by yet another infant, his son Ptolemy VII Neos Philopator. But Euergetes soon returned, killed his young nephew, seized the throne and as Ptolemy VIII soon proved himself a cruel tyrant. On his death in 116 BC he left the kingdom to his wife Cleopatra III and her son Ptolemy IX Philometor Soter II. The young king was driven out by his mother in 107 BC, who reigned jointly with Euergetes's youngest son Ptolemy X Alexander I. In 88 BC Ptolemy IX again returned to the throne, and retained it until his death in 80 BC. He was succeeded by Ptolemy XI Alexander II, the son of Ptolemy X. He was lynched by the Alexandrian mob after murdering his stepmother, who was also his cousin, aunt and wife. These sordid dynastic quarrels left Egypt so weakened that the country became a de facto protectorate of Rome, which had by now absorbed most of the Greek world. Ptolemy XI was succeeded by a son of Ptolemy IX, Ptolemy XII Neos Dionysos, nicknamed Auletes, the flute-player. By now Rome was the arbiter of Egyptian affairs, and annexed both Libya and Cyprus. In 58 BC Auletes was driven out by the Alexandrian mob, but the Romans restored him to power three years later. He died in 51 BC, leaving the kingdom to his ten-year-old son, Ptolemy XIII Theos Philopator, who reigned jointly with his 17-year-old sister and wife, Cleopatra VII. The demise of the Ptolemies' power coincided with the rise of the Roman Empire. Having little choice, and seeing one city after another falling to Macedon and the Seleucid Empire, the Ptolemies decided to ally with the Romans, a pact that lasted over 150 years. During the rule of the later Ptolemies, Rome gained more and more power over Egypt, and was even declared guardian of the Ptolemaic Dynasty. Cleopatra's father, Ptolemy XII, had to pay tribute to the Romans to keep them away from his Kingdom. Upon his death, the fall of the Dynasty seemed even closer. As children, Cleopatra and her siblings witnessed the defeat of their guardian, Pompey, by Julius Caesar through civil war. Meanwhile, Cleopatra and her brother/husband Ptolemy XIII were both attempting to gain control of Egypt's throne. In the middle of all this turmoil, Julius Caesar left Rome for Alexandria in 48 BC. During his stay in the Palace, he received 22-year-old Cleopatra, allegedly wrapped in a rug. She counted on Caesar's support to alienate Ptolemy XIII. With the arrival of Roman reinforcements, and after a few battles in Alexandria, Ptolemy XIII was defeated at the Battle of the Nile. He later drowned in the river, although the circumstances of his death are unclear. In the summer of 47 BC, having married her younger brother Ptolemy XIV, Cleopatra embarked with Caesar for a two-month trip along the Nile. Together, they visited Dendara, where Cleopatra was being worshiped as Pharaoh, an honor beyond Caesar's reach. They became lovers, and she bore him a son, Caesarion, who was later proclaimed with many titles like king of kings. In 45 BC, Cleopatra and Caesarion left Alexandria for Rome, where they stayed in a palace built by Caesar in their honor. In 44 BC, Caesar was murdered in Rome by several Senators. With his death, Rome split between supporters of Mark Antony and Octavian. Cleopatra was watching in silence, and when Mark Antony seemed to prevail, she supported him and, shortly after, they too became lovers. Mark Antony's alliance with Cleopatra angered Rome even more. The senators called her a sorceress, and accused her of all sorts of evil. The Romans became even more furious as Antony was giving away parts of their Empire - at the donations of Alexandria ceremony in autumn 34 BC - Tarsus, Cyrene, Crete, Cyprus, and Israel - one after the other to Cleopatra and her children. Octavian was able to somehow gain possession of Mark Antony's will, which expressed his desire to be buried in Alexandria, rather than taken to Rome in the event of his death. It was the boiling point when Octavian declared war on the "Foreign Queen", and off the coast of Greece in the Adriatic Sea they met in at Actium, where the forces of Marcus Vipsanius Agrippa defeated the Navy of Cleopatra and Antony. Octavian waited for a year before he claimed Egypt as a Roman province. He arrived in Alexandria and easily defeated Mark Antony outside the city, near present-day Camp César. Following this defeat, and facing certain death at the hands of Octavian, Antony committed suicide by falling on his own sword. Octavian entered Alexandria in 30 BC. Cleopatra was captured and taken to him, but Octavian had no interest in any relation, reconciliation, or even negotiation with the Egyptian Queen. Realizing that her end was close, she decided to put an end to her life. It is not known for sure how she killed herself, but many believe she used a poisonous snake as her death instrument. With the death of Cleopatra, the dynasty of Ptolemies came to an end. Alexandria remained capital of Egypt, but Egypt became a Roman province. In 30 BC, following the death of Cleopatra VII, the Roman Empire declared that Egypt was a province (Aegyptus), and that it was to be governed by a prefect selected by the Emperor from the Equestrian and not a governor from the Senatorial order, to prevent interference by the Roman Senate. The main Roman interest in Egypt was always the reliable delivery of grain to the city of Rome. To this end the Roman administration made no change to the Ptolemaic system of government, although Romans replaced Greeks in the highest offices. But Greeks continued to staff most of the administrative offices and Greek remained the language of government except at the highest levels. Unlike the Greeks, the Romans did not settle in Egypt in large numbers. Culture, education and civic life largely remained Greek throughout the Roman period. The Romans, like the Ptolemies, respected and protected Egyptian religion and customs, although the cult of the Roman state and of the Emperor was gradually introduced. Ptolemy I, perhaps with advice from Demetrius of Phalerum, founded the Museum and Library of Alexandria The Museum was a research centre supported by the king. It was located in the royal sector of the city. The scholars were housed in the same sector and funded by the Ptolemaic rulers. They had access to the Library. The chief librarian served also as the crown prince's tutor. For the first hundred and fifty years of its existence this library and research centre drew the top Greek scholars. This was a key academic, literary and scientific centre. Greek culture had a long but minor presence in Egypt long before Alexander the Great founded the city of Alexandria. It began when Greek colonists, encouraged by the many Pharaohs, set up the trading post of Naucratis, which became an important link between the Greek world and Egypt's grain. As Egypt came under foreign domination and decline, the Pharaohs depended on the Greeks as mercenaries and even advisors. When the Persians took over Egypt, Naucratis remained an important Greek port and the colonist population were used as mercenaries by both the rebel Egyptian princes and the Persian kings, who later gave them land grants, spreading the Greek culture into the valley of the Nile. When Alexander the Great arrived, he established Alexandria on the site of the Persian fort of Rhakortis. Following Alexander's death, control passed into the hands of the Lagid (Ptolemaic) dynasty; they built Greek cities across their empire and gave land grants across Egypt to the veterans of their many military conflicts. Hellenistic civilization continued to thrive even after Rome annexed Egypt after the battle of Actium and did not decline until the Islamic conquests. Hellenistic art is richly diverse in subject matter and in stylistic development. It was created during an age characterized by a strong sense of history. For the first time, there were museums and great libraries, such as those at Alexandria and Pergamon. Hellenistic artists copied and adapted earlier styles, and also made great innovations. Representations of Greek gods took on new forms. The popular image of a nude Aphrodite, for example, reflects the increased secularization of traditional religion. Also prominent in Hellenistic art are representations of Dionysos, the god of wine and legendary conqueror of the East, as well as those of Hermes, the god of commerce. In strikingly tender depictions, Eros, the Greek personification of love, is portrayed as a young child. Most of the Ptolemaic magical stele were connected with matters of health. They were commonly of limestone; the Greeks tended to use marble or bronze for private sculpture. The most striking change in depiction of figures is the range from idealizing to nearly grotesque realism in portrayal of men. Previously Egyptian depictions tended toward the idealistic but stiff, not with an attempt at likeness. Likeness was still not the goal of art under the Ptolemies. The influence of Greek sculpture under the Ptolemies was shown in its emphasis on the face more than in the past. Smiles suddenly appear. Toward the end of the Ptolemaic period, the headdress sometimes gives way to tousled hair. One significant change in Ptolemaic art is the sudden re-appearance of women, who had been absent since about the Twenty-sixth Dynasty. Some of this must have been due to the importance of women, such as the series of Cleopatras, who acted as co-regents or sometimes occupied the throne by themselves. Although women were present in artwork, they were shown less realistically than men in the this era. Even with the Greek influence on art, the notion of the individual portrait still had not supplanted Egyptian artistic norms during the Ptolemaic Dynasty. Ways of presenting text on columns and reliefs became formal and rigid during the Ptolemaic Dynasty. When Ptolemy I Soter made himself king of Egypt, he created a new god, Serapis, which was a combination of two Egyptian gods: Apis and Osiris, plus the main Greek gods: Zeus, Hades, Asklepios, Dionysios, and Helios. Serapis had powers over fertility, the sun, corn, funerary rites, and medicine. Many people started to worship this god. In the time of the Ptolemies, the cult of Serapis included the worship of the new Ptolemaic line of pharaohs. Alexandria supplanted Memphis as the preeminent religious city. The wife of Ptolemy II, Arsinoe II was often depicted in the form of the Greek goddess Aphrodite, but she wore the crown of lower Egypt, with ram's horns, ostrich feathers, and other traditional Egyptian indicators of royalty and/or deity. She wore the vulture headdress only on the religious portion of a relief. Cleopatra VII, the last of the Ptolemaic line, was often depicted with characteristics of the goddess Isis. She often had either a small throne as her headdress or the more traditional sun disk between two horns. The traditional table for offerings disappeared from reliefs during the Ptolemaic period. Male gods were no longer portrayed with tails in attempt to make them more humanlike. The wealthy and connected of Egyptian society seemed to put more stock in magical stela during the Ptolemaic period. These were religious objects produced for private individuals, something uncommon in earlier Egyptian times. The Greeks now formed the new upper classes in Egypt, replacing the old native aristocracy. In general, the Ptolemies undertook changes that went far beyond any other measures that earlier foreign rulers had imposed. They used the religion and traditions to increase their own power and wealth. Although they established a prosperous kingdom, enhanced with fine buildings, the native population enjoyed few benefits, and there were frequent uprisings. These expressions of nationalism reached a peak in the reign of Ptolemy IV Philopator (221–205 BC) when others gained control over one district and ruled as a line of native "pharaohs." This was only curtailed nineteen years later when Ptolemy V Epiphanes (205–181 BC) succeeded in subduing them, but the underlying grievances continued and there were riots again later in the dynasty. Family conflicts affected the later years of the dynasty when Ptolemy VIII Euergetes II fought his brother Ptolemy VI Philometor and briefly seized the throne. The struggle was continued by his sister and niece (who both became his wives) until they finally issued an Amnesty Decree in 118 BC. Ptolemaic Egypt was noted for its extensive series of coinage in gold, silver and bronze. It was especially noted for its issues of large coins in all three metals, most notably gold pentadrachm and octadrachm, and silver tetradrachm, decadrachm and pentakaidecadrachm. This was especially noteworthy as it would not be until the introduction of the Guldengroschen in 1486 that coins of substantial size (particularly in silver) would be minted in significant quantities. Ptolemaic Egypt, along with the other Hellenistic states outside of the Greek mainland after Alexander the Great, had its armies based on the Macedonian phalanx and featured Macedonian and native troops fighting side by side. The Ptolemaic military was filled with diverse peoples from across their territories. At first most of the military was made up of a pool of Greek settlers who, in exchange for military service, were given land grants. These made up the majority of the army. With the many wars the Ptolemies were involved in, their pool of Macedonian troops dwindled and there was little Greek immigration from the mainland so they were kept in the royal bodyguard and as generals and officers. Native troops were looked down upon and distrusted due to their disloyalty and frequent tendency to aid local revolts. However, with the decline of royal power, they gained influence and became common in the military. The Ptolemies used the great wealth of Egypt to their advantage by hiring vast amounts of mercenaries from across the known world. Black Ethiopians are also known to have served in the military along with the Galatians, Mysians and others. With their vast amount of territory spread along the Eastern Mediterranean such as Cyprus, Crete, the islands of the Aegean and even Thrace, the Ptolemies required a large navy to defend these far-flung strongholds from enemies like the Seleucids and Macedonians. While ruling Egypt, the Ptolemaic Dynasty built many Greek settlements throughout their Empire, to either Hellenize new conquered peoples or reinforce the area. Egypt had only three main Greek cities—Alexandria, Naucratis, and Ptolemais. Of the three Greek cities, Naucratis, although its commercial importance was reduced with the founding of Alexandria, continued in a quiet way its life as a Greek city-state. During the interval between the death of Alexander and Ptolemy's assumption of the style of king, it even issued an autonomous coinage. And the number of Greek men of letters during the Ptolemaic and Roman period, who were citizens of Naucratis, proves that in the sphere of Hellenic culture Naucratis held to its traditions. Ptolemy II bestowed his care upon Naucratis. He built a large structure of limestone, about 330 feet (100 m) long and 60 feet (18 m) wide, to fill up the broken entrance to the great Temenos; he strengthened the great block of chambers in the Temenos, and re-established them. At the time when Sir Flinders Petrie wrote the words just quoted the great Temenos was identified with p91the Hellenion. But Mr. Edgar has recently pointed out that the building connected with it was an Egyptian temple, not a Greek building. Naucratis, therefore, in spite of its general Hellenic character, had an Egyptian element. That the city flourished in Ptolemaic times "we may see by the quantity of imported amphorae, of which the handles stamped at Rhodes and elsewhere are found so abundantly. "The Zeno papyri show that it was the chief port of call on the inland voyage from Memphis to Alexandria, as well as a stopping-place on the land-route from Pelusium to the capital. It was attached, in the administrative system, to the Saïte nome. A major Mediterranean port of Egypt, in ancient times and still today, Alexandria was founded in 331 BC by Alexander the Great. According to Plutarch, the Alexandrians believed that Alexander the Great's motivation to build the city was his wish to "found a large and populous Greek city that should bear his name." Located 20 miles (32 km) west of the Nile's westernmost mouth, the city was immune to the silt deposits that persistently choked harbors along the river. Alexandria became the capital of the Hellenized Egypt of King Ptolemy (1) I (reigned 323—283 BC). Under the wealthy Ptolemy dynasty, the city soon surpassed Athens as the cultural center of the Hellenic world. Laid out on a grid pattern, Alexandria occupied a stretch of land between the sea to the north and Lake Mareotis to the south; a man-made causeway, over three-quarters of a mile long, extended north to the sheltering island of Pharos, thus forming a double harbor, east and west. On the east was the main harbor, called the Great Harbor; it faced the city's chief buildings, including the royal palace and the famous Library and Museum. At the Great Harbor's mouth, on an outcropping of Pharos, stood the lighthouse, built c. 280 BC. Now vanished, the lighthouse was reckoned as one of the Seven Wonders of the World for its unsurpassed height (perhaps 460 feet); it was a square, fenestrated tower, topped with a metal fire basket and a statue of Zeus the Savior. The Library, at that time the largest in the world, contained several hundred thousand volumes and housed and employed scholars and poets. A similar scholarly complex was the Museum (Mouseion, "hall of the Muses"). During Alexandria's brief literary golden period, c. 280–240 BC, the Library subsidized three poets—Callimachus, Apollonius, and Theocritus—whose work now represents the best of Hellenistic literature. Among other thinkers associated with the Library or other Alexandrian patronage were the mathematician Euclid (ca. 300 BC), the inventor Archimedes (287 BC – c. 212 BC), and the polymath Eratosthenes (ca. 225 BC). Cosmopolitan and flourishing, Alexandria possessed a varied population of Greeks, Egyptians and other Oriental peoples, including a sizable minority of Jews, who had their own city quarter. Periodic conflicts occurred between Jews and ethnic Greeks. According to Strabo, Alexandria had been inhabited during Polybius' lifetime by local Egyptians, foreign mercenaries and the tribe of the Alexandrians, whose origin and customs Polybius identified as Greek. The city enjoyed a calm political history under the Ptolemies. It passed, with the rest of Egypt, into Roman hands in 30 BC, and became the second city of the Roman Empire. The second Greek city founded after the conquest in Egypt was Ptolemais, 400 miles (640 km) up the Nile, where there was a native village called Psoï, in the nome called after the ancient Egyptian city of Thinis. If Alexandria perpetuated the name and cult of the great Alexander, Ptolemais was to perpetuate the name and cult of the founder of the Ptolemaic time. Framed in by the barren hills of the Nile Valley and the Egyptian sky, here a Greek city arose, with its public buildings and temples and theatre, no doubt exhibiting the regular architectural forms associated with Greek culture, with a citizen-body Greek in blood, and the institutions of a Greek city. If there is some doubt whether Alexandria possessed a council and assembly, there is none in regard to Ptolemais. It was more possible for the kings to allow a measure of self-government to a people removed at that distance from the ordinary residence of the court. We have still, inscribed on stone, decrees passed in the assembly of the people of Ptolemais, couched in the regular forms of Greek political tradition: It seemed good to the boule and to the demos: Hermas son of Doreon, of the deme Megisteus, was the proposer: Whereas the prytaneis who were colleagues with Dionysius the son of Musaeus in the 8th year, etc. The Ptolemaic kingdom was diverse in the people who settled and made Egypt their home on this time. During this period, Macedonian troops under Ptolemy I Soter were given land grants and brought their families encouraging tens of thousands of Greeks to settle the country making themselves the new ruling class. Native Egyptians continued having a role, yet a small one in the Ptolemaic government mostly in lower posts and outnumbered the foreigners. During the reign of the Ptolemaic Pharaohs, many Jews were imported from neighboring Judea by the thousands for being renowned fighters and established an important presence there. Other foreign groups settled during this time and even Galatian mercenaries were invited. Of the aliens who had come to settle in Egypt, the ruling group, Greeks, were the most important element. They were partly spread as allotment-holders over the country, forming social groups, in the country towns and villages, side by side with the native population, partly gathered in the three Greek cities — the old Naucratis, founded before 600 BC (in the interval of Egyptian independence after the expulsion of the Assyrians and before the coming of the Persians), and the two new cities, Alexandria by the sea, and Ptolemais in Upper Egypt. Alexander and his Seleucid successors were great as the founders of Greek cities all over their dominions. Greek culture was so much bound up with the life of the city-state that any king who wanted to present himself to the world as a genuine champion of Hellenism had to do something in this direction, but the king of Egypt, whilst as ambitious as any to shine as a Hellene, would find Greek cities, with their republican tradition and aspirations to independence, inconvenient elements in a country that lent itself, as no other did, to bureaucratic centralization. The Ptolemies therefore limited the number of Greek city-states in Egypt to Alexandria, Ptolemais, and Naucratis. Outside of Egypt, they had Greek cities under their dominion—including the old Greek cities in the Cyrenaica, in Cyprus, on the coasts and islands of the Aegean— but they were smaller than the three big ones in Egypt. There were indeed country towns with names such as Ptolemais, Arsinoe, and Berenice, in which Greek communities existed with a certain social life; there were similar groups of Greeks in many of the old Egyptian towns, but they were not communities with the political forms of a city-state. Yet if they had no place of political assembly, they would have their gymnasium, the essential sign of Hellenism, serving something of the purpose of a university for the young men. Far up the Nile at Ombi was found in 136–135 BC a gymnasium of the local Greeks, which passes resolutions and corresponds with the king. And in 123 BC, when there is trouble in Upper Egypt between the towns of Crocodilopolis and Hermonthis, the negotiators sent from Crocodilopolis are the young men attached to the gymnasium, who, according to the Greek tradition, eat bread and salt with the negotiators from the other town. All Greek dialects of the Greek world gradually became assimilated in the Koine Greek dialect that was the common language of the Hellenistic world. Generally the Greeks of the Ptolemaic Egypt felt like a representative of a higher civilization yet were curious about the native culture of Egypt. Arabs under the Ptolemies Arab nomads of the eastern desert penetrated in small bodies into the cultivated land of the Nile, as they do today. The Greeks called all the land on the eastern side of the Nile "Arabia", and villages were to be found here and there with a population of Arabs who had exchanged the life of tent-dwellers for that of settled agriculturists. Apollonius tells of one such village, Poïs, in the Memphite nome, two of whose inhabitants send a letter on September 20, 152 BC. The letter is in Greek; it had to be written for the two Arabs by the young Macedonian Apollonius, the Arabs being unable apparently to write. Apollonius writes their names as Myrullas and Chalbas, the first probably, and the second certainly, Semitic. A century earlier Arabs farther west, in the Fayûm, organized under a leader of their own, and working mainly as herdsmen on the dorea of Apollonius the dioiketes; but these Arabs bear Greek and Egyptian names. In 1990, more than 2,000 papyri written by Zeno of Caunus from the time of Ptolemy II Philadelphus were discovered, which contained at least 19 references to Arabs in the area between the Nile and the Red Sea, and mentioned their jobs as police officers in charge of "ten person units", while some others were mentioned as shepherds. Arabs in Ptolemaic kingdom had provided camel convoys to the armies of some Ptolemaic leaders during their invasions, but they didn't have allegiance towards any of the kingdoms of Egypt or Syria, and also managed to raid and attack both sides of the conflict between Ptolemaic Kingdom and its enemies. Jews under the Ptolemies The Jews who lived in Egypt had originally immigrated from Israel. The Jews absorbed Greek, the dominant language of Egypt at the time, while heavily mixing it with Hebrew It was during this period that the Septuagint, the Greek translation of the Jewish scriptures, appeared. The Septuagint was written by Seventy Jewish Translators under royal compulsion during Ptolemy II's reign. This is confirmed by historian Flavius Josephus, who writes that Ptolemy, desirous to collect every book in the habitable earth, applied Demetrius Phalereus to the task of organizing an effort with the Jewish high priests to translate the Jewish books of the Law for his library. This testimony of Josephus places the origins of the Septuagint in the 3rd century BC, as that is the time wherein Demetrius and Ptolemy II lived. According to Jewish Legend, the seventy translators wrote their translations independently from memory, and the resultant works were identical at every letter. The early Ptolemies increased cultivatable land through irrigation and introduced crops such as cotton and better wine-producing grapes. They also increased the availability of luxury goods through foreign trade. They enriched themselves and absorbed Egyptian culture. Ptolemy and his descendants adopted Egyptian royal trappings and added Egypt's religion to their own, worshiping Egyptian gods and building temples to them, and even being mummified and buried in sarcophagi covered with hieroglyphs. List of Ptolemaic rulers - Buraselis, Stefanou and Thompson ed; The Ptolemies, the Sea and the Nile: Studies in Waterborne Power. - North Africa in the Hellenistic and Roman Periods, 323 BC to AD 305, R.C.C. Law, The Cambridge History of Africa, Vol. 2 ed. J. D. Fage, Roland Anthony Oliver, (Cambridge University Press, 1979), 154. - Diodorus Siculus, Bibliotheca historica, 18.21.9 - Lewis, Naphtali (1986). Greeks in Ptolemaic Egypt: Case Studies in the Social History of the Hellenistic World. Oxford: Clarendon Press. pp. 5. ISBN 0-19-814867-4. - Department of Ancient Near Eastern Art. "The Achaemenid Persian Empire (550–330 B.C.)". In Heilbrunn Timeline of Art History. New York: The Metropolitan Museum of Art, 2000–. http://www.metmuseum.org/toah/hd/acha/hd_acha.htm (October 2004) Source: The Achaemenid Persian Empire (550–330 B.C.) \| Thematic Essay \| Heilbrunn Timeline of Art History \| The Metropolitan Museum of Art - Hemingway, Colette, and Seán Hemingway. "The Rise of Macedonia and the Conquests of Alexander the Great". In Heilbrunn Timeline of Art History. New York: The Metropolitan Museum of Art, 2000–. http://www.metmuseum.org/toah/hd/alex/hd_alex.htm (October 2004) Source: The Rise of Macedonia and the Conquests of Alexander the Great \| Thematic Essay \| Heilbrunn Timeline of Art History \| The Metropolitan Museum of Art - Grabbe, L. L. (2008). A History of the Jews and Judaism in the Second Temple Period. Volume 2 – The Coming of the Greeks: The Early Hellenistic Period (335 – 175 BC). T&T Clark. ISBN 978-0-567-03396-3.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Ptolemy II Philadelphus [308-246 BC. Mahlon H. Smith. Retrieved 2010-06-13. - Burstein (2007), p. 7 - Maccabees. (2015, September 22). In Wikipedia, The Free Encyclopedia. Retrieved 21:01, September 24, 2015, from https://en.wikipedia.org/w/index.php?title=Maccabees&oldid=682248927 - Cleopatra: A Life - Peters (1970), p. 193 - Peters (1970), p. 194 - Peters (1970), p. 195f - Antiquities Experts. "Egyptian Art During the Ptolemaic Period of Egyptian History". http://antiquitiesexperts.com/egypt_ptol.html. External link in \|website=(help); Missing or empty - Phillips, Heather A., "The Great Library of Alexandria?". Library Philosophy and Practice, August 2010 - Arabs in Antiquity: Their History from the Assyrians to the Umayyads, Prof. Jan Retso, Page: 301 - A History of the Arabs in the Sudan: The inhabitants of the northern Sudan before the time of the Islamic invasions. The progress of the Arab tribes through Egypt. The Arab tribes of the Sudan at the present day, Sir Harold Alfred MacMichael, Cambridge University Press, 1922, Page: 7 - History of Egypt, Sir John Pentland Mahaffy, Pages: 20-21 - Solomon Grayzel "A History of the Jews" p. 56 - Solomon Grayzel "A History of the Jews" pp. 56-57 - Flavius Josephus "Antiquities of the Jews" Book 12 Ch. 2 - Bingen, Jean. Hellenistic Egypt. Edinburgh: Edinburgh University Press, 2007 (hardcover, ISBN 0-7486-1578-4; paperback, ISBN 0-7486-1579-2). Hellenistic Egypt: Monarchy, Society, Economy, Culture. Berkeley: University of California Press, 2007 (hardcover, ISBN 0-520-25141-5; paperback, ISBN 0-520-25142-3). - Bowman, Alan Keir. 1996. Egypt After the Pharaohs: 332 BC–AD 642; From Alexander to the Arab Conquest. 2nd ed. Berkeley: University of California Press - Burstein, Stanley Meyer (December 1, 2007). The Reign of Cleopatra. University of Oklahoma Press. ISBN 0806138718. Retrieved April 6, 2015.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Chauveau, Michel. 2000. Egypt in the Age of Cleopatra: History and Society under the Ptolemies. Translated by David Lorton. Ithaca: Cornell University Press - Ellis, Simon P. 1992. Graeco-Roman Egypt. Shire Egyptology 17, ser. ed. Barbara G. Adams. Aylesbury: Shire Publications, ltd. - Hölbl, Günther. 2001. A History of the Ptolemaic Empire. Translated by Tina Saavedra. London: Routledge Ltd. - Lloyd, Alan Brian. 2000. "The Ptolemaic Period (332–30 BC)". In The Oxford History of Ancient Egypt, edited by Ian Shaw. Oxford and New York: Oxford University Press. 395–421 - Susan Stephens, Seeing Double. Intercultural Poetics in Ptolemaic Alexandria (Berkeley, 2002). - A. Lampela, Rome and the Ptolemies of Egypt. The development of their political relations 273-80 B.C. (Helsinki, 1998). - Peters, F. E. (1970). The Harvest of Hellenism. New York: Simon & Schuster.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - J. G. Manning, The Last Pharaohs: Egypt Under the Ptolemies, 305-30 BC (Princeton, 2009).<\|endoftext\|>	3.890625
697	Light travelling to us from space is distorted by small temperature variations in the Earth’s turbulent atmosphere. We see this when we look up at the stars. Large optical disturbances in starlight appear as high-frequency brightness variations, otherwise known as “twinkling”. While enjoyable to the eye, twinkling makes science very difficult because it turns small, detailed images into large, blurry blobs. Telescopes serve two primary purposes, improving our sensitivity and resolution. Sensitivity refers to how faint an object can be and still be detected. Resolution refers to how finely we can see the detail in an object we’re observing. We can theoretically increase both the sensitivity and the resolution of our telescopes by increasing the size of their primary mirrors. However, due to blurring from Earth’s atmosphere, once a ground-based telescope’s primary mirror diameter is larger than a few inches, increasing its size no longer increases its resolution. One solution to atmospheric blurring is to launch the telescope into space. There have been many successful telescopes launched into space that have returned invaluable information on our Sun and the universe we live in. Unfortunately, space missions are very expensive and impose strict size, weight, and maintenance requirements on their payloads. Adaptive Optics is another solution to atmospheric blurring. An Adaptive Optics system corrects the optical disturbances that light encounters as it traverses Earth’s atmosphere, allowing a ground-based telescope to achieve the same resolution as a space telescope of the same size! An Adaptive Optics system works by sensing optical disturbances in the incoming light with a device called a wavefront sensor and correcting those disturbances with a deformable mirror. A deformable mirror is a mirror made of a flexible material with many push-pull actuators connected to its back surface. By commanding the actuators to exert forces on the mirror’s surface, we can bend the deformable mirror to almost any arbitrary shape. Optical disturbances in light manifest themselves as differences in the distance that each light ray travels before it reaches our camera. Once our wavefront sensor has measured these path length differences, a computer control system calculates the mirror shape needed to correct them, and commands the deformable mirror to assume a shape that “undoes” the effects of the atmospheric blurring. Due to the constantly evolving nature of atmospheric turbulence, an Adaptive Optics system must execute hundreds or thousands of these measure-correct cycles per second. The DKIST Adaptive Optics system contains a Deformable Mirror with 1600 actuators and a wavefront sensor that samples the incoming wavefront in 1521 locations. Our control system uses Field-Programmable Gate Arrays to compute and correct optical disturbances in the image 2000 times per second. By providing a correction with such a high level of detail, the Adaptive Optics system will enable scientists to use the full power of the DKIST’s four-meter diameter primary mirror, resulting in the highest-resolution images of the sun ever seen! While operational, the DKIST Adaptive Optics system monitors itself and automatically adjusts its own configuration to optimize its performance. Additionally, all its setup and calibration procedures are automated via Python scripts, making it much easier to operate than typical Adaptive Optics systems. This increased ease of operation will result in more observing time, maximizing the scientific value of the telescope. National Solar Observatory<\|endoftext\|>	4.15625
5,672	The Carpathian Basin, in which Hungary lies, has been populated for hundreds of thousands of years. Bone fragments found at Vértesszőlős, about 5km southeast of Tata, in the 1960s are believed to be half a million years old. These findings suggest that Palaeolithic and later Neanderthal humans were attracted to the area by the hot springs and the abundance of reindeer, bears and mammoths. During the Neolithic period (3500-2500 BC), climatic changes forced much of the indigenous wildlife to migrate northward. As a result the domestication of animals and the first forms of agriculture appeared, simultaneously with the rest of Europe. Remnants of the Körös culture in the Szeged area of the southeast suggest that these goddess-worshipping people herded sheep, fished and hunted. Indo-European tribes from the Balkans stormed the Carpathian Basin in horse-drawn carts in about 2000 BC, bringing with them copper tools and weapons. After the introduction of the more durable metal bronze, forts were built and a military elite began to develop. Over the next millennium, invaders from the west (Illyrians, Thracians) and east (Scythians) brought iron, but it was not in common use until the Celts arrived at the start of the 4th century BC. They introduced glass and crafted some of the fine gold jewellery that can still be seen in museums throughout Hungary. Some three decades before the start of the Christian era the Romans conquered the area west and south of the Danube River and established the province of Pannonia – later divided into Upper (Superior) and Lower (Inferior) Pannonia. Subsequent victories over the Celts extended Roman domination across the Tisza River as far as Dacia (today’s Romania). The Romans brought writing, viticulture and stone architecture, and established garrison towns and other settlements, the remains of which can still be seen in Óbuda (Aquincum in Roman times), Szombathely (Savaria), Pécs (Sophianae) and Sopron (Scarabantia). They also built baths near the region’s thermal waters and their soldiers introduced the new religion of Christianity. The great migrations The first of the so-called Great Migrations of nomadic peoples from Asia reached the eastern outposts of the Roman Empire late in the 2nd century AD, and in 270 the Romans abandoned Dacia altogether. Within less than two centuries they were also forced to flee Pannonia by the Huns, whose short-lived empire was established by Attila; he had previously conquered the Magyars near the lower Volga River and for centuries these two groups were thought – erroneously – to share a common ancestry. Attila remains a very common given name for males in Hungary, however. Germanic tribes such as the Goths, Gepids and Longobards occupied the region for the next century and a half until the Avars, a powerful Turkic people, gained control of the Carpathian Basin in the late 6th century. They in turn were subdued by Charlemagne in 796 and converted to Christianity. By that time, the Carpathian Basin was virtually unpopulated except for groups of Turkic and Germanic tribes on the plains and Slavs in the northern hills. The Magyars & the conquest of the Carpathian basin The origin of the Magyars is a complex issue, not in the least helped by the similarity in English of the words ‘Hun’ and ‘Hungary’, which are not related. One thing is certain: Magyars are part of the Finno-Ugric group of peoples who inhabited the forests somewhere between the middle Volga River and the Ural Mountains in western Siberia as early as 4000 BC. By about 2000 BC population growth had forced the Finnish-Estonian branch of the group to move westward, ultimately reaching the Baltic Sea. The Ugrians migrated from the southeastern slopes of the Urals into the valleys, and switched from hunting and fishing to primitive farming and raising livestock, especially horses. The Magyars’ equestrian skills proved useful half a millennium later when climatic changes brought drought, forcing them to move north to the steppes. On the plains, the Ugrians turned to nomadic herding. After 500 BC, by which time the use of iron had become commonplace, some of the tribes moved westward to the area of Bashkiria in central Asia. Here they lived among Persians and Bulgars and began referring to themselves as Magyars (from the Finno-Ugric words mon, ‘to speak’, and e, ‘man’). Several centuries later another group split away and moved south to the Don River under the control of the Khazars, a Turkic people. Here they lived among various groups under a tribal alliance called onogur, or ’10 peoples’. This is the derivation of the word ‘Hungary’ in English and ‘Ungarn’ in German. Their penultimate migration brought them to what modern Hungarians call the Etelköz, the region between the Dnieper and lower Danube Rivers just north of the Black Sea. Small nomadic groups of Magyars probably reached the Carpathian Basin as early as the mid-9th century AD, acting as mercenaries for various armies. It is believed that while the men were away on a campaign in about 889, the Pechenegs, a fierce people from the Asiatic steppe, allied themselves with the Bulgars and attacked the Etelköz settlements. When they were attacked again in about 895, seven tribes under the leadership of Árpád – the gyula (chief military commander) – upped stakes. They crossed the Verecke Pass (in today’s Ukraine) into the Carpathian Basin. The Magyars met almost no resistance and the tribes dispersed in three directions: the Bulgars were quickly dispatched eastward; the Germans had already taken care of the Slavs in the west; and Transylvania was wide open. Known for their ability to ride and shoot, and no longer content with being hired guns, the Magyars began plundering and pillaging. Their raids took them as far as Spain, northern Germany and southern Italy, but in the early 10th century they began to suffer a string of defeats. In 955 they were stopped in their tracks for good by the German king Otto I at the Battle of Augsburg. This and subsequent defeats – the Magyars’ raids on Byzantium ended in 970 – left the tribes in disarray, and they had to choose between their more powerful neighbours – Byzantium to the south and east or the Holy Roman Empire to the west – to form an alliance. In 973 Prince Géza, the great-grandson of Árpád, asked the Holy Roman emperor Otto II to send Catholic missionaries to Hungary. Géza was baptised along with his son Vajk, who took the Christian name Stephen (István), after the first martyr. When Géza died, Stephen ruled as prince. Three years later, he was crowned ‘Christian King’ Stephen I, with a crown sent from Rome by Otto’s erstwhile tutor, Pope Sylvester II. Hungary the kingdom – and the nation – was born. King Stephen i & the Árpád dynasty Stephen set about consolidating royal authority by siezing the land of the independent-minded clan chieftains and establishing a system of megye (counties) protected by fortified vár (castles). The crown began minting coins and, shrewdly, Stephen transferred much land to his most loyal (mostly Germanic) knights. The king sought the support of the church throughout and, to hasten the conversion of the population, ordered that one in every 10 villages build a church. He also established 10 episcopates, two of which – Kalocsa and Esztergom – were made archbishoprics. Monasteries were set up around the country and staffed by foreign – notably Irish – scholars. By the time Stephen died in 1038 – he was canonised less than half a century after his death – Hungary was a nascent Christian nation, increasingly westward-looking and multiethnic. Despite this apparent consolidation, the next two and a half centuries until 1301 – the reign of the House of Árpád – would test the kingdom to its limit. The period was marked by continuous struggles between rival pretenders to the throne, weakening the young nation’s defences against its more powerful neighbours. There was a brief hiatus under King Ladislas I (László; r 1077-95), who ruled with an iron fist and fended off attacks from Byzantium; and also under his successor Koloman the Booklover (Könyves Kálmán; r 1095-1116), who encouraged literature, art and the writing of chronicles until his death in 1116. Tensions flared again when the Byzantine emperor made a grab for Hungary’s provinces in Dalmatia and Croatia, which it had acquired by the early 12th century. Béla III (r 1172-96) successfully resisted the invasion and had a permanent residence built at Esztergom, which was then the alternative royal seat to Székesfehérvár. Béla’s son, Andrew II (András; r 1205-35), however, weakened the crown when, to help fund his crusades, he gave in to local barons’ demands for more land. This led to the Golden Bull, a kind of Magna Carta signed at Székesfehérvár in 1222, which limited some of the king’s powers in favour of the nobility. When Béla IV (r 1235-70) tried to regain the estates, the barons were able to oppose him on equal terms. Fearing Mongol expansion and realising he could not count on the support of his subjects, Béla looked to the west and brought in German and Slovak settlers. He also gave asylum to Turkic Cuman (Kun) tribes displaced by the Mongols in the east. In 1241 the Mongols arrived in Hungary and swept through the country, burning it virtually to the ground and killing an estimated one-third to one-half of its two million people. To rebuild the country as quickly as possible Béla, known as the ‘second founding father’, again encouraged immigration, inviting Germans to settle in Transdanubia, Saxons in Transylvania and Cumans on the Great Plain. He also built a string of defensive hilltop castles, including the ones at Buda and Visegrád. But in a bid to appease the lesser nobility, he handed them large tracts of land. This strengthened their position and demands for more independence even further; by the time of Béla’s death in 1270, anarchy had descended upon Hungary. The rule of his reprobate son and heir Ladislas the Cuman (so-called because his mother was a Cuman princess) was equally unsettled. The Árpád line died out in 1301 with the death of Andrew III, who left no heir. The struggle for the Hungarian throne following the death of Andrew III involved several European dynasties, but it was Charles Robert (Károly Róbert) of the French House of Anjou who, with the pope’s blessing, finally won out in 1308 and ruled for the next three and a half decades. Charles Robert was an able administrator who managed to break the power of the provincial barons (though much of the land remained in private hands), sought diplomatic links with his neighbours and introduced a stable gold currency called the florin (or forint). In 1335 Charles Robert met the Polish and Bohemian kings at the new royal palace in Visegrád to discuss territorial disputes and to forge an alliance that would smash Vienna’s control of trade. Under Charles Robert’s son, Louis I the Great (Nagy Lajos; r 1342-82), Hungary returned to a policy of conquest. A brilliant military strategist, Louis acquired territory in the Balkans as far as Dalmatia and Romania and as far north as Poland. He was crowned king of Poland in 1370, but his successes were short-lived; the menace of the Ottoman Turks had begun. As Louis had no sons, one of his daughters, Mary (r 1382-87), succeeded him. This was deemed unacceptable by the barons, who rose up against the ‘petticoat throne’. Within a short time Mary’s husband, Sigismund (Zsigmond; r 1387-1437) of Luxembourg, was crowned king. Sigismund’s 50-year reign brought peace at home, and there was a great flowering of Gothic art and architecture in Hungary. But while he managed to procure the coveted crown of Bohemia and was made Holy Roman emperor in 1433, he was unable to stop the Ottoman onslaught and was defeated by the Turks at Nicopolis (now Bulgaria) in 1396. There was an alliance between Poland and Hungary in 1440 that gave Poland the Hungarian crown. When Vladislav I (Úlászló) of the Polish Jagiellon dynasty was killed fighting the Turks at Varna in 1444, János Hunyadi was declared regent. A Transylvanian general born of a Wallachian (Romanian) father, János Hunyadi began his career at the court of Sigismund. His 1456 decisive victory over the Turks at Belgrade (Hungarian: Nándorfehérvár) checked the Ottoman advance into Hungary for 70 years and assured the coronation of his son Matthias (Mátyás), the greatest ruler of medieval Hungary. Wisely, Matthias (r 1458-90), nicknamed Corvinus (the Raven) from his coat of arms, maintained a mercenary force of 8000 to 10, 000 men by taxing the nobility, and this ‘Black Army’ conquered Moravia, Bohemia and even parts of lower Austria. Not only did Matthias Corvinus make Hungary one of central Europe’s leading powers, but under his rule the nation enjoyed its first golden age. His second wife, the Neapolitan princess Beatrice, brought artisans from Italy who completely rebuilt and extended the Gothic palace at Visegrád; the beauty and sheer size of the Renaissance residence was beyond compare in the Europe of the time. But while Matthias, a fair and just king, busied himself with centralising power for the crown, he ignored the growing Turkish threat. His successor Vladislav II (Úlászló; r 1490-1516) was unable to maintain even royal authority, as the members of the diet (assembly), which met to approve royal decrees, squandered royal funds and expropriated land. In May 1514, what had begun as a crusade organised by the power-hungry archbishop of Esztergom, Tamás Bakócz, turned into a peasant uprising against landlords under the leadership of one György Dózsa. The revolt was brutally repressed by noble leader John Szapolyai (Zápolyai János). Some 70, 000 peasants were tortured and executed; Dózsa himself was fried alive on a red-hot iron throne. The retrograde Tripartitum Law that followed the crackdown codified the rights and privileges of the barons and nobles, and reduced the peasants to perpetual serfdom. By the time Louis II (Lajos) took the throne in 1516 at the tender age of nine, he couldn’t count on either side. The battle of Mohács & Turkish occupation The defeat of Louis’ ragtag army by the Ottoman Turks at Mohács in 1526 is a watershed in Hungarian history. On the battlefield near this small town in Southern Transdanubia a relatively prosperous and independent medieval Hungary died, sending the nation into a tailspin of partition, foreign domination and despair that would be felt for centuries afterward. It would be unfair to lay all the blame on the weak and indecisive teenage King Louis or on his commander-in-chief, Pál Tomori, the archbishop of Kalocsa. Bickering among the nobility and the brutal response to the peasant uprising a dozen years before had severely diminished Hungary’s military might, and there was virtually nothing left in the royal coffers. By 1526 the Ottoman sultan Suleiman the Magnificent occupied much of the Balkans, including Belgrade, and was poised to march on Buda and then Vienna with a force of 100, 000 men. Unable – or, more likely, unwilling – to wait for reinforcements from Transylvania under the command of his rival John Szapolyai, Louis rushed south with a motley army of 26, 000 men to battle the Turks and was soundly thrashed in less than two hours. Along with bishops, nobles and an estimated 20, 000 soldiers, the king was killed – crushed by his horse while trying to retreat across a stream. John Szapolyai, who had sat out the battle in Tokaj, was crowned king six weeks later. Despite grovelling before the Turks, Szapolyai was never able to exploit the power he had sought so single-mindedly. In many ways greed, self-interest and ambition had led Hungary to defeat itself. After Buda Castle fell to the Turks in 1541, Hungary was torn into three parts. The central section, including Buda, went to the Turks, while parts of Transdanubia and what is now Slovakia were governed by the Austrian House of Habsburg and assisted by the Hungarian nobility based at Bratislava. The principality of Transylvania, east of the Tisza River, prospered as a vassal state of the Ottoman Empire, initially under Szapolyai’s son John Sigismund (Zsigmond János; r 1559-71). Though heroic resistance continued against the Turks throughout Hungary, most notably at Kőszeg in 1532, Eger 20 years later and Szigetvár in 1566, this division would remain in place for more than a century and a half. The Turkish occupation was marked by constant fighting among the three divisions; Catholic ‘Royal Hungary’ was pitted against both the Turks and the Protestant Transylvanian princes. Gábor Bethlen, who ruled Transylvania from 1613 to 1629, tried to end the warfare by conquering ‘Royal Hungary’ with a mercenary army of Heyduck peasants and some Turkish assistance in 1620. But both the Habsburgs and the Hungarians themselves viewed the ‘infidel’ Ottomans as the greatest threat to Europe since the Mongols and blocked the advance. As Ottoman power began to wane in the 17th century, Hungarian resistance to the Habsburgs, who had used ‘Royal Hungary’ as a buffer zone between Vienna and the Turks, increased. A plot inspired by the palatine Ferenc Wesselényi was foiled in 1670 and a revolt (1682) by Imre Thököly and his army of kuruc (anti-Habsburg mercenaries) was quelled. But with the help of the Polish army, Austrian and Hungarian forces liberated Buda from the Turks in 1686. An imperial army under Eugene of Savoy wiped out the last Turkish army in Hungary at the Battle of Zenta (now Senta in Serbia) 11 years later. Peace was signed with the Turks at Karlowitz (now in Serbia) in 1699. The expulsion of the Turks did not result in a free and independent Hungary, and the policies of the Habsburgs’ Counter-Reformation and heavy taxation further alienated the nobility. In 1703 the Transylvanian prince Ferenc Rákóczi II assembled an army of kuruc forces against the Austrians at Tiszahát in northeastern Hungary. The war dragged on for eight years and in 1706 the rebels ‘dethroned’ the Habsburgs as the rulers of Hungary. Superior imperial forces and lack of funds, however, forced the kuruc to negotiate a separate peace with Vienna behind Rákóczi’s back. The 1703-11 war of independence had failed, but Rákóczi was the first leader to unite Hungarians against the Habsburgs. The armistice may have brought the fighting to an end, but Hungary was now little more than a province of the Habsburg Empire. Five years after Maria Theresa ascended the throne in 1740, the Hungarian nobility pledged their ‘lives and blood’ to her at the diet in Bratislava in exchange for tax exemptions on their land. Thus began the period of ‘enlightened absolutism’ that would continue under the rule of Maria Theresa’s son Joseph II (r 1780-90). Under both Maria Theresa and Joseph, Hungary took great steps forward economically and culturally. Depopulated areas in the east and south were settled by Romanians and Serbs, while German Swabians were sent to Transdanubia. Joseph’s attempts to modernise society by dissolving the all-powerful (and corrupt) religious orders, abolishing serfdom and replacing ‘neutral’ Latin with German as the official language of state administration were opposed by the Hungarian nobility, and he rescinded most (but not all) of these orders on his deathbed. Dissenting voices could still be heard and the ideals of the French Revolution of 1789 began to take root in certain intellectual circles in Hungary. In 1795 Ignác Martonovics, a former Franciscan priest, and six other prorepublican Jacobites were beheaded at Vérmező (Blood Meadow) in Buda for plotting against the crown. Liberalism and social reform found their greatest supporters among certain members of the aristocracy, however. Count György Festetics (1755-1819), for example, founded Europe’s first agricultural college at Keszthely. Count István Széchenyi (1791-1860), a true Renaissance man and called ‘the greatest Hungarian’ by his contemporaries, advocated the abolition of serfdom and returned much of his own land to the peasantry. The proponents of gradual reform were quickly superseded by a more radical faction who demanded more immediate action. The group included Miklós Wesselényi, Ferenc Deák and Ferenc Kölcsey, but the predominant figure was Lajos Kossuth (1802-94). It was this dynamic lawyer and journalist who would lead Hungary to its greatest-ever confrontation with the Habsburgs. The 1848-49 war of independence Early in the 19th century the Habsburg Empire began to weaken as Hungarian nationalism increased. Suspicious of Napoleon’s motives and polcies, the Hungarians ignored French appeals to revolt against Vienna and certain reforms were introduced: the replacement of Latin, the official language of administration, with Magyar; a law allowing serfs alternative means of discharging their feudal obligations of service; and increased Hungarian representation in the Council of State. The reforms carried out were too limited and far too late, however, and the Diet became more defiant in its dealings with the crown. At the same time, the wave of revolution sweeping Europe spurred on the more radical faction. In 1848 the liberal Count Lajos Batthyány was made prime minister of the new Hungarian ministry, which counted Deák, Kossuth and Széchenyi among its members. The Habsburgs also reluctantly agreed to abolish serfdom and proclaim equality under the law. But on 15 March a group calling itself the Youth of March, led by the poet Sándor Petőfi, took to the streets to press for even more radical reforms and revolution. Habsburg patience was wearing thin. In September 1848 the Habsburg forces, under the governor of Croatia, Josip Jelačić, launched an attack on Hungary, and Batthyány’s government was dissolved. The Hungarians hastily formed a national defence commission and moved the government seat to Debrecen, where Kossuth was elected governor-president. In April 1849 the parliament declared Hungary’s full independence and the Habsburgs were ‘dethroned’ for the second time. The new Habsburg emperor, Franz Joseph (r 1848-1916), was not at all like his feeble-minded predecessor Ferdinand V (r 1835-48). He quickly took action, seeking the assistance of Russian tsar Nicholas I, who obliged with 200, 000 troops. Support for the revolution was waning rapidly, particularly in areas of mixed population where the Magyars were seen as oppressors. Weak and vastly outnumbered, the rebel troops were defeated by August 1849. A series of brutal reprisals ensued. In October Batthyány and 13 of his generals – the so-called ‘Martyrs of Arad’ – were executed, and Kossuth went into exile in Turkey. (Petőfi died in battle in July of that year.) Habsburg troops then went around the country systematically blowing up castles and fortifications lest they be used by resurgent rebels. The dual monarchy Hungary was again merged into the Habsburg Empire as a conquered province and ‘neoabsolutism’ was the order of the day. Passive local resistance and disastrous military defeats for the Habsburgs in 1859 and 1865, however, pushed Franz Joseph to the negotiating table with liberal Hungarians under Deák’s leadership. The result was the Act of Compromise of 1867 (German: Ausgleich), which created the Dual Monarchy of Austria (the empire) and Hungary (the kingdom) – a federated state with two parliaments and two capitals: Vienna and Pest (Budapest when Buda, Pest and Óbuda were merged in 1873). Only defence, foreign relations and customs were shared. Hungary was even allowed to raise a small army. Read more: http://www.lonelyplanet.com/hungary/history#ixzz3dPfZpfPp<\|endoftext\|>	3.734375
256	MySQL Data types Introduction MySQL Data types - Introduction MySQL Data types :Definition : Data type is the characteristic of columns and variables that defines what types of data values they can store. The characteristic indicating whether a data item represents a number, date, character string, etc. Data types are used to indicate the type of the field we are creating into the table. MySQL data types supports in three important categories: Before creating a table, identify whether a column should be a text, number, or date type. Each column in a table is made of a data type. The size of the value should be the smallest value depending upon the largest input value. For example, if the number of students in a school are in hundreds set the column as an unsigned three-digit SMALLINT (allowing for up to 999 values). We should be concise in inserting a string of five characters long into a char(3) field, the final two characters will be truncated. It is better to set the maximum length for text and number columns as well as other attributes such as UNSIGNED Square brackets ('[' and ']') indicate optional parts of type definitions. Now we slightly move to the overview of MySQL data types.<\|endoftext\|>	3.734375
538	# Notes Class 9 Mathematics Chapter 13 Surface Areas and Volumes Please refer to Surface Areas and Volumes Class 9 Mathematics Notes and important questions below. The Class 9 Mathematics Chapter wise notes have been prepared based on the latest syllabus issued for the current academic year by CBSE. Students should revise these notes and go through important Class 9 Mathematics examination questions given below to obtain better marks in exams ## Surface Areas and Volumes Class 9 Mathematics Notes and Questions The below Class 9 Surface Areas and Volumes notes have been designed by expert Mathematics teachers. These will help you a lot to understand all the important topics given in your NCERT Class 9 Mathematics textbook. Refer to Chapter 13 Surface Areas and Volumes Notes below which have been designed as per the latest syllabus issued by CBSE and will be very useful for upcoming examinations to help clear your concepts and get better marks in examinations. 1. Surface Area of a Cuboid and a Cube 2. Surface Area of a Right Circular Cylinder 3. Surface Area of a Right Circular Cone 4. Surface Area of a Sphere 5. Volume of a Cuboid 6. Volume of a Cylinder 7. Volume of a Right Circular Cone 8. Volume of a Sphere · Polyhedrons Shapes: (i) Cube: Cube whose edge = a Diagonal of Cube = √3a Lateral Surface Area of Cube = 4a2 Total Surface Area of Cube = 6a2 Volume of Cube = a3 (ii) Cuboid: Cuboid whose length = l, breadth = b and height = h Diagonal of Cuboid =√ l2 + b2 + h2 Lateral Surface Area of Cuboid = 2(l + b) h Total Surface Area of Cuboid = 2 (lb + bh + hl) Volume of Cuboid = lbh · Non-polyhedrons: (i) Cylinder: Cylinder whose radius = r, height = h Curved Surface Area of Cylinder = 2πrh Total Surface Area of Cylinder = 2πrh(r + h) Volume of Cylinder = πr2h (ii) Cone: Cone having height = h, radius = r and slant height = l Slant height of Cone (l) = √r2 + h2 Curved Surface Area of Cone = πrl Total Surface Area of Cone = πr(r + l) Volume of Cone = 1/3πr2h (iii) Sphere:<\|endoftext\|>	4.40625
3,696	# 3. ## Presentation on theme: "3."— Presentation transcript: 3 Unit 1: Energy and Motion Table of Contents 3 Unit 1: Energy and Motion Chapter 3: Forces 3.1: Newton’s Second Law 3.2: Gravity 3.3: The Third Law of Motion Force, Mass, and Acceleration Newton’s Second Law 3.1 Force, Mass, and Acceleration Newton’s first law of motion states that the motion of an object changes only if an unbalanced force acts on the object. Newton’s second law of motion describes how the forces exerted on an object, its mass, and its acceleration are related. Force and Acceleration Newton’s Second Law 3.1 Force and Acceleration What’s different about throwing a ball horizontally as hard as you can and tossing it gently? When you throw hard, you exert a much greater force on the ball. Force and Acceleration Newton’s Second Law 3.1 Force and Acceleration The hard-thrown ball has a greater change in velocity, and the change occurs over a shorter period of time. Force and Acceleration Newton’s Second Law 3.1 Force and Acceleration Recall that acceleration is the change in velocity divided by the time it takes for the change to occur. So, a hard-thrown ball has a greater acceleration than a gently thrown ball. Newton’s Second Law 3.1 Mass and Acceleration If you throw a softball and a baseball as hard as you can, why don’t they have the same speed? The difference is due to their masses. Newton’s Second Law 3.1 Mass and Acceleration If it takes the same amount of time to throw both balls, the softball would have less acceleration. The acceleration of an object depends on its mass as well as the force exerted on it. The heavier the object the less acceleration Force, mass, and acceleration are related. Newton’s Second Law 3.1 Newton’s second law of motion – F= MA Force equals the mass x acceleration Force- Newtons (N) Mass- Kg Acceleration- M/S² Sample Problems: Calculating Net Force with the Second Law Newton’s Second Law 3.1 Calculating Net Force with the Second Law Newton’s second law also can be used to calculate the net force if mass and acceleration are known. To do this, the equation for Newton’s second law must be solved for the net force, F. Calculating Net Force with the Second Law Newton’s Second Law 3.1 Calculating Net Force with the Second Law To solve for the net force, multiply both sides of the equation by the mass: The mass, m, on the left side cancels, giving the equation: Friction 3.1 Suppose you give a skateboard a push with your hand. Newton’s Second Law 3.1 Friction Suppose you give a skateboard a push with your hand. According to Newton’s first law of motion, if the net force acting on a moving object is zero, it will continue to move in a straight line with constant speed. Does the skateboard keep moving with constant speed after it leaves your hand? Newton’s Second Law 3.1 Friction Friction- force that opposes the motion of two surfaces that are touching each other. Friction depends on the roughness of surfaces force pressing the surfaces together. Newton’s Second Law 3.1 Static Friction Suppose you have filled a cardboard box with books and want to move it. It’s too heavy to lift, so you start pushing on it, but it doesn’t budge. If the box doesn’t move, then it has zero acceleration. Newton’s Second Law 3.1 Static Friction Static friction frictional force that prevents two surfaces from moving. Newton’s Second Law 3.1 Sliding Friction If you stop pushing, the box quickly comes to a stop. This is because as the box slides across the floor, another forcesliding frictionopposes the motion of the box. Sliding friction opposes the motion of two surfaces sliding past each other. Newton’s Second Law 3.1 Rolling Friction As a wheel rolls over a surface, the wheel digs into the surface, causing both the wheel and the surface to be deformed. Newton’s Second Law 3.1 Rolling Friction Static friction acts over the deformed area where the wheel and surface are in contact, producing a frictional force called rolling fiction. Rolling friction- is the frictional force between a rolling object and the surface it rolls on. Newton’s Second Law 3.1 Air Resistance air resistance opposes the motion of objects that move through the air. Air resistance causes objects to fall with different accelerations and different speeds. Newton’s Second Law 3.1 Air Resistance Air resistance acts in the opposite direction to the motion of an object through air. If the object is falling downward, air resistance acts upward on the object. Newton’s Second Law 3.1 Air Resistance The amount of air resistance on an object depends on the speed, size, and shape of the object. Air resistance, not the object’s mass, is why feathers, leaves, and pieces of paper fall more slowly than pennies, acorns, and apples. Newton’s Second Law 3.1 Terminal Velocity As an object falls, the downward force of gravity causes the object to accelerate. However, as an object falls faster, the upward force of air resistance increases. This causes the net force on a sky diver to decrease as the sky diver falls. Newton’s Second Law 3.1 Terminal Velocity Finally, the upward air resistance force becomes large enough to balance the downward force of gravity. This means the net force on the object is zero. Then the acceleration of the object is also zero, and the object falls with a constant speed called the terminal velocity. Newton’s Second Law 3.1 Terminal Velocity The terminal velocity is the highest speed a falling object will reach. The terminal velocity depends on the size, shape, and mass of a falling object. Section Check 3.1 Question 1 Newton’s second law of motion states that _________ of an object is in the same direction as the net force on the object. A. acceleration B. momentum C. speed D. velocity Section Check 3.1 Answer The answer is A. Acceleration can be calculated by dividing the net force in newtons by the mass in kilograms. Question 2 3.1 The unit of force is __________. A. joule B. lux Section Check 3.1 Question 2 The unit of force is __________. A. joule B. lux C. newton D. watt Section Check 3.1 Answer The answer is C. One newton = 1 kg · m/s2 Question 3 Answer 3.1 What causes friction? Section Check 3.1 Question 3 What causes friction? Answer Friction results from the sticking together of two surfaces that are in contact. Gravity 3.2 What is gravity? Gravity is an attractive force between any two objects that depends on the masses of the objects and the distance between them. The Law of Universal Gravitation Gravity 3.2 The Law of Universal Gravitation Isaac Newton formulated the law of universal gravitation, which he published in 1687. The Law of Universal Gravitation Gravity 3.2 The Law of Universal Gravitation G = Universal gravitational constant M1 and M2 = mass of objects d = distance between objects F= force of gravity between objects The law of universal gravitation enables the force of gravity to be calculated between any two objects if their masses and the distance between them is known. Gravity 3.2 The Range of Gravity According to the law of universal gravitation, the gravitational force between two masses decreases rapidly as the distance between the masses increases. Gravity 3.2 The Range of Gravity No matter how far apart two objects are, the gravitational force between them never completely goes to zero. Gravity 3.2 Finding Other Planets In the 1840s the most distant planet known was Uranus. The motion of Uranus calculated from the law of universal gravitation disagreed slightly with its observed motion. Some astronomers suggested that there must be an undiscovered planet affecting the motion of Uranus. Gravity 3.2 Finding Other Planets Using the law of universal gravitation and Newton’s laws of motion, two astronomers independently calculated the orbit of this planet. As a result of these calculations, the planet Neptune was found in 1846. Earth’s Gravitational Acceleration Gravity 3.2 Earth’s Gravitational Acceleration Earth’s gravity acceleration- 9.8 m/s2. Represented by the symbol – g Gravity 3.2 Weight Weight- gravitational force exerted on an object Weight and Mass 3.2 Weight and mass are not the same. Gravity 3.2 Weight and Mass Weight and mass are not the same. Weight is a force and mass is a measure of the amount of matter an object contains. Gravity 3.2 Weight and Mass The weight of an object usually is the gravitational force between the object and Earth. The weight of an object can change, depending on the gravitational force on the object. Gravity 3.2 Weight and Mass The table shows how various weights on Earth would be different on the Moon and some of the planets. Gravity 3.2 Projectile Motion If you’ve tossed a ball to someone, you’ve probably noticed that thrown objects don’t always travel in straight lines. They curve downward. Horizontal and Vertical Motions Gravity 3.2 Horizontal and Vertical Motions When you throw a ball, the force exerted by your hand pushes the ball forward. This force gives the ball horizontal motion. No force accelerates it forward, so its horizontal velocity is constant, if you ignore air resistance. Horizontal and Vertical Motions Gravity 3.2 Horizontal and Vertical Motions However, when you let go of the ball, gravity can pull it downward, giving it vertical motion. The ball has constant horizontal velocity but increasing vertical velocity. Horizontal and Vertical Motions Gravity 3.2 Horizontal and Vertical Motions Gravity exerts an unbalanced force on the ball, changing the direction of its path from only forward to forward and downward. The result of these two motions is that the ball appears to travel in a curve. Click image to view movie Gravity 3.2 Horizontal and Vertical Distance If you were to throw a ball as hard as you could from shoulder height in a perfectly horizontal direction, would it take longer to reach the ground than if you dropped a ball from the same height? Click image to view movie Horizontal and Vertical Distance Gravity 3.2 Horizontal and Vertical Distance Surprisingly, it wouldn’t. Both balls travel the same vertical distance in the same amount of time. Gravity 3.2 Centripetal Force centripetal acceleration Acceleration toward the center of a curved or circular path is called. Gravity 3.2 Centripetal Force centripetal force The net force exerted toward the center of a curved path is called a. Force pushes objects to the outside Centripetal Force and Traction Gravity 3.2 Centripetal Force and Traction When a car rounds a curve on a highway, a centripetal force must be acting on the car to keep it moving in a curved path. This centripetal force is the frictional force, or the traction, between the tires and the road surface. Centripetal Force and Traction Gravity 3.2 Centripetal Force and Traction Anything that moves in a circle is doing so because a centripetal force is accelerating it toward the center. Gravity Can Be a Centripetal Force 3.2 Gravity Can Be a Centripetal Force Imagine whirling an object tied to a string above your head. The string exerts a centripetal force on the object that keeps it moving in a circular path. Gravity Can Be a Centripetal Force 3.2 Gravity Can Be a Centripetal Force Earth’s gravity exerts a centripetal force on the Moon that keeps it moving in a nearly circular orbit. Section Check 3.2 Question 1 Gravity is an attractive force between any two objects and depends on __________. Answer Gravity is an attractive force between any two objects and depends on the masses of the objects and the distance between them. Question 2 3.2 Which is NOT one of the four basic forces? A. gravity Section Check 3.2 Question 2 Which is NOT one of the four basic forces? A. gravity B. net C. strong nuclear D. weak nuclear Section Check 3.2 Answer The answer is B. The fourth basic force is the electromagnetic force, which causes electricity, magnetism, and chemical interactions between atoms and molecules. Section Check 3.2 Question 3 Which of the following equations represents the law of universal gravitation? A. F = G(m1m2/d2) B. G = F(m1m2/d2) C. F = G(m1 - m2/d2) D. F = G(d2/m1m2) Section Check 3.2 Answer The answer is A. In the equation, G is the universal gravitational constant and d is the distance between the two masses, m1 and m2. Newton’s Third Law 3.3 Newton’s third law of motion The Third Law of Motion 3.3 Newton’s Third Law Newton’s third law of motion For every force… there is an equal and opposite force The Third Law of Motion 3.3 Action and Reaction When you jump on a trampoline, for example, you exert a downward force on the trampoline. Simultaneously, the trampoline exerts an equal force upward, sending you high into the air. Action and Reaction Forces Don’t Cancel The Third Law of Motion 3.3 Action and Reaction Forces Don’t Cancel According to the third law of motion, objects are experiencing unbalanced forces Thus, even though the forces are equal, they are not balanced because they act on different objects. Action and Reaction Forces Don’t Cancel The Third Law of Motion 3.3 Action and Reaction Forces Don’t Cancel For example, a swimmer “acts” on the water, the “reaction” of the water pushes the swimmer forward. Thus, a net force, or unbalanced force, acts on the swimmer so a change in his or her motion occurs. The Third Law of Motion 3.3 Rocket Propulsion In a rocket engine, burning fuel produces hot gases. The rocket engine exerts a force on these gases and causes them to escape out the back of the rocket. Newton’s third law, the gases exert a force on the rocket and push it forward. The Third Law of Motion 3.3 Momentum momentum that is related to how much force is needed to change its motion. momentum of an object is the product of its mass and velocity. Momentum 3.3 Momentum = symbol p The unit for momentum is kg · m/s. The Third Law of Motion 3.3 Momentum Momentum = symbol p The unit for momentum is kg · m/s. Force and Changing Momentum The Third Law of Motion 3.3 Force and Changing Momentum Recall that acceleration is the difference between the initial and final velocity, divided by the time. Also, from Newton’s second law, the net force on an object equals its mass times its acceleration. Force and Changing Momentum The Third Law of Motion 3.3 Force and Changing Momentum Calculating force from momentum mvf is the final momentum mvi is the initial momentum. Law of Conservation of Momentum The Third Law of Motion 3.3 Law of Conservation of Momentum The momentum of an object doesn’t change unless its mass, velocity, or both change. Momentum, however, can be transferred from one object to another. The law of conservation of momentum- if a group of objects exerts forces only on each other, their total momentum doesn’t change. The Third Law of Motion 3.3 When Objects Collide A collision depend on the momentum of each object. When the first puck hits the second puck from behind, it gives the second puck momentum in the same direction. The Third Law of Motion 3.3 When Objects Collide If the pucks are speeding toward each other with the same speed, the total momentum is zero. Question 1 Answer 3.3 According to Newton’s third law of motion, Section Check 3.3 Question 1 According to Newton’s third law of motion, what happens when one object exerts a force on a second object? Answer According to Newton’s law, the second object exerts a force on the first that is equal in strength and opposite in direction. Section Check 3.3 Question 2 The momentum of an object is the product of its __________ and __________. A. mass, acceleration B. mass, velocity C. mass, weight D. net force, velocity Section Check 3.3 Answer The correct answer is B. An object’s momentum is the product of its mass and velocity, and is given the symbol p. Section Check 3.3 Question 3 When two objects collide, what happens to their momentum? Section Check 3.3 Answer According to the law of conservation of momentum, if the objects in a collision exert forces only on each other, their total momentum doesn’t change, even when momentum is transferred from one object to another. Help 3 To advance to the next item or next page click on any of the following keys: mouse, space bar, enter, down or forward arrow. Click on this icon to return to the table of contents Click on this icon to return to the previous slide Click on this icon to move to the next slide Click on this icon to open the resources file. Click on this icon to go to the end of the presentation. End of Chapter Summary File<\|endoftext\|>	4.5625
607	What is a partition? Let’s assume that a company is moving to new place, and it has just one big room. There are no walls and everyone works in that one big room. There are chances that the head of the office wants a separate room so he will build an internal wall to separate him from other workers. Now the room will be partitioned into separate cubicles where each employee will work. The hard drive of computer operates in the same way. The disk can be partitioned into separate smaller parts which are the property of different people. On hard drives, the word “separation” refers to a different space. Partitioning Schemes for Linux: Some of the standard partitions schemes for most Linux system installs is as follows: - There will be a 12-20 GB partition for the operating system, which is mounted as / known as root. - To augment the RAM, a smaller partition can be used, which is installed and referred to as a swap. - There will be a larger partition for personal use, which is mounted as /home. The exact requirement for the partition will be based on the needs of the user, but usually, people start with the swap. If the user in involved in multimedia editing, and he has a RAM that is small, then the user can go for a greater amount of swap. If there is large amount of memory on the drive then you can economize on it. Some of the Linux distributions will have some issues regarding standby or hibernating mode if they don’t have the required amount of swap. The basic rule is that you will select between 1.5 to 2 times the amount of RAM as the swap space on the drive and then the user will put this partition in a place that is easy to access. A maximum amount of 20 GB will be enough for the root partition even there is much software installed on the system. The ext3 or ext4 are used as the file system on most of the Linux distributions these days. The ext3 and ext4 have a built-in self-cleaning mechanism. There should be 25 to 30 percent of free space should be available on the partition to work in the best way. The personal files or data of the user is stored in /home. It is functionally equal to the users partition in Windows which will house the music, documents, your application settings, downloads and much more. If the system has a separate partition for /home, then it will be the best because he doesn’t have to back up the data or any other files or folders in this partition whenever he is upgrading or reinstalling the system. The best thing about it is that your UI related settings and the programs will be saved, and you don’t have to reset the setting after upgrading your system. These were some of the best-known schemes for Linux partitioning that you can use on your Linux operating system.<\|endoftext\|>	3.828125
197	Genetics and DNA 1. DNA origami. Most suitable for ages 10+ This hands-on activity allows you to create your own paper model of a DNA double helix. Download: DNA folding template and Sanger DNA instructions 2. Mitosis. Most suitable for ages 14 + A fun card game to review the processes of mitosis and understand the process of cell division. Download: Mitosis game cards 3. Extracting DNA. Most suitable for ages 8+ DNA is the blueprint for life. It spells out the instructions to all living things to tell them how to become what they are, and how their cells should work. See DNA in fruit with this simple experiment. Download: Extracting DNA instructions 4. What is DNA? Most suitable for ages 6 - 10 A simple colouring exercise for young participants, plus some simple facts about DNA. Download: DNA colouring worksheet \| DNA answers (coming soon)<\|endoftext\|>	4.15625
2,072	# PUFM 1.4 Subtraction Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education. When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”. — Thomas H. Parker & Scott J. Baldridge Elementary Mathematics for Teachers Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?” Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse. ## The Advantage of Number Bonds This connection between addition and subtraction is represented in many textbooks by the “four-fact family.” The idea is that if students know one of the facts in the family, then they know all of them. Many students never see the connection, however, and think of these equations as four separate little bits of abstract information, all of which have to be memorized. This can overload their minds and make them give up on math. A four-fact family looks like this: 4 + 2 = 6 2 + 4 = 6 6 – 4 = 2 6 – 2 = 4 Number bonds connect to the student’s understanding at a deeper level, showing all four of the fact family relationships in a single picture. You can use either the circles (like a pile of small items which can be pulled apart and then slid back together) or a bar model diagram — whichever you prefer. ## Study Teaching Materials In this section, our textbook only refers to Singapore Primary Math 3A pp. 18-23. It seems to me that the following would also be helpful, if you have these books: 1A pp. 38-51 and 65-67; 1B pp. 10-13, 32-39, 82-87, and 94-96; 2A pp. 22-51; 2B pp. 6-19; 3A pp. 18-23 (which introduces the Singapore math models or bar model diagrams), and 36-38. Page numbers are from my 3rd edition books. If you have a different edition or another textbook series, look for headings like these: • Subtraction • Comparing Numbers • Mental Calculation • Word Problems ## Many Ways to Understand Subtraction With two models (set and measurement) and three interpretations of subtraction, we have a total of six situations to represent in story problems. Remember that the set model refers to things that are discrete and countable, such as strawberries and stuffed animals. The measurement model refers to things that are continuous and easily split into smaller pieces (including fractional amounts), such as fabric or fruit punch. As with other topics, our students need to think their way through lots and lots and lots of word problems of all sorts, so they learn to recognize subtraction in all its disguises. Can you come up with a word problem for each type of situation? • Part-Whole Interpretation: (Set model) Part of group A is X. How many are not-X? (Measurement model) Part of measurement B is Y. How large is not-Y? • Take-Away Interpretation: (Set model) Some X of group A is taken away or used. How many are left? (Measurement model) Part Y of measurement B is taken away or used up. How much is left? • Comparison Interpretation: (Set model) How many more in set A than in set X? (Measurement model) How much larger is measurement B than measurement Y? Notice that the part-whole interpretation is the most general. Both the take-away model and the comparison model can be seen as special cases of part-whole: • One part is taken away, and one part is left. • One part is the smaller amount, and one part is the difference. The comparison interpretation is by far the most difficult for children. The other situations give them some sort of story to guide their imaginations, but in a comparison, nothing happens — nothing is taken away or changed, so there’s no action to guide a child’s thinking. If your children get stuck, try modifying the question. Instead of asking who has more, ask something like, “How many would we have to give Tommy so he would have the same number as Annika?” ## Bar Model Diagrams With subtraction, we see the first significant advantage of Singapore math models, which are commonly called “bar diagrams.” Bar models make it easy to see comparison relationships, which are more abstract than “take away” situations and therefore more difficult for children to wrap their minds around. Bar model diagrams will continue to be powerful problem-solving tools as our children work their way through the elementary curriculum. If you have students who thrive on hands-on work, you might consider using Cuisenaire rods as a manipulative to help students develop an intuitive feel for bar diagrams. Here’s a fun Cuisenaire rod game for practicing number bonds: Eight is Having a Party! ## Thinking Strategies for Subtraction Beginners solve simple subtraction problems by counting backward, but we want to help our children progress to more conceptual thinking strategies. All of these mental math strategies look more difficult when written out than they are in real life. When children practice them over and over, they get very fast. Sometimes it’s hard for my old brain to keep up. Remember what I said in the last post: Our goal at this level is NOT for our children to memorize a series of math facts, but to develop confidence in working with numbers. If we stress fact memorization too early, we will short-circuit the child’s learning process. Once children “know” an answer, they don’t bother to think about it — but it is in the “thinking about it” stage that they build a logical foundation for understanding all numbers. Let’s examine several different ways to think about the calculation: $54 - 18 = \, ?$ Counting Down The goal of this strategy is not simply to count backwards, but to take away the number in easy chunks. First take away the easy 10 (leaving 8 more to take away). Then how many do we take away to make 40? And how many are left to take away after that? • $54 - 18 = (54 - 10) - 8$ $\rightarrow 44 - 8 = (44 - 4) - 4$ $\rightarrow 40 - 4 = 36$ Counting Up This strategy uses the definition of subtraction as a “mirror image” of addition. How many would we add to 18, to make 54? First add 2 to make 20, then add 34 more, so we’re adding a total of 36. • $54 - 18 = \, ?$ $\rightarrow 54 = 18 \, + \, ?$ $\rightarrow 18 + 2 + 34 = 18 + 36$ Make It Easier (Compensation) “Simplify the amount subtracted.” When we add the exact same amount to each number, the difference between the numbers stays the same, but we can make the actual calculation a whole lot easier. Since 18 + 2 = 20, and that will be easy to subtract, we can add 2 to BOTH of the numbers without changing the difference between them. • $54 - 18 = (54 + 2) - (18 + 2)$ $\rightarrow 56 - 20 = 36$ If Only, If Only… This strategy is another version of compensation, and I like it better than the one above. We find an easy number to take away, then adjust to make our answer correct: “If only we were taking away 20, that would be so much easier!” But we took away too much — what should we add to balance it? • $54 - 18 = 54 - 20 + 2$ $\rightarrow 34 + 2 = 36$ ## Place Value Considerations Remember, place value is “invisible” to us as adults, so we have to pay special attention to these issues. For mental math, when we work the bigger parts (tens or hundreds) first and the smaller parts (ones) last, our answer is much easier to keep in short-term memory because it appears in the order we normally say it. ## Homework Set 4 My favorite homework problem in this set is #6, making up word problems. In fact, I recommend going through any homework page that is merely calculations and making up stories to represent each problem. Remember to use the different models (set/measurement) and interpretations (part-whole/take-away/comparison) in your stories. • Better yet, take turns with your child to make up stories, and then write down your favorites in your math journal! My least favorite homework question was #7c. Why would anyone ever think of different strategies for this problem? 8 + 10 = 18 is one of those nearly-automatic calculations based on place value, and there is no reason in the universe to think about alternate strategies for it. Doing so can only confuse our students. I think the question is a typo or editing error. They probably meant to write something like: “List as many different strategies as possible that can be used to calculate $103 - 78$ .” • How many strategies can you think of? • Which of them is your favorite? This post is part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post. Or start at the beginning.]<\|endoftext\|>	4.4375
1,385	rockets are excellent devices for investigating Newtons Three Laws of Motion.The rocket will remain on the launch pad until an unbalanced force is exerted propelling the rocket upward (First Law).The amount of force depends upon how much air you pumped inside the rocket (Second Law).You can increase the force further by adding a small amount of water to the rocket.This increases the mass the rocket expels by air pressure.Finally the action force of the air (and water) as it rushes out of the nozzle creates an equal and opposite reaction force propelling the rocket upward (Third Law). In this assignment, you will make a .5 liter bottle rocket, that is propelled by water and high pressure CO2. You can use any design that you would like, but a couple of success design types will be highlighted in this assignment. and make a rocket that will achieve maximum altitude when released from a Materials and Tools: ·.5 Liter / 20 oz / 16 ozsoda pop bottle (Dr. Pepper is best) As much as possible, pressurize your .5 L pop bottle. The best way to do this is with a pop bottle pump, or you can simply blow into the bottle and try to make it as hard as possible.Place the plastic bottle top on the bottle, losing as little pressure as possible. Cut out the paper measuring tape printed on the sheet with the transition cone and measuring tape. You need this to decide where to place your fins. Are you going to use 2, 3, or 4 fins?There are 360 degrees around your bottle. You fins need to be placed evenly apart. You will use the measuring tape and a marker to mark where you will be putting your fins.(1800, 1200, or 900). The fins should also go 6.0 cm from the opening (flange) of the bottle. The placement and usage of fins is critical to the success of your rocket. It is critical that the fins be placed straight on the bottle, and they be evenly spaced on the rocket. To make straight vertical lines for the fin placement, hold the bottle up against a door frame. Use a permanent marker to draw the placement markings. Make these lines 5.0 cm long. Lightly sand inside of each of the bottle feet where the air tube (paper towel roll) will fit. The sanding is done in preparation for the attachment of the air tube. You will also sand the areas where the fins will be attacked. THE AIR TUBE The stratoblaster box is a holding fixture and alignment tube. The essential thing here is that the AIR TUBE MUST GO ON STRAIGHT. You can make your own holding box from a shoe box, or come up with another method, but the tube has to Using a glue gun, run a small bead of glue on one end of the air tube. Stick the air tube to the sanded bottom of the plastic bottle, in the holding fixture. Make sure that the holding fixture is properly aligned. Let the glue set for a few minutes. After the glue has set, remove the assembly from the holding fixture. Run a bead of the glue around the base of the tube, where it meets the plastic bottle. This will secure the air tube to the body of the bottle rocket. Check the rocket again in the holding fixture, to assure that the air tube and body stay aligned and straight. If possible, repressurize your bottle rocket. The tighter the surface body, the easier it will be to attach your fins How are you going to design your fins? Will you use balsa wood, cardboard, or some other material? The fins must be firm, hard, but not TOO heavy. Styrofoam may work, as some types of plastics ( remember you have to cut these? What shape do you want to make your fins? Sand both sides and the bottom of the fins. These fin edges need to be smooth where the glue will be applied. Use a glue gun to run a thin bead of glue along the base of one of the fins. Press the fin into place along the mark on the rocket body. Make sure the fin is perfectly aligned along the mark. Let the glue set for a minute. Then run a bead of glue along the other side of the fin. Repeat this process for as many fins as you are using. Cut out the transition cone from the printed sheet with the cone template on it. Crease the transition cone along the dotted line. What are you going to use for your cone material?It needs to be something lightweight, flexible, and somewhat sturdy? Cardboard will not make a cone very well, maybe something like construction paper, maybe something like thin foam rubber, or manila folder paper. Use the transition cone template to make a transition cone out of the material that you want to use. Roll the transition cone across the edge of a table. Roll the transition cone semi-tightly in your hand. Then carefully unroll it and align the dotted line to the opposite edge. The small opening will be 2.5 cm in diameter. Tape the cone along the seam using transparent tape. Slide the transition cone over the air tube that is connected to the body of the rocket. IT should fit snugly. If it does not fit snugly, spot glue the cone to the air tube using a glue gun. Smooth the glue as much as possible. Allow the glue to cool before finishing the rocket. The NOSE CONE A ping pong ball works extremely well as the nose cone for your rocket. You are welcome to try and experiment with other tops for your rocket if you wish.The ping pong ball is an outstanding performer in this rocket style. Using your glue gun, glue one end of a piece of string or fishing line to the ping pong ball. Your string needs to be no longer than 10.0 Take the other end of the string and thread it through a hole in the back of a old thread spool. Knot it or glue it to the back end of the spool so that the ping pong ball nose cone will stay in the spool. Place the spool into the air tube with the ping pong ball sitting on top. Decorate Your Rocket using paint, stickers, and<\|endoftext\|>	3.96875
1,358	#### Chapter 13 Linear Equations in Two Variables R.D. Sharma Solutions for Class 9th MCQ's Exercise 13.3 Mark the correct alternative in each of the following: 1. If (4, 19) is a solution of the equation y = ax + 3, then a= (a) 3 (b) 4 (c) 5 (d) 6 Solution We are given (4,19)as the solution of equation. y = ax+3 Substituting x= 4 and y= 19, we get 19 = 4a+3 ⇒ 4a=19-3 ⇒ 4a=16 ⇒ a=4 Therefore, the correct answer is (b). 2. If (a, 4) lies on the graph of 3x + y = 10, then the value of a is (a) 3 (b) 1 (c) 3 (d) 4 Solution We are given (a, 4) lies on the graph of linear equation 3x+ y= 10. So, the given co-ordinates are the solution of the equation 3x+ y= 10. Therefore, we can calculate the value of a by substituting the value of given co-ordinates in equation 3x+ y= 10. Substituting x= a and y= 4 in equation 3x+ y= 10, we get 3×a + 4 = 10 ⇒ 3a = 10-4 ⇒ 3a=6 ⇒ a=6/3 ⇒ a=2 No, option is correct. 3. The graph of the linear equation 2x − y = 4 cuts x- axis at (a) (2, 0) (b) (−2, 0) (c) (0, −4) (d) (0, 4) Solution Given, 2x- y = 4 we get, y = 2x-4 We will substitute y =0 in y = 2x-4 to get the co-ordinates for the graph of 2x- y = 4 at x axis 0 = 2x-4 ⇒ 2x=0+4 ⇒ 2x=4 x=2 Co-ordinates for the graph of 2x- y = 4 are (2,0). Therefore, the correct answer is (a). 4. How many linear equations are satisfied by x = 2 and y = −3? (a) Only one (b) Two (c) Three (d) Infinitely many Solution There are infinite numbers of linear equations that are satisfied by x= 2 and y = -3 as (i) Every solution of the linear equation represent a point on the line. (ii) Every point on the line is the solution of the linear equation. Therefore, the correct answer is (d). 5. The equation x − 2 = 0 on number line is represented by (a) a line (b) a point (c) infinitely many lines (d) two lines Solution The equation x - 2 = 0 is represented by a point on the number line. Therefore, the correct answer is (b). 6. x = 2, y = −1 is a solution of the linear equation (a) x + 2y = 0 (b) x + 2y = 4 (c) 2x + y = 0 (d) 2x + y = 5 Solution We are given x=2; y = -1 as the solution of linear equation, which we have to find. The equation is x+2y = 0 which can he proved by Substituting x=2 and y= -1 in the equation x+2y= 0, we get 2 + 2×(-1) = 0 2-2 = 0 0 = 0 RHS = LHS Therefore, the correct answer is (a). 7. If (2k − 1, k) is a solution of the equation 10x − 9y = 12, then k = (a) 1 (b) 2 (c) 3 (d) 4 Solution We are given (2k-1, k)as the solution of equation 10x - 9y = 12 Substituting x=2k-1 and y = k , we get 10 ×(2k-1)- 9×k =12 ⇒ 20k - 10-9k =12 ⇒ 11k = 12+10 ⇒ 11k = 22 ⇒ k = 22/11 ⇒ k = 2 Therefore, the correct answer is (b). 8. The distance between the graph of the equations x = −3 and x = 2 is (a) 1 (b) 2 (c) 3 (d) 5 Solution Distance between the graph of equations x= -3 and x = 2. D = Distance of co-ordinate on negative side of x axis + Distance of co-ordinate on positive side of x axis Distance of co-ordinate on negative side of x axis = x = 3 units Distance of co-ordinate on positive side of x axis = x = 2 units D = 3+2 D = 5 units Therefore, the correct answer is d 9. The distance between the graphs of the equations y = −1 and y = 3 is (a) 2 (b) 4 (c) 3 (d) 1 Solution Here, you can see, the distance is 4 units. The distance can be calculated by subtracting y coordinates. 3-(-1) =4. So, (b) is the correct answer. 10. If the graph of the equation 4x + 3y = 12 cuts the coordinate axes at A and B, then hypotenuse of right triangle AOB is of length (a) 4 units (b) 3 units (c) 5 units (d) none of these Solution X<\|endoftext\|>	4.6875
707	# Difference between revisions of "2020 CIME II Problems/Problem 7" ## Problem 7 Let $ABC$ be a triangle with $AB=340$, $BC=146$, and $CA=390$. If $M$ is a point on the interior of segment $BC$ such that the length $AM$ is an integer, then the average of all distinct possible values of $AM$ can be expressed in the form $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. $[asy] size(3.5cm); defaultpen(fontsize(10pt)); pair A,B,C,M; A=dir(95); B=dir(-117); C=dir(-63); M=(2B+3C)/5; draw(A--B--C--A); draw(A--M,dashed); dot("A",A,N); dot("B",B,SW); dot("C",C,SE); dot("M",M,NW); label("340",A--B,W); label("390",A--C,E); label("146",B--C,S); [/asy]$ ## Solution Given that the length $AM$ is an integer and that it lies on the interior of segment $BC$, the shortest possible length of $AM$ is the length of the altitude dropped straight down from vertex $A$. This can be calculated as $\frac{2[\triangle ABC]}{BC}$, which is equal to $$\frac{2[\triangle ABC]}{146}$$ or $$\frac{[\triangle ABC]}{73}.$$ The area of triangle $ABC$ can be found using Heron's formula. It is just $$\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{438 \cdot 98 \cdot 292 \cdot 48}=24528.$$ The shortest possible length of $AM$ is $$\frac{24528}{73}=336.$$ $AM$ can be anything greater than or equal to $336$, but the condition that point $M$ lies in the interior of segment $BC$ limits the values that we can reach. Starting at $B$ and heading east (we cannot get $340$ because $M$ is strictly between $B$ and $C$), we reach the integers $$AM=339, 338, 337, 336,$$ and then as we move further east the length of $AM$ will start to increase. We then reach $$AM=337, 338, 339, 340,..., 386, 387, 388, 389.$$ We cannot get $390$ because then $M=C$ which is not allowed. The distinct possible values of $AM$ are $$336, 337, 338,..., 388, 389.$$ The average is $$\frac{336+389}{2}=\frac{725}{2}.$$ The answer is $725+2=\boxed{727}.$ 2020 CIME II (Problems • Answer Key • Resources) Preceded byProblem 6 Followed byProblem 8 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All CIME Problems and Solutions<\|endoftext\|>	4.71875
2,014	# 0.3 Gravity and mechanical energy (Page 4/9) Page 4 / 9 A ball is dropped from the balcony of a tall building. The balcony is $15\phantom{\rule{2pt}{0ex}}m$ above the ground. Assuming gravitational acceleration is $9,8\phantom{\rule{2pt}{0ex}}m·s{}^{-2}$ , find: 1. the time required for the ball to hit the ground, and 2. the velocity with which it hits the ground. 1. It always helps to understand the problem if we draw a picture like the one below: 2. We have these quantities: $\begin{array}{ccc}\hfill \Delta x& =& 15\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\hfill \\ \hfill {v}_{i}& =& 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \\ \hfill g& =& 9,8\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}\hfill \end{array}$ 3. Since the ball is falling, we choose down as positive. This means that the values for ${v}_{i}$ , $\Delta x$ and $a$ will be positive. 4. We can use [link] to find the time: $\Delta x={v}_{i}t+\frac{1}{2}g{t}^{2}$ 5. $\begin{array}{ccc}\hfill \Delta x& =& {v}_{i}t+\frac{1}{2}g{t}^{2}\hfill \\ \hfill 15& =& \left(0\right)t+\frac{1}{2}\left(9,8\right){\left(t\right)}^{2}\hfill \\ \hfill 15& =& 4,9\phantom{\rule{3.33333pt}{0ex}}{t}^{2}\hfill \\ \hfill {t}^{2}& =& 3,0612...\hfill \\ \hfill t& =& 1,7496...\hfill \\ \hfill t& =& 1,75\phantom{\rule{3.33333pt}{0ex}}s\hfill \end{array}$ 6. Using [link] to find ${v}_{f}$ : $\begin{array}{ccc}\hfill {v}_{f}& =& {v}_{i}+gt\hfill \\ \hfill {v}_{f}& =& 0+\left(9,8\right)\left(1,7496...\right)\hfill \\ \hfill {v}_{f}& =& 17,1464...\hfill \end{array}$ Remember to add the direction: ${v}_{f}=17,15\phantom{\rule{2pt}{0ex}}m·s{}^{-1}$ downwards. By now you should have seen that free fall motion is just a special case of motion with constant acceleration, and we use the same equations as before. The only difference is that the value for the acceleration, $a$ , is always equal to the value of gravitational acceleration, $g$ . In the equations of motion we can replace $a$ with $g$ . ## Gravitational acceleration 1. A brick falls from the top of a $5\phantom{\rule{2pt}{0ex}}m$ high building. Calculate the velocity with which the brick reaches the ground. How long does it take the brick to reach the ground? 2. A stone is dropped from a window. It takes the stone $1,5\phantom{\rule{2pt}{0ex}}s$ to reach the ground. How high above the ground is the window? 3. An apple falls from a tree from a height of $1,8\phantom{\rule{2pt}{0ex}}m$ . What is the velocity of the apple when it reaches the ground? ## Potential energy The potential energy of an object is generally defined as the energy an object has because of its position relative to other objects that it interacts with. There are different kinds of potential energy such as gravitional potential energy, chemical potential energy, electrical potential energy, to name a few. In this section we will be looking at gravitational potential energy. Potential energy Potential energy is the energy an object has due to its position or state. Gravitational potential energy is the energy of an object due to its position above the surface of the Earth. The symbol $PE$ is used to refer to gravitational potential energy. You will often find that the words potential energy are used where gravitational potential energy is meant. We can define potential energy (or gravitational potential energy, if you like) as: $PE=mgh$ where PE = potential energy measured in joules (J) m = mass of the object (measured in kg) g = gravitational acceleration ( $9,8\phantom{\rule{2pt}{0ex}}m·s{}^{-2}$ ) h = perpendicular height from the reference point (measured in m) A suitcase, with a mass of $1\phantom{\rule{2pt}{0ex}}\mathrm{kg}$ , is placed at the top of a $2\phantom{\rule{2pt}{0ex}}m$ high cupboard. By lifting the suitcase against the force of gravity, we give the suitcase potential energy. This potential energy can be calculated using [link] . what is the stm is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? what is a peer What is meant by 'nano scale'? What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq what is Nano technology ? write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? why we need to study biomolecules, molecular biology in nanotechnology? ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. why? what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology what does nano mean? nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now it is a goid question and i want to know the answer as well Maciej Abigail for teaching engĺish at school how nano technology help us Anassong How can I make nanorobot? Lily Do somebody tell me a best nano engineering book for beginners? there is no specific books for beginners but there is book called principle of nanotechnology NANO how can I make nanorobot? Lily what is fullerene does it is used to make bukky balls are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers!<\|endoftext\|>	4.5
889	Well, we’re off to see the Wizard again, my friends. This time it’s to explore the possibilities of primordial black holes colliding with stars and all the implications therein. If this theory is correct, then we should be able to observe the effects of dark matter first hand – proof that it really does exist – and deeper understand the very core of the Universe. Are primordial black holes blueprints for dark matter? Postdoctoral researchers Shravan Hanasoge of Princeton’s Department of Geosciences and Michael Kesden of NYU’s Center for Cosmology and Particle Physics have utilized computer modeling to visualize a primordial black hole passing through a star. “Stars are transparent to the passage of primordial black holes (PBHs) and serve as seismic detectors for such objects.” says Kesden. “The gravitational field of a PBH squeezes a star and causes it to ring acoustically.” If primordial black holes do exist, then chances are great that these type of collisions happen within our own galaxy – and frequently. With ever more telescopes and satellites observing the stellar neighborhoods, it only stands to reason that sooner or later we’re going to catch one of these events. But, the most important thing is simply understanding what we’re looking for. The computer model developed by Hanasoge and Kesden can be used with these current solar-observation techniques to offer a more precise method for detecting primordial black holes than existing tools. “If astronomers were just looking at the Sun, the chances of observing a primordial black hole are not likely, but people are now looking at thousands of stars,” Hanasoge said.”There’s a larger question of what constitutes dark matter, and if a primordial black hole were found it would fit all the parameters — they have mass and force so they directly influence other objects in the Universe, and they don’t interact with light. Identifying one would have profound implications for our understanding of the early Universe and dark matter.” Sure. We haven’t seen DM, but what we can see are galaxies that are hypothesized to have extended dark-matter halos and to study the effects the gravity has on their materials – like gaseous regions and stellar members. If these new models are correct, primordial black holes should be heavier than existing dark matter and when they collide with a star, should cause a rippling effect. “If you imagine poking a water balloon and watching the water ripple inside, that’s similar to how a star’s surface appears,” Kesden said. “By looking at how a star’s surface moves, you can figure out what’s going on inside. If a black hole goes through, you can see the surface vibrate.” Using the Sun as a model, Kesden and Hanasoge calculated the effects a PBH might have and then gave the data to NASA’s Tim Sandstrom. Using the Pleiades supercomputer at the agency’s Ames Research Center in California, the team was then able to create a video simulation of the collisional effect. Below is the clip which shows the vibrations of the Sun’s surface as a primordial black hole — represented by a white trail — passes through its interior. “It’s been known that as a primordial black hole went by a star, it would have an effect, but this is the first time we have calculations that are numerically precise,” comments Marc Kamionkowski, a professor of physics and astronomy at Johns Hopkins University. “This is a clever idea that takes advantage of observations and measurements already made by solar physics. It’s like someone calling you to say there might be a million dollars under your front doormat. If it turns out to not be true, it cost you nothing to look. In this case, there might be dark matter in the data sets astronomers already have, so why not look?” I’ll race you to the door… Original Story Source: Princeton University News. For Further Reading: Transient Solar Oscillations Driven by Primordial Black Holes.<\|endoftext\|>	3.8125
866	Have you ever wondered why some people seem to have eyes that don't look straight ahead? Maybe you've noticed that sometimes their eyes go in different directions when they're looking at an object or at you. Let's find out why some eyes don't see straight and how kids with this problem get help to fix it. Why Are Some Eyes Not Straight? Strabismus (say: struh-BIZ-mus) is the term used for eyes that are not straight and do not focus on the same object. Eyes can turn in (toward the nose), out (toward the ear), up, or down if the muscles that move the eyes don't work right or if the eyes are not able to focus properly. Strabismus is also sometimes called crossed eyes (when the eyes turn in) or walleye (when they turn out). You may have heard someone describe an eye that is not straight as a lazy eye, but that is not what lazy eye means. It means that a person's vision is weak or lazy. The medical term for lazy eye is amblyopia (say: am-blee-OH-pee-uh). Strabismus and amblyopia are closely related and often (but not always) occur together. Sometimes strabismus can cause amblyopia, and sometimes it's the other way around with amblyopia causing strabismus. When a kid has both conditions, it may be difficult to say which came first. The Eyes Have It! So what happens when a kid has an eye that isn't straight? To understand, first you need to know a little bit about the eye. The eye is like a camera, and the back of the eye, called the retina, is the film. Objects that your eyes see are projected onto the retina, and these pictures are sent to the brain by way of nerve signals. The brain detects these signals, puts them together to form an image, and that's how you see. It's amazing that it all happens so quickly — in a split second! When a kid has strabismus, the eyes don't focus together on the same object and each eye sends a different picture to the brain. As a result, the brain might see two images (double vision) or the object looks blurry. Kids' brains are really smart, and they don't like getting two different pictures instead of one. To fix the problem, the brain may ignore the picture coming from the one eye so it gets only one clear image. By shutting off the bad eye in favor of the good one, the bad eye gets weak or lazy (causing amblyopia), and without treatment the bad eye might eventually even become blind. What Will the Doctor Do? Usually doctors diagnose strabismus when a baby or young kid has a regular checkup. If a doctor or parent suspects a kid has strabismus, he or she will be sent to a special eye doctor called an ophthalmologist (say: af-thal-MAH-luh-jist). The doctor will examine the kid's eyes and ask him or her to read a chart on the wall or look into a microscope-type machine. None of these tests hurt. If the doctor thinks a kid has strabismus, there are different ways to treat it. Some kids just need to wear glasses. Others may need surgery to straighten their eyes. Kids who have amblyopia may need to wear a patch over the good eye to strengthen the weaker one. For any kid who has strabismus or amblyopia, starting treatment as soon as possible is the best way to improve vision. What's It Like? Sometimes kids who have strabismus get teased. Like teasing for any reason, this causes hurt feelings. If you know someone with strabismus, be a friend and do not tease him or her. And if you are the one who has strabismus, talk with a parent, teacher, or counselor if you are being teased. The good news is that this condition almost always can be treated, so someday you can say so long to strabismus!<\|endoftext\|>	3.78125
569	# Lesson 13 Building a Volume Formula for a Pyramid • Let’s create a formula for the volume of any pyramid or cone. ### Problem 1 Find the volume of a pyramid whose base is a square with side lengths of 6 units and height of 8 units. ### Problem 2 A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height. 1. Find the volume of the cylinder. 2. Find the volume of the cone. 3. What fraction of the cylinder’s volume is the cone’s volume? ### Problem 3 Each solid in the image has height 4 units. The area of each solid’s base is 8 square units. A cross section has been created in each by dilating the base using the apex as a center with scale factor $$k=0.25$$. 1. Calculate the area of each of the 2 cross sections. 2. Suppose a new cross section was created in each solid, both at the same height, using some scale factor $$k$$. How would the areas of these 2 cross sections compare? Explain your reasoning. ### Problem 4 Select the most specific and accurate name for the solid in the image. A: triangular pyramid B: regular prism C: square prism D: right triangular prism (From Unit 5, Lesson 12.) ### Problem 5 A solid can be constructed with 4 triangles and 1 rectangle. What is the name for this solid? A: rectangular pyramid B: triangular pyramid C: right triangular prism D: rectangular prism (From Unit 5, Lesson 12.) ### Problem 6 Find the volume of the solid produced by rotating this two-dimensional shape using the axis shown. (From Unit 5, Lesson 11.) ### Problem 7 This zigzag crystal vase has a height of 20 centimeters. The cross sections parallel to the base are always rectangles that are 12 centimeters wide by 6 centimeters long. 1. If we assume the crystal itself has no thickness, what would be the volume of the vase? 2. The crystal is actually 1 centimeter thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning. (From Unit 5, Lesson 10.) ### Problem 8 A trapezoid has an area of 10 square units. What scale factor would be required to dilate the trapezoid to have an area of 90 square units? A: 9 B: 6 C: 3 D: $$\frac13$$ (From Unit 5, Lesson 5.)<\|endoftext\|>	4.4375
369	# Find the Value of 6/(Sqrt5 - Sqrt3) It Being Given that Sqrt3 = 1.732 And Sqrt5 = 2.236 - Mathematics Advertisement Remove all ads Advertisement Remove all ads Advertisement Remove all ads Find the value of 6/(sqrt5 - sqrt3) it being given that sqrt3 = 1.732 and sqrt5 = 2.236 Advertisement Remove all ads #### Solution We know that rationalization factor for sqrt5 - sqrt3 is sqrt5 + sqrt3. We will multiply denominator and numerator of the given expression 6/(sqrt5 - sqrt3) by sqrt5 + sqrt3 to get 6/(sqrt5 - sqrt3) xx (sqrt5 + sqrt3)/(sqrt5 + sqrt3) = (6sqrt5 + 6sqrt3)/((sqrt5)^2 - (sqrt3)^3) = (6sqrt5 + 6sqrt3)/(5 - 3) = (6sqrt5 + 6sqrt3)/2 = 3sqrt5 + 3sqrt3 Putting the values of sqrt5 and sqrt3 we get 3sqrt5 + 3sqrt3 = 3(2.236) + 3(1.732) = 6.708 + 5.196 = 11.904 Hence value of the given expression is 11.904 Concept: Operations on Real Numbers Is there an error in this question or solution? #### APPEARS IN RD Sharma Mathematics for Class 9 Chapter 3 Rationalisation Exercise 3.2 \| Q 7 \| Page 15 #### Video TutorialsVIEW ALL [1] Share Notifications View all notifications Forgot password?<\|endoftext\|>	4.4375
2,596	Upcoming SlideShare × # Functions And Relations 2,214 views Published on Published in: Education 0 Likes Statistics Notes • Full Name Comment goes here. Are you sure you want to Yes No • Be the first to comment • Be the first to like this Views Total views 2,214 On SlideShare 0 From Embeds 0 Number of Embeds 1 Actions Shares 0 59 0 Likes 0 Embeds 0 No embeds No notes for slide ### Functions And Relations 1. 1. Functions and Relations <ul><li>Objectives </li></ul><ul><li>To understand and use the notation of sets, including the symbols ∈, ⊆, ∩, ∪, ∅ and . </li></ul><ul><li>To use the notation for sets of numbers. </li></ul><ul><li>To understand the concept of relation. </li></ul><ul><li>To understand the terms domain and range. </li></ul><ul><li>To understand the concept of function. </li></ul><ul><li>To understand the term one-to-one. </li></ul><ul><li>To understand the terms implied domain, restriction of a function, hybrid function, and odd and even functions. </li></ul><ul><li>To understand the modulus function. </li></ul><ul><li>To understand and use sums and products of functions. </li></ul><ul><li>To define composite functions. </li></ul><ul><li>To understand and find inverse functions. </li></ul><ul><li>To apply a knowledge of functions to solving problems. </li></ul> 2. 2. Set Notation <ul><li>A set is a collection of objects e.g A = {3,4}. </li></ul><ul><li>The objects in the set are known as the elements or members of the set. </li></ul><ul><li>For example, you are ‘elements’ of our class ‘set’. </li></ul><ul><li>3 ∈ A means ‘3 is a member of set A ’ or ‘3 belongs to A ’. </li></ul><ul><li>6 ∉ A means ‘6 is not an element of A’ . </li></ul> 3. 3. Set Notation <ul><li>If x ∈ B implies x ∈ A, then B is a subset of A , we write B ⊆ A . This expression can also be read as ‘ B is contained in A ’ or ‘ A contains B ’. </li></ul><ul><li>The set ∅ is called the empty set or null set. </li></ul><ul><li>A ∩ B is called the i n tersection of A and B . Thus x ∈ A ∩ B if and only if x ∈ A and x ∈ B . </li></ul><ul><li>A ∩ B = ∅ if the sets A and B have no elements in common. </li></ul><ul><li>A ∪ B , is the u nion of A and B. If elements are in both A and B they are only included in the union once. </li></ul><ul><li>The set difference of two sets A and B is denoted A B ( A but not B ) </li></ul><ul><li>Example 1 : A = {1, 2, 3, 7}; B = {3, 4, 5, 6, 7} </li></ul><ul><li>Find: a) A ∩ B b) A ∪ B c) A B d) B A </li></ul><ul><li>Solution: a) A ∩ B = {3, 7} </li></ul><ul><li>b) A ∪ B = {1, 2, 3, 4, 5, 6, 7} </li></ul><ul><li>c) A B = {1, 2} </li></ul><ul><li>d) B A = {4, 5, 6} </li></ul> 4. 4. Sets of numbers <ul><li>N: Natural numbers {1, 2, 3, 4, . . .} </li></ul><ul><li>Z: Integers {. . . ,−2,−1, 0, 1, 2, . . .} </li></ul><ul><li>Q: Rational numbers – can be written as a fraction. Each rational number may be written as a terminating or recurring decimal. </li></ul><ul><li>The real numbers that are not rational numbers are called irrational (e.g. π and √2). </li></ul><ul><li>R: Real numbers. (How can a number not be real?) </li></ul><ul><li>It is clear that N ⊆ Z ⊆ Q ⊆ R and this may be represented by the diagram: </li></ul> 5. 5. Sets of numbers <ul><li>The following are also subsets of the real numbers for which there are special notations: </li></ul><ul><li>R + = { x : x > 0} </li></ul><ul><li>R − = { x : x < 0} </li></ul><ul><li>R {0} is the set of real numbers excluding 0. </li></ul><ul><li>Z + = { x : x ∈ Z, x > 0} </li></ul><ul><li>Note: </li></ul><ul><li>{ x : 0 < x < 1} is the set of all real numbers between 0 and 1. </li></ul><ul><li>{ x : x > 0 , x rational} is the set of all positive rational numbers . </li></ul><ul><li>{2 n : n = 0, 1, 2, . . .} is the set of all even numbers. </li></ul> 6. 6. Representing sets of numbers on a number line <ul><li>Among the most important subsets of R are the intervals. </li></ul><ul><li>(-2, 4) means all ‘real’ numbers between (but not including) -2 and 4. </li></ul><ul><li>[3, 7] means all ‘real’ numbers between 3 and 7 inclusive. </li></ul><ul><li>[4, ∞) means all ‘real’ numbers greater than or equal to 4. </li></ul><ul><li>(-∞, 3) means all ‘real’ numbers less than 3. </li></ul> 7. 7. Representing sets of numbers on a number line <ul><li>Example 2: Illustrate each of the following intervals of the real numbers on a number line: </li></ul><ul><li>a [−2, 3] b (−3, 4] c (−∞, 5] d (−2, 4) e (−3,∞) </li></ul> 8. 8. Describing relations and functions <ul><li>An ordered pair , denoted ( x, y ), is a pair of elements x and y in which x is considered to be the first element and y the second (it doesn’t mean they have to be in numerical order). </li></ul><ul><li>A relation is a set of ordered pairs. The following are examples of relations: </li></ul><ul><li>S = {(1, 1), (1, 2), (3, 4), (5, 6)} </li></ul><ul><li>T = {(−3, 5), (4, 12), (5, 12), (7,−6)} </li></ul><ul><li>The domain of a relation S is the set of all first elements of the ordered pairs in S. </li></ul><ul><li>The range of a relation S is the set of all second elements of the ordered pairs in S. </li></ul><ul><li>In the above examples: </li></ul><ul><li>domain of S = {1, 3, 5}; range of S = {1, 2, 4, 6} </li></ul><ul><li>domain of T = {−3, 4, 5, 7}; range of T = {5, 12, −6} </li></ul><ul><li>A relation may be defined by a rule which pairs the elements in its domain and range. </li></ul><ul><li>Let’s watch an example. </li></ul> 9. 9. Describing relations and functions <ul><li>Example 3: Sketch the graph of each of the following relations and state the domain and range of each. </li></ul><ul><li>a {( x, y ): y = x 2 } </li></ul><ul><li>b {( x, y ): y ≤ x + 1} </li></ul><ul><li>c {(−2 , −1) , (−1 , −1) , (−1 , 1) , (0 , 1) , (1 , −1)} </li></ul><ul><li>d {( x, y ): x 2 + y 2 = 1} </li></ul><ul><li>e {( x, y ): 2 x + 3 y = 6 , x ≥ 0} </li></ul><ul><li>f {( x, y ): y = 2 x − 1 , x ∈ [−1 , 2]} </li></ul> 10. 10. Describing relations and functions <ul><li>A function is a relation such that no two ordered pairs of the relation have the same ﬁrst element. </li></ul><ul><li>For instance, in Example 3, a, e and f are functions but b, c and d are not. </li></ul><ul><li>Functions are usually denoted by lower case letters such as f, g, h. </li></ul><ul><li>The definition of a function tells us that for each x in the domain of f there is a unique element, y , in the range. </li></ul><ul><li>The element y is denoted by f ( x ) (read ‘ f of x ’). </li></ul> 11. 11. Describing relations and functions <ul><li>Example 4: If f ( x ) = 2 x 2 + x, find f (3) , f (−2) and f ( x − 1) . </li></ul><ul><li>Solution </li></ul><ul><li>f (3) = 2(3) 2 + 3 = 21 </li></ul><ul><li>f (−2) = 2(−2) 2 − 2 = 6 </li></ul><ul><li>f ( x − 1) = 2( x − 1) 2 + x − 1 </li></ul><ul><li>= 2( x 2 − 2 x + 1) + ( x − 1) </li></ul><ul><li>= 2 x 2 − 3 x + 1 </li></ul> 12. 12. Describing relations and functions <ul><li>Example 5: For each of the following, sketch the graph and state the range: </li></ul><ul><li>a f : [−2 , 4] -> R, f ( x ) = 2 x − 4 </li></ul><ul><li>b g : (−1 , 2] -> R, g ( x ) = x 2 </li></ul> 13. 13. Exercise 1 14. 14. Exercise 1<\|endoftext\|>	4.4375
1,189	Post -by Gautam Shah Fresco (Italian =fresh), is a method of loading colours on a wet or green plaster. The term is mainly applied to art of creating paintings, but may include craft of painting the architectural entities like walls and ceilings. It was a slow and labourious process, but for many centuries, it was the only viable process for rendering a painting on a masonry surface and for applying architectural colour. Fresco technique of coating was not suitable for wood, or other soft surfaces. Metals and alloys were new materials at that time, and being mostly non ferrous did not require any additional finish. In TRUE OR REAL FRESCO, the colourant pigments are applied on the top layer of a multi layered plaster system. For structural and levelling purposes one or more layers of plaster were applied. The final layer was thinner, and this was scratched to imprint the outline or sketch of the scheme. The sketch was drawn directly or imprinted by using a full size replica drawn on cloth (and later paper), called a cartoon. The cartoons were often reused in different works of art. The outlines of the various figures and forms were than filled-in with indicative dark water-based colours. A topping or finishing plaster was laid over the drawing in small sections, and each section of wet plaster was loaded with final colour scheme. As the plaster dried, the lime in the plaster absorbed the carbon dioxide from the air, forming a surface of calcium carbonate. This film of calcium carbonate impregnated with the colours became part of the plaster mass. The colours of a fresco are usually thin, translucent, and light, often with a chalky or pastel look. It was not possible to achieve saturated hues due to dominant presence of white of Lime. It was very necessary to finish the section of the painting before the plaster set. Most of the paintings, as a result lacked the finer detailing and perforce consisted of bare essentials. As the work was carried out in zones, segment by segment, it became the ‘style of painting’. In early works of Fresco painting such compartmentalization is very apparent. In later art works paintings were overdrawn to create graduated or tonal variations. Defective portions were removed by scratching the plaster and redone, and patchy effects were inevitable. The artist had to be well aware as to the amount of colour the plaster will hold or absorb. Too much pigment caused the surface to become chalky or powdery. In later day Frescos, the fixing of colour was controlled with use of variety of fixers such as sizing compounds, starch, gums, and plant excreted resins. These fixers were mostly hygroscopic, and so used to ‘run’ in wet weather or developed fungus. Plant resins had a few other problems that such fixers on drying provided comparatively a flat or dull-matt finish. Plant resins were acidic in nature and not always suitable for alkaline masonry surfaces. Fresco painting was known to the ancient Egyptians, Cretans, and Greeks. The Romans also practised fresco painting, examples of which are found in Herculaneum and Pompeii. In early Christian times (2nd C AD) Frescos were used to decorate the walls of catacombs, or underground burial chambers. The art of fresco underwent a great revival in Italy during the 13th and 14th C., begun by the Florentine painters Cimabue and Giotto, who painted numerous fine works in churches in Assisi, Florence, and Pisa. In the 15th C. the art flourished in Florence, notably in the work of Masaccio, Benozzo Gozzoli, and Ghirlandaio. Fresco painting reached its peak in the 16th C., with the supreme achievements of Raphael in the Vatican Palace and with The Last Judgement and Genesis frescoes by Michelangelo in the Vatican’s Sistine Chapel. Fresco painting was widely practiced in Europe in the 18th C., with nobility of style replaced by elegance and illusionist effects. One outstanding fresco painter in this period was Giovanni Battista Tiepolo in Italy. In FRESCO SECCO or lime-painting, the dry plaster was rubbed with a pumice stone to remove the crust, then washed with a thin mixture of water and lime. The colours were applied to this surface. Secco colours dry out lighter than their tone at the time of application, producing a pale, mat, chalky or ‘a distempered wall’ like effect. The fresco secco is inferior to true fresco. The colours are not clear, and the painting is less durable. The pigments are fused with the surface, but not completely absorbed in it, and may flake in time. Secco painting was the prevailing medium during medieval and early Renaissance period, and was revived in 18th C. Europe. Sgraffito (Italian Graffiare, “to scratch”) is a form of fresco painting for exterior walls. A rough plaster undercoat is covered by thin layers of plaster, each stained with a different lime-fast colour. These coats were than covered by a fine-grain mortar finishing surface. The plaster was then engraved with sharp tools to varying depths to reveal the underlying layers of various colours. The surface of modern sgraffito fresco is often enriched with textures and with mosaics of stone, glass, plastic, and metal tesserae. Sgraffito has been a traditional folk art in Europe since the Middle Ages and was practiced as a fine art in 13th-century Germany. It has been recently revived in northern Europe.<\|endoftext\|>	3.84375
842	Sickle cell disease is caused by the mutation of just one errant gene that causes blood cells stuffed with hemoglobin to become distorted into sickle shapes. The misshapen cells get stuck in blood vessels, causing strokes, organ damage and episodes of agonizing pain as muscles are starved of oxygen. While scientists since the time of Linus Pauling have known what causes sickle-cell disease (SCD), the only potential cure is a dangerous and expensive bone marrow transplant. But that may be about to change. Late last year, the Bill & Melinda Gates Foundation awarded Boston Children’s Hospital a $1.5 million grant to initiate basic work to expand gene therapy treatment of sickle cell disease in developing countries. r. Worldwide, about 300,000 infants are born with the sickle cell disease each year, a figure projected to grow to more than 400,000 by 2050. The disorder is most common in sub-Saharan Africa, where an estimated 70 percent of children with it die before adulthood. It is also the most common genetic blood disorder in the U.S. The Promise of Gene Editing Gene editing is actually a group of technologies that give scientists the ability to change an organism’s DNA. These technologies allow genetic material to be added, removed, or altered at particular locations in the genome. The patient’s cells in the affected tissues would be either edited within the body (in vivo) or outside (ex vivo) and returned to the patient. The Bill and Melinda Gates Foundation has supported research in gene editing for years. Gates has spoken widely on the case for using CRISPR and other gene-editing techniques on a global scale to meet growing demand for food and to improve disease prevention, particularly for malaria. Researchers have published successes with CRISPR to treat animals with an inherited liver disease and with muscular dystrophy. Using a process called somatic gene editing, scientists are exploring ways to treat diseases caused by a single mutated gene such as cystic fibrosis, Huntington’s, as well as SCD. Challenges Still Exist With the Gates money, David A. Williams, chief scientific officer and senior vice president of Boston Children’s Hospital and President of Dana-Farber/Boston Children’s Cancer and Blood Disorders Center, hopes to adapt methods used in the current Dana-Farber/Boston Children’s clinical gene therapy trial for SCD to solve current bioengineering and manufacturing constraints. Currently, most gene therapy trials for sickle cell disease are ex vivo. However, ex vivo gene therapy is a complicated, multi-step process that takes weeks and requires hospitalization. In contrast, in vivo therapy would be quicker and with less burden to the patient. The goal of this work is to develop methods that allow in vivo gene therapy applications for SCD in areas of the world where health care is less developed than in the United States and Europe. “Ultimately, an in vivo approach, in which a gene or inhibitory RNA is delivered directly to the body, is likely to be optimal for broadening global access to gene therapy for sickle cell disease,” said Williams, in a press release announcing the funding. Under the grant, Williams and colleagues will conduct research in gene delivery methods: “We will look at new technologies of non-viral methods for introducing the therapeutic gene into stem cells that could help standardize gene therapies and make them more available and affordable.” Beyond Sickle Cell Disease Gene therapies are currently confined to a few research hospitals in the U.S. and other developed countries. The long-term goal of the research is to make this potentially curative therapy available to patients in developing countries, according to the news releases. By entering the SCD space, the Gates organization joins with luminary funder The Doris Duke Charitable Trust in providing substantial philanthropic support for biomedical research of SCD. The Doris Duke Sickle Cell Disease/Advancing Cures award has provided millions in grants grants to advance curative approaches for sickle cell disease, including gene modification.<\|endoftext\|>	3.90625
280	who sank the boat The content in this unit links to the Australian Curriculum: Foundation (English). Text structure and organisation \|Understand that texts can take many forms, can be very short (for example an exit sign) or quite long (for example an information book or a film) and that stories and informative texts have different purposes (ACELA1430) (ENe-7B)\| \|Expressing and developing ideas\| Literature and context \|Recognise that texts are created by authors who tell stories and share experiences that may be similar or different to students’ own experiences (ACELT1575) (ENe-11D)\| \|Responding to literature\| Recognise some different types of literary texts and identify some characteristic features of literary texts, for example beginnings and endings of traditional texts and rhyme in poetry (ACELT1785) (ENe-10C) \|Creating literature\|\|Retell familiar literary texts through performance, use of illustrations and images (ACELT1580) (ENe-10C)\| \|Create short texts to explore, record and report ideas and events using familiar words and beginning writing knowledge (ACELY1651) (ENe-2A)\| Source for content descriptions above: Australian Curriculum, Assessment and Reporting Authority (ACARA).<\|endoftext\|>	3.921875
773	Conceptual Change Theory: How to Teach Science Certain science-related school subjects require a lot of cognitive effort. This is mainly due to the fact that the students need to understand the content deeply, something that current educational methodologies don’t help the students achieve. In this regard, the theory of conceptual change aims to help teachers teach science better. Students have intuitive theories of the world. Children don’t arrive at school as empty vessels. Before they receive information that describes the world around them, they have already theorized their own explanations. More often than not, their intuitive theories are incorrect and condition the newly learned material. It’s important that teachers take this into account when it comes to educating children. Stages of the conceptual change theory In this article, we’ll demonstrate how to develop a deep understanding in science. For this, we must understand the three stages of the conceptual change theory: - Recognition of an anomaly. - Building of a new model. - Use of the new model. Recognition of an anomaly This is the first step to help students develop a deep comprehension of a concept. The teacher‘s job is to break apart the students’ intuitive theories. The students must abandon their old ideas and discover that they were wrong. If the students don’t let go of their intuitive theories, their learning process will be affected and they’ll reject the new information. On many occasions, this stems from the superficial learning of science that won’t make the student feel obligated to leave their intuitive theory behind. Direct experimentation is one way to help students realize their intuitive theory is incorrect with their own senses. Thus, it allows them to recognize the anomaly. Teachers can also help students realize they were wrong by attacking erroneous ideas in a healthy and respectful dialogue. This method is very useful and can help the students see facts in a critical way. Building of a new model Once the students’ intuitive theory is broken apart, the next step is to give them a new explanation. One essential aspect of this step is that they have to build the new model themselves. If a teacher simply presents the new concept to the class, it would be difficult for the students to really understand it. They’ll most likely develop superficial learning based on memorization. Constructivist paradigms suggest that the student is the one who needs to build their knowledge. Thus, the teachers’ role is to guide students while they explore different possibilities. This is a complex process, but it yields amazing results. It’s more complicated in a classroom because it means that the teacher must apply this to many students at one time. A great way to do this is through debates. The students themselves create their own models to refute and argue their points of view. In this case, the teacher’s role is to prepare the material and necessary resources for the debate. This step is the most difficult one in the conceptual change theory since it’s where students start to really understand topics. Use of the new model It wouldn’t make much sense to break apart erroneous views if you don’t build a model to use in future problems. Therefore, the last step of the process is to teach students to use their theory. For this, students must do exercises where they can practice this new model. On the other hand, it’s essential to integrate this new model with previous knowledge. The conceptual change theory is a valid teaching method that yields incredible results. If we want to make sure that students really understand the material and know how to apply it in a critical and constructive way, we should use this type of educational tool in the classroom.<\|endoftext\|>	3.671875
606	# Sprint 17466 and solutions April 9, 2019 ### Monday April 8 sprint. ##### Problem 1: There are 10! ways to arrange the 10 items. Since 9 items aren't distinct, divide the 10! ways by the 9! ways of arranging the indistinct items. The answer is computed by evaluating (10!)/(9!). ##### Problem 2: Since the house numbers are 7 digits long, there are 7 slots. There are 5 stencils so there are 5 choices for each slot. Stencils are reusable so each of the digits can appear in each slot. The answer is computed by evaluating $$5^7$$. ##### Problem 3: Starting with a single cell, after d divisions there will be 2^d cells. For example, if the cell divides 3 times, there will be 2^3 cells. If you start with 5 cells instead of 1, there will be 5 times as many cells after 3 divisions or 5 * 2^3. In general, if you start with n cells there are n2^d cells after d divisions. The cells divide every 2 hours so there are 12 divisions per day. They divide for 10 days so there are a total of 120 divisions. The answer is computed by evaluating 82^120. ##### Problem 4: Solve a simpler problem to understand this problem. Assume the building is only 4 stories high. That means he has to climb 3 stories to reach the top. If the building was 5 stories, he would have to climb 4 stories. In general, the number of stories he climbs is 1 less than the the story number he climbs to. To climb to the 13th story, he climbs 12 stories. When he reaches his goal, the 21st story, he will have climbed 20 stories. The answer then is the reduced form of 12/20 or 3/5. ##### Problem 5: Work backwards. Since she spent 1/3 of her money at the BBQ, she still had 2/3 of the money left over. The first question then is $10 is 2/3 of what or $10 = 2/3 x Solve for x by multiplying both sides by the reciprocal of 2/3 or 3/2. That means she had $15 before she entered Nob Hill. She spent 1/6 of her money at Nob Hill so the$15 is 5/6 of the money she had when she entered Nob Hill or $15 = 5/6 x Solve for x by mulitplying both sides by the reciprocal of 5/6 or 6/5. That means she had$18 when she entered Nob Hill so she spent $18-$15 at Nob Hill or \$3.<\|endoftext\|>	4.5625
298	There has been a definite, palpable impact on the Earth’s climate by global warming. The unusual high frequency of heatwaves indicate that there has to be a human influence. As much of the USA sizzles through another scorching summer and the Midwest endures a historic drought, NASA’s climatologist James Hansen states that the future he predicted in 1988 has finally arrived. Hansen and his colleagues have published a paper entitled Perceptions of Climate Change in the journal PNAS. They used seasonal temperature records form 1951-80, which was a relatively stable period, as a baseline. Then, they analyzed the frequency and scale of subsequent temperature anomalies. They conclude that on average, the Earth’s temperature has warmed by one 0.5-0.6 °C. Since then, the shift has impacted many parts of the world. Extremely hot summers, with a 3.5°C warmer temperature, have affected 10% of the world since 2006, which is an order of magnitude higher than the period of 1951-80. The likelihood that these events would have occurred without global warming is minuscule, states Hansen. The study is purely statistical and doesn’t try to explain how climate changes have affected extremely hot summers. It should help people understand global warming and the profound effect humans have on the climate system, states Kevin Trenberth, a climatologist at the US National Center for Atmospheric Research in Boulder, Colorado.<\|endoftext\|>	4
685	The European greenfinch, or just greenfinch (Chloris chloris), is a small passerine bird in the finch family Fringillidae. This bird is widespread throughout Europe, north Africa and south west Asia. It is mainly resident, but some northernmost populations migrate further south. The greenfinch has also been introduced into both Australia and New Zealand. In Malta, it is considered a prestigious song bird, and it has been trapped for many years. It has been domesticated, and many Maltese people breed them. The greenfich was described by Linnaeus in 1758 in the 10th edition of his Systema Naturae under the binomial name of Loxia chloris. The scientific name is from khloris, the Ancient Greek name for this bird, from khloros, "green". The finch family, Fringillidae, is divided into two subfamilies, the Carduelinae, containing around 28 genera with 141 species and the Fringillinae containing a single genus, Fringilla, with 3 species. The finch family are all seed-eaters with stout conical bills. They have similar skull morphologies, nine large primaries, twelve tail feathers and no crop. In all species the female bird builds the nest, incubates the eggs and broods the young. Fringilline finches raise their young almost entirely on arthropods while the cardueline finches raise their young on regurgitated seeds. Phylogenetic analysis based on DNA sequence data indicated that the greenfinches were not closely related to other members of the genus Carduelis. They have therefore been placed in a separate genus Chloris. The greenfinch is 15 cm (5.9 in) long with a wing span of 24.5 to 27.5 cm (9.6 to 10.8 in). It is similar in size and shape to a house sparrow, but is mainly green, with yellow in the wings and tail. The female and young birds are duller and have brown tones on the back. The bill is thick and conical. The song contains a lot of trilling twitters interspersed with wheezes, and the male has a "butterfly" display flight. Behaviour and ecology Woodland edges, farmland hedges and gardens with relatively thick vegetation are favoured for breeding. It nests in trees or bushes, laying 3 to 6 Eggs. This species can form large flocks outside the breeding season, sometimes mixing with other finches and buntings. They feed largely on seeds, but also take berries. Breeding season occurs in spring, starting in the second half of March, until June, with fledging young in early July. Incubation lasts about 13–14 days, by the female. Male feeds her at the nest during this period. Chicks are covered with thick, long, greyish-white down at hatching. They are fed on insect larvae by both adults during the first days, and later, by frequent regurgitated yellowish past of seeds. They leave the nest about 13 days later but they are not able to fly. Usually, they fledge 16–18 days after hatching. This species produces two or three broods per year.<\|endoftext\|>	3.6875
1,598	The construction of identity is a communal endeavor; we contribute to the identity of ourselves and those we encounter through our attitudes and historical biases. Stuart Hall discusses identity as “an ever-unfinished conversation.” Much of his work concerning identity relies on the construction of identity in the context of confrontation of difference. He speaks to his experience of race in Kingston, Jamaica and London, England. In Jamaica, where the majority of the community was black, there was no recognition of race; however, once he arrived in London he was greeted with the long-standing conception of race surrounding the color of his skin. As a member of a historically marginalized group, he experienced the biases and perceptions of hegemonic groups that were attributed to his identity. In his book Orientalism, Edward Said discusses the importance of recognizing hegemonic biases in constructing the identity of an “Other.” Said describes these assumptions as results of a highly motivated process; we view the “Other” through a lens that distorts reality and alienates one culture from another. He emphasizes the manner in which hegemonic groups distinguish themselves from minorities as a means of maintaining power. By patronizing and “Other-ing” marginalized groups hegemonic groups perpetuate harmful ethnocentric ideas.Most importantly, these cultural theorists recognized identity as an ongoing product of history and culture. With all this in mind, we turn to a historically marginalized community that continues to navigate hegemonic spaces with creativity, passion, and grace. Since at least 1000 BCE, deaf individuals and communities have been perpetually marginalized by hegemonic groups, or groups that hold social and political power. This held true through the Ancient Greeks (384-322 BCE); prominent philosopher Aristotle claimed that deaf individuals are “born incapable to reason” and “could not be educated [because] without hearing, people could not learn.” Deaf individuals have been excluded from participating in government; the unnamed deaf son of King Croesus of Lydia was not recognized as a legitimate heir. Furthermore, St. Augustine, apologist of the Catholic church, proselytized that deaf children were a result of God’s anger at the sins of their parents. Much of the deaf community’s value had been historically defined by hearing individuals. Until the sixteenth century, there was little educational opportunity for deaf individuals; the first book of well-known signs was not published until 1620 by Juan Pablo Bonet. Since then, deaf schools have opened, hundreds of sign languages have developed, and the Americans with Disabilities Act was passed in 1990 to grant equal access to the deaf, hard-of-hearing, wheelchair bound, and other differently abled communities. While significant advances have been made in the United States and Europe regarding protection under the law and increased accessibility for deaf individuals, there is still a lot of ground to cover both domestically and internationally for the Deaf community. Part of navigating hegemonic spaces is creating space that intentionally addresses the needs and desires of a given marginalized group. The Deaf community of New York City provides multiple platforms for the deaf population to share in American Sign Language (hereafter ASL) and performance. Deaf events and events with accommodations for deaf individuals throughout the city can be found here through DeafNYC. These include a variety of events, anywhere from Deaf coffee to plays with translators, infrared hearing systems, captioned performances, to slam poetry performed by members of the Deaf community. One of the many notable events is ASL Slam, a monthly event that showcases featured ASL poets and provides a platform for aspiring ASL performers. Founded in 2005, ASL Slam maintains the mission to provide a space for artists in the Deaf community to present and cultivate their work in front of a live audience. As a community, ASL Slam seeks to and successfully promotes a safe and supportive environment for new and experimental artists in various forms, including performance art, literature, improvisation, visual art and music. They host varying types of events including open slams, featured artists, and interviews with notable members of the Deaf community. This installment of ASL Slam featured a founding member of ASL and star of the documentary Deaf jam, Aneta Brodski. The majority of her stage time was devoted to an interview about her experiences as a deaf woman with a remarkably intersectional identity. Not only is she deaf, but she was born in Uzbekistan to deaf Russian Jewish parents, lived in Israel, and immigrated to the United States where she attended the Lexington School for the Deaf in Queens, New York. At Lexington, she was exposed to ASL poetry and quickly developed a compelling poetic voice in three-dimensional poetry. Aneta emphasized two major points concerning the deaf community: education and accessibility. Many schools with deaf children continue to use teaching methods that do not adequately address the needs of non-hearing students. The most common teaching method is Oralism, the education of deaf students through oral language by using lip reading, speech, and mimicking the mouth shapes and breathing patterns of speech instead of using sign language within the classroom. The topic of Oralism is one of heated debate among parents and educators of deaf and hard of hearing children. Many believe this practice is not in place for the education of students, but rather the assimilation of deaf individuals into the hearing community, that Oralism exists for the convenience hearing people rather than the benefit of deaf students. For this reason, Aneta found school frustrating and often stayed home where she could communicate with her family. Much of her identity as a deaf woman was shaped by the institutional biases concerning hard of hearing and deaf students in schools with hearing teachers. Once she took an ASL poetry workshop, she was able to communicate in more meaningful, intentional, and artistic ways with other deaf students. So much of the success of the Deaf community has been rooted in the accessibility of language, in New York today and past Deaf communities. In Martha’s Vineyard from 1694 through the nineteenth century, as few as one in twenty-five and as many as one in four residents were deaf due to hereditary hearing issues. The community developed its own sign language, MVSL; members of the community who did not know sign language were considered ignorant. Knowledge and practice of sign language in this so-called deaf utopia was so commonplace that most town meetings were conducted in sign language so as to accommodate the needs of the community. ASL Slam is an excellent example of an intentionally accommodating space. Despite being a space for the Deaf community, interviews and emcees sign with vocalizing translators. As a perpetually marginalized community, in terms of accessibility and accommodation, deaf individuals recognize the importance of understanding and communication in shared spaces. As Aneta continues to advocate for deaf rights, her main concerns are education and accessibility in public spaces. This includes a sign-language based education system, requiring family of deaf children to learn and use sign language in the home, intentional spaces for Deaf students to explore more creative means of expression, and public accommodations including signing translators and availability of captioning in public spaces, including entertainment (plays, movie theaters etc). Aneta has traveled to Ecuador and Australia in pursuit of these goals. She has not only visited Deaf schools and communities, but also has made efforts to bring these issues to the attention of public officials, with the intent of creating more accommodating spaces for deaf individuals. The Deaf community has navigated a society in which their value has been defined by structures and individuals that do not understand them. They have taken space into their own hands and constructed a network of support in the face of historical adversity with enthusiasm and grace. To find Deaf events in your area, check out Deaf Standard Time and Deaf Websites.<\|endoftext\|>	3.671875
355	Scientists have found evidence for a huge mountain range that sustained an explosion of life on Earth 600 million years ago. The mountain range was similar in scale to the Himalayas and spanned at least 2,500 kilometers of modern west Africa and northeast Brazil, which at that time were part of the supercontinent Gondwana. "Just like the Himalayas, this range was eroded intensely because it was so huge. As the sediments washed into the oceans they provided the perfect nutrients for life to flourish," said Professor Daniela Rubatto of the Research School of Earth Sciences at The Australian National University (ANU). The discovery is earliest evidence of Himalayan-scale mountains on Earth. "Although the mountains have long since washed away, rocks from their roots told the story of the ancient mountain range's grandeur," said co-researcher Professor Joerg Hermann. "The range was formed by two continents colliding. During this collision, rocks from the crust were pushed around 100 kilometers deep into the mantle, where the high temperatures and pressures formed new minerals." As the mountains eroded, the roots came back up to the surface, to be collected in Togo, Mali and northeast Brazil, by Brazilian co-researcher Carlos Ganade de Araujo, from the University of Sao Paulo and Geological Survey of Brazil. Dr Ganade de Araujo recognized the samples were unique and brought the rocks to ANU where, using world-leading equipment, the research team accurately identified that the rocks were of similar age, and had been formed at similar, great depths. Tell us what you think of Chemistry 2011 -- we welcome both positive and negative comments. Have any problems using the site? Questions?<\|endoftext\|>	3.9375
2,140	Suppose in our library shelf we want to arrange about $35$ books which are of three different subjects. These can be arranged in such a way that the books on the same subjects are together. When we start arranging, our question is in how many ways can this be arranged? From $10$ students, a committee of $6$ is to be arranged which contain a President, a Vice President and a secretary. In how many ways this can be arranged? When we try to solve these problems practically, it will be very difficult and time consuming. Hence we use the method of permutation and combinations to make these arrangements. In this section let us see about the meaning of the terms Permutations and Combinations and the method of evaluating them using appropriate formulae. ## Permutation vs Combination What is Permutation? The different arrangements which can be made out of a given number of things by taking some or all at a time are called permutations. Examples: 1) What are the two digit numbers that can be formed using the digits $2, 5, 7$. Solution: Here out of three digits, we formed two digit numbers, (i.e) we form numbers by taking two digits at a time. The numbers that can be formed are, $25, 27, 52, 57, 72, 75$. 2) What are the different numbers that can be formed using all the digits, $9, 8$ and $5$. Solution: Here we form numbers by taking all at a time. (i. e) we form three digit numbers by taking all the digits at a time. The numbers that can be formed are $589, 598, 859, 895, 958$ and $985$ NOTE: Hence we observe that in permutations there will be selection and then arrangements. What is Combination? Each of the different groups or selections which can be formed by taking some or all of a number of objects irrespective of their arrangements, is called a combination. Example: 1) John as three pens each of blue, red and green. In how many ways two pens can be selected from the three pens. Solution: John as three colored pens each of blue, red and green. The different ways of selecting two pens are, blue and red ; red and green (or) blue and green. Therefore there are three ways of selecting two pens from three pens. 2) A bag has an yellow marble, black marble and a blue marble. In how many ways three marbles can be selected? Solution: Since the bag contains three marbles each of yellow, black and blue, the selection of thee marbles contain all the three colors yellow, black and blue. Therefore, the selection can be done only one way. NOTE: Hence we observe that in combinations there will be only selection (and no arrangement). ## Formulas Factorial: Factorial notation is used to express the product of first $n$ natural numbers. (i. e) $n!$ = $1.2.3.4 . . . . . . . . . . n$ Example: $5 !$ = $1.2.3.4.5$ = $120$ $10 !$ = $1.2.3.4.5.6.7.8.9.10$ = $3628800$ Permutations Formula: Formula 1: The permutations of n objects by taking $r$ at $a$ time is, $P (n, r)$ = $nPr$ = $n (n - 1) (n - 2)$ . . . . . . . . $(n - r + 1)$ Formula 2: The above permutation can be expressed using factorial notation as follows. $P (n, r)$ = $nPr$ = $\frac{n!}{(n-r)!}$ Example: In how many ways three digits numbers can be formed using the digits, $3, 4, 5, 7$ and $9$. Solution: We have $5$ digits, $3, 4, 5, 7$ and $9$. The number of three digit numbers that can be formed is permutation of five things taken three at a time. (i. e), $5P3$ = $5 ( 5 - 1) (5 - 2)$ [ by Formula $1, nPr$ = $n (n-1) (n-2)$ . . . . . . $(n - r + 1)$] = $5.4.3$ = $60$ Factorial Method: Using the factorial formula, we have $nPr$ = $\frac{n!}{(n-r)!}$ $P (5, 3)$ = $5P3$ = $\frac{5!}{(5-3)!}$ = $\frac{5.4.3.2.1}{2.1}$ = $5.4.3$ = $60$ There will be $60$ three digit numbers that can be formed. Combinations Formula: Formula 1: Combination of $n$ objects by taking $r$ at a time is $nCr$ = $C\ (n, r)$ = $\frac{nPr}{r!}$ Formula 2: Combination of $n$ objects by taking $r$ at a time is $nCr$ = $C\ (n, r)$ = $\frac{n!\;}{(n-r)!\;r!}$ Example: In how many ways $3$ balls can be selected from a box containing $10$ balls. Solution: Since the above situation is only selected we have combination. Therefore, $C\ (n, r)$ = $C(6, 3)$ = $\frac{nPr}{r!}$ = $\frac{6P3}{3!}$ = $\frac{6.5.4}{3.2.1}$ = $20$ (or) $C\ (6, 3)$ = $\frac{n!}{(n-r)!\;r!}$ = $\frac{6!}{3!\;3!}$ = $\frac{1.2.3.4.5.6}{1.2.3\;.1.2.3}$ = $20$ ## Examples ### Solved Examples Question 1: How many four letter words, with or without meaning can be formed out of the letters of the word, "MATHEMAGIC", if repetition of letters is not allowed. Solution: The word "MATHEMAGIC" contain the letters, M, A, T, H, E, G, I, C which are 8 letters. Here we need to select four letters and arrange them. Hence we have selection and arrangement which is permutation. Therefore the required number of words = Permutations of 8 letters by taking four at a time = P (8, 4) = 8 (8 - 1) (8 - 2) (8 - 3) = 8 . 7. 6. 5 = 1680 Question 2: In how many ways can 10 books be arranged on a shelf so that a particular pair of books shall be a. always together. b. never together. Solution: a. Since a particular pair of books is always together. If we keep these together as one pair, then we have to arrange 9 books on the shelf. This can be done in P (9, 9) ways = 9 ! ways Since the pair of books together can be arranged in 2 ways, the total number of ways of arranged the ten books so that a particular pair of books is always together = 2 x 9 ! b. The ten books can be arranged in P (10, 10) ways = 10 ! ways From (a), the ten books can be arranged in 9! . 2 ways by arranging particular pair of books together. Therefore, the number of ways of arranging the 10 books so that a particular pair is never together is, =10 ! - 2 x 9 ! = 10 x 9 ! - 2 x 9! = (10 - 2) x 9 ! = 8 x 9 ! the number of ways of arranging the 10 books so that a particular pair is never together is = 8 x 9 ! Question 3: How many diagonals are there in an octagon? Solution: A polygon of 8 sides has 8 vertices. By joining any two of these vertices, we obtain either a side or a diagonal of the polygon. Here we have only a selection (and no arrangement) , hence we have combination. Number of all straight lines obtained by joining 2 vertices at a time = C (8 , 2) = $\frac{8!}{6!\;.2!}$ = $\frac{8.7.6.5.4.3.2.1}{1.2.3.4.5.6.1.2}$ = 28 Since the number of sides = 8, Number of diagonals of the octagon = 28 - 8 = 20 ## Practice Problems ### Practice Problems Question 1: In how many ways 7 boys and 5 girls be arranged for a group photograph, if the girls are to sit on chairs in a row and the boys are to stand in a row behind them? Question 2: Find the number of ways in which the letters of the word, "LAPTOP", can be arranged such that the vowels occupy only even position. Question 3: How many three digit numbers can be formed with the digits, 3, 4, 5, 6, 7, 8, when the digits may be repeated any number of times in any arrangement. Question 4: Out of 6 men and 4 women a committee of 54 is to be formed containing at least one woman. In how many ways the committee can be formed. Question 5: A code word is to consists of two distinct alphabets followed by two distinct numbers between 1 and 5. How may such code words are there?<\|endoftext\|>	4.9375
549	HAT IS ICE? Ice is the solid state of water (H20 molecule). In a glacier, ice is mixed with air bubbles, making it 0.9 times denser than water. For that simple reason, ice floats on water. GLACIER ICE FORMATION During winter, snow piles up and compresses. Its hexagonal crystals start to deform due to compaction, releasing air, giving crystals a more granular shape. This brings us to the second stage of snow: névé. As new layers of snow accumulate, the weight of said layers compresses the snow into glacial ice. HOW LONG DOES IT TAKE TO FORM A GLACIER? It varies considerably from one glacier to the next. It can take just a dozen years for temperate glaciers like the Patagonian glaciers, or up to hundreds of years for cold glaciers like the ones in Antarctica. Contrary to popular belief, the warmer the glacier is, the quicker the ice forms, because the snow crystal needs moderate temperatures (above 0ºC, 32ºF) in order to fuse into glacial ice. In Antarctica, temperatures are so low that the snow compaction process takes much longer. There are two phenomena that cause movement: sliding and internal deformation. - Sliding is produced by friction between the base of the glacier and the rocky substrate, which creates a thin film of water that allows movement. It can also be caused by water leaking from the upper layers down to the base of the glacier. - Internal deformation is produced by the pressure (approximately 650 tons per cubic meter) exerted by the weight of the ice. This tension leads to deformation, which causes the glacier to move. ANATOMY OF A GLACIER The accumulation zone is the top of the glacier, where snow accumulates. The ablation zone is the bottom of the glacier, where there is a loss in glacial mass. The equilibrium line separates the accumulation zone from the ablation zone. A moraine is an accumulation of rock, sand or clay that is picked up and transported by glaciers as they advance. There are several kinds of moraines: Lateral Moraine: as its name states, it consists of sediment deposited on the sides of a glacier. Medial Moraine: is the junction of two glaciers merging their lateral moraine deposits. Terminal Moraine: this moraine marks the furthest advance of a glacier and the point where it starts to recede. Internal Moraine: is an accumulation of sediment which falls into crevasses and is trapped in the ice, giving the ice a “dirty” appearance.<\|endoftext\|>	4.0625
280	# 2003 AIME II Problems/Problem 10 ## Problem Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? ## Solution Call the two integers $b$ and $b+60$, so we have $\sqrt{b}+\sqrt{b+60}=\sqrt{c}$. Square both sides to get $2b+60+2\sqrt{b^2+60b}=c$. Thus, $b^2+60b$ must be a square, so we have $b^2+60b=n^2$, and $(b+n+30)(b-n+30)=900$. The sum of these two factors is $2b+60$, so they must both be even. To maximize $b$, we want to maximixe $b+n+30$, so we let it equal $450$ and the other factor $2$, but solving gives $b=196$, which is already a perfect square, so we have to keep going. In order to keep both factors even, we let the larger one equal $150$ and the other $6$, which gives $b=48$. This checks, so the solution is $48+108=\boxed{156}$.<\|endoftext\|>	4.53125
1,249	1.1 BACKGROUND TO THE STUDY The foundational history of the use of photography could be traced to 1950’s when the photo type block for copying line subjects, engraving manuscripts and photographic reproduction into books was invented. Ogedengbe (2002) reports that colour photo prints can be a source of help to students in Secondary Schools to develop the ability to compose effectively and increase the performance level in their studies. Rheinhold Thiele’s (2000) supports the adoption of photograph unto teaching and learning. She was a press photographer who made photo prints of six hundred and eleven (611) subjects of the Natal Campaign for the Navy and the Army. She also covered the Russo-Japanese war and the Boer war. And many of the documented photographs that were left over were later adopted for educational purpose with more research works into the uses of photography especially for teaching and learning purposes. Learners’ academic performances have been increased through the use of photography. Bower and Spaulding (2000) in a research study on simple reading materials concluded that a sequence of picture can be effectively used for communicating new idea and to increase learners’ performance if the pictures are used together with verbal face to face explanation. Fonsesca and Karle (1960) explained that recognizable and familiar objects presented in a sequence of photo prints enhance comprehension and better performance. Travers (2003) shows that photographic prints can be used to improve the performance of a lawn tennis player. According to Hudson and Hector (1963) photography and pictorial illustration accompanied with verbal explanation have increased the level of performance of cocoa farmer in Ghana. Sofowora (1994) contributes that the comprehension of a process involving series of actions is possible if photographs are used. Chaplain (1960) working on a science education research project contributes that instruction when given largely with numerous sequence of photo prints helps to convey adequate scientific information and instruction. Photographic study thus offers an excellent opportunity for learning the challenge of accurate description. Arundale (2005) reports from his study that learners were able to retain a great part of the lesson content when oral teaching are linked with a sequence of photographs. It also shows that the students in the experimental group performed better and were able to recall the various experiences they acquired in their projects than those in the control group. Spaulding (1955) was of the opinion that colour photo prints are real life situation that help the retention of what has been learnt. Adeosun (1986) explained that students that were exposed to pictures were able to retain and recall a greater part of the lesson content than those that were taught without the use of picture. Franden (1961) showed that a sequence of pictures and pictorial illustration make learning real, more meaningful and reinforces the learning materials for easy assimilation and retention. John and Litcher (2000) in their experiment concluded that four months after exposing the students to multi-racial pictures the attitude of the students changed towards other races. Working with photographs also adds another layer of complexity to the lessons, because every photograph was created at one point in time, in a particular place, of a chosen subject, by a particular photographer, for a specific purpose, and using a particular technology. All these elements has made learning an unforgettable experience when adopting the photography teaching method. 1.2 STATEMENT OF THE PROBLEM Several secondary schools in Nigeria are facing serious problems in the area of teaching and learning. Among the problems are poor methods of teaching, lack of relevant instructional materials; negative attitude of the parents and the students towards the subject and shortage of qualified teachers. Several other studies conducted outside Nigeria revealed that the use of photography teaching method will enhance teaching and learning. The researcher is of the believe that photography teaching method can be used as a panacea to the problem of poor performance in secondary school. 1.3 OBJECTIVES OF THE STUDY The following are the objectives of this study: - To examine the effectiveness of photography as a teaching method in Secondary schools in Nigeria. - To determine the relationship between photography teaching method and student’s academic performance in secondary schools in Nigeria. - To determine the relationship between photography teaching method and student’s retention level in secondary schools in Nigeria. 1.4 RESEARCH QUESTIONS - What is the effectiveness of photography as a teaching method in Secondary schools in Nigeria? - What is the relationship between photography teaching method and student’s academic performance in secondary schools in Nigeria? - What is the relationship between photography teaching method and student’s retention level in secondary schools in Nigeria? HO: There is no significant relationship between photography teaching method and student performance in secondary schools in Nigeria. HA: There is significant relationship between photography teaching method and student performance in secondary schools in Nigeria. 1.6 SIGNIFICANCE OF THE STUDY The following are the significance of this study: - The results from this study will enlighten the administrators in the education sector and the general public on the benefits of photography teaching method and its impact on performance and effectiveness. - This research will be a contribution to the body of literature in the area of the effect of personality trait on student’s academic performance, thereby constituting the empirical literature for future research in the subject area. 1.7 SCOPE/LIMITATIONS OF THE STUDY This study is limited to secondary schools in Egor Local Government Area of Edo State. It will also cover the effectiveness of photography as a teaching method as reflected in the academic performance of secondary school students. LIMITATION OF STUDY Financial constraint– Insufficient fund tends to impede the efficiency of the researcher in sourcing for the relevant materials, literature or information and in the process of data collection (internet, questionnaire and interview). Time constraint– The researcher will simultaneously engage in this study with other academic work. This consequently will cut down on the time devoted for the research work - Format: Microsoft Word - Pages: 78 - Price: ₦3000 - Chapters: 1-5<\|endoftext\|>	3.6875
723	Coordinate Geometry : Exercise >> # Number System Question: 1 - Is zero a Question 1: Is zero a rational number? Answer:- Zero can be written in the form p/q , where p and q are integers and q is not equal to 0. Therefore, zero is a rational number. Question: 2 :- Find six rational number between 3 and 4. You can notice that by calculating averages between two numbers we get a number which is exactly between these two numbers. This way you can go on calculating infinite numbers of numbers. Question3: Find five rational numbers between 3/5 and 4/5 Question4: State if following statements are true or false: (a) Every natural number is a whole number. (b) Every integer is a whole number. (c) Every rational number is a whole number. Answer: (a) As natural number is all numbers starting from 1 and the whole number includes zero as well so this statement is true. On the other hand every whole number is not natural number as zero is not a natural number. (b) Only positive integers are whole numbers. (c) Rational numbers are not whole numbers as they are not complete. Question5: Write the following in decimal form and comment on their kind of decimal expression. Question6: Express the following in the form p/q , where p and q are integers and p is not 0 Put 9 for every non-zero digit in the denominator and zero for zero in the denominator. Question7: What can the maximum number of digits be in the repeating block of digits in the decimal expression of 1/17 ? A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The period of the repeating decimal, 1⁄p, where p is prime, is either p − 1 (the first group) or a divisor of p − 1 (the second group). Examples of fractions of the first group are: • 1⁄7 = 0.142857 ; 6 repeating digits • 1⁄17 = 0.0588235294117647 ; 16 repeating digits • 1⁄19 = 0.052631578947368421 ; 18 repeating digits • 1⁄23 = 0.0434782608695652173913 ; 22 repeating digits • 1⁄29 = 0.0344827586206896551724137931 ; 28 repeating digits • 1⁄97 = 0.01030927 83505154 63917525 77319587 62886597 93814432 98969072 16494845 36082474 22680412 37113402 06185567 ; 96 repeating digits Question8: What property a rational number must satisfy to have terminating decimal expression Answer: If the denominator is either 2 or 5 as its factor then the result will be terminating decimal. As 10 is the product of 2 and 5 so to have terminating decimal 2 or 5 are required. If there is a prime number other than 2 or 5 in the denominator then the decimal can or cannot be treminating. Question9: Simplify the following expressions: Coordinate Geometry : Exercise >> Number System will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.<\|endoftext\|>	4.59375
455	A new way to test the theory of how stars work, which involves making precise measurements of the energy of neutrinos from the Sun, has been proposed by an American astrophysicist. John Bahcall of the Institute for Advanced Study in Princeton, New Jersey, is the theorist who first suggested the idea of measuring the flux of solar neutrinos, about 30 years ago. While theorists are still trying to explain the puzzling results of those measurements, he is already thinking about possibilities for the next generation of solar neutrino experiments. Bahcall’s latest idea concerns one specific nuclear reaction inside the Sun. This occurs when a nucleus of beryllium-7 captures an electron and is transformed into a nucleus of lithium-7, emitting a neutrino. Under laboratory conditions, the neutrinos emitted in this reaction have an energy of 861.84 kiloelectronvolts (keV). But Bahcall calculates on the basis of the standard astrophysical theory of the conditions at the heart of the Sun, the solar neutrinos produced in this reaction will have a higher energy of 862.27 keV (Physical Review Letters, vol 71, p 2369). This particular reaction occurs only near the centre of the Sun, within the innermost 4 per cent of the Sun’s mass. So confirmation of the neutrino energies predicted by Bahcall would provide a test of how accurately those standard theories describe conditions at the heart of the Sun, and give a direct indication of the temperature there. Several experiments have been proposed which might, Bahcall says, be able to measure the position and strength of this neutrino ‘line’. One possibility involves running the solar reaction in reverse, by capturing the neutrinos with lithium-7 which would then be converted into beryllium-7 and emit an electron. The energy of the emitted electron, which is relatively easy to measure, is an indication of the energy of the incoming neutrino. The experiment could be running by the end of the 1990s.<\|endoftext\|>	3.921875
794	# Into Math Grade 2 Module 2 Lesson 2 Answer Key Write Equations to Represent Even Numbers We included HMH Into Math Grade 2 Answer Key PDF Module 2 Lesson 2 Write Equations to Represent Even Numbers to make students experts in learning maths. ## HMH Into Math Grade 2 Module 2 Lesson 2 Answer Key Write Equations to Represent Even Numbers I Can write an equation to model an even number as the sum of two equal addends. How can you show Sofia’s toy cars in two equal groups? Explanation: 4 + 4 = 8 There are 8 toy cars with sofia. Have children work in groups. Read the following: Sofia has an even number of toy cars. How can you show her toy cars in two equal groups? Complete the equation to show the equal groups and the number of toy cars Sofia has. Build Understanding Von finds 14 shells at the beach. He sorts the shells into two equal groups. How many shells are in each group? A. How can you make a concrete model to show Von’ s shells as two equal groups? B. An even number of objects can be shown as two equal groups. Is 14 an even number? Circle Yes or No. C. How can you write an addition equation to show 14 osa sum of two equal addends? ___ + ___ = ____ D. How many šhells are in each group? ____ shells Explanation: 7 + 7 = 14 There are 7 shells in each group. Turn and Talk How did you show Von’s shells as two equal groups? Explain. Step It Out 1. Mr. Lee plants 12 seeds in a garden. He sorts the seeds into two equal groups. How many seeds are in each group? A. Show 12 as two equal groups. B. Write an addition equation to show 12 as a sum of two equal addends. ___ = ___ + ____ C. Solve. There are ____ seeds in each group. Explanation: 6 + 6 = 12 There are 6 seeds in each group. Check Understanding Math Board Question 1. Monica and Billy have 8 puzzles in all. They each have the same number of puzzles. How many puzzles do Monica and Billy each have? Write an addition equation to show the equal groups. Solve. ___ = ____ + ___ ____ puzzles 4 + 4 = 8 puzzles Explanation: Monica and Billy have 8 puzzles in all They each have the same number of puzzles 8 = 4 + 4. Question 2. There are three different kinds of bluebirds in the United States. But all bluebird eggs are light blue in color. Noah and Elisa see 10 bluebirds in all. They each see the same number of bluebirds. How many bluebirds do Noah and Elisa each see? Write an addition equation to show the equal groups. Solve. ____ = ___ + ____ ____ bluebirds 10 = 5 + 5 Explanation: There are three different kinds of bluebirds in the United States. But all bluebird eggs are light blue in color. Noah and Elisa see 10 bluebirds in all. They each see the same number of bluebirds 10 = 5 + 5. Model with Mathematics Write an addition equation to show the number as the sum of two equal addends. Question 3. 6 ___ = ____ + ____ 6 = 3 + 3 Explanation: Sum of 3 and 3 is 6. Question 4. 18 ____ = ___ + ____ 18 = 9 + 9 Explanation: The sum of 9 and 9 is 18. Question 5. Open Ended Can you show the number 5 as a sum of two equal addends? Explain.<\|endoftext\|>	4.78125
1,075	# 泰勒和麦克劳林（幂）级数计算器 Enter a function: Enter a point: For Maclaurin series, set the point to 0. Order n= Evaluate the series and find the error at the point The point is optional. If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below. ## Solution Your input: calculate the Taylor (Maclaurin) series of $\sin{\left(x \right)}$ up to $n=5$ A Maclaurin series is given by $f\left(x\right)=\sum\limits_{k=0}^{\infty}\frac{f^{(k)}\left(a\right)}{k!}x^k$ In our case, $f\left(x\right) \approx P\left(x\right) = \sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k=\sum\limits_{k=0}^{5}\frac{f^{(k)}\left(a\right)}{k!}x^k$ So, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula. $f^{(0)}\left(x\right)=f\left(x\right)=\sin{\left(x \right)}$ Evaluate the function at the point: $f\left(0\right)=0$ 1. Find the 1st derivative: $f^{(1)}\left(x\right)=\left(f^{(0)}\left(x\right)\right)^{\prime}=\left(\sin{\left(x \right)}\right)^{\prime}=\cos{\left(x \right)}$ (steps can be seen here). Evaluate the 1st derivative at the given point: $\left(f\left(0\right)\right)^{\prime }=1$ 2. Find the 2nd derivative: $f^{(2)}\left(x\right)=\left(f^{(1)}\left(x\right)\right)^{\prime}=\left(\cos{\left(x \right)}\right)^{\prime}=- \sin{\left(x \right)}$ (steps can be seen here). Evaluate the 2nd derivative at the given point: $\left(f\left(0\right)\right)^{\prime \prime }=0$ 3. Find the 3rd derivative: $f^{(3)}\left(x\right)=\left(f^{(2)}\left(x\right)\right)^{\prime}=\left(- \sin{\left(x \right)}\right)^{\prime}=- \cos{\left(x \right)}$ (steps can be seen here). Evaluate the 3rd derivative at the given point: $\left(f\left(0\right)\right)^{\prime \prime \prime }=-1$ 4. Find the 4th derivative: $f^{(4)}\left(x\right)=\left(f^{(3)}\left(x\right)\right)^{\prime}=\left(- \cos{\left(x \right)}\right)^{\prime}=\sin{\left(x \right)}$ (steps can be seen here). Evaluate the 4th derivative at the given point: $\left(f\left(0\right)\right)^{\prime \prime \prime \prime }=0$ 5. Find the 5th derivative: $f^{(5)}\left(x\right)=\left(f^{(4)}\left(x\right)\right)^{\prime}=\left(\sin{\left(x \right)}\right)^{\prime}=\cos{\left(x \right)}$ (steps can be seen here). Evaluate the 5th derivative at the given point: $\left(f\left(0\right)\right)^{\left(5\right)}=1$ Now, use the calculated values to get a polynomial: $f\left(x\right)\approx\frac{0}{0!}x^{0}+\frac{1}{1!}x^{1}+\frac{0}{2!}x^{2}+\frac{-1}{3!}x^{3}+\frac{0}{4!}x^{4}+\frac{1}{5!}x^{5}$ Finally, after simplifying we get the final answer: $f\left(x\right)\approx P\left(x\right) = x- \frac{1}{6}x^{3}+\frac{1}{120}x^{5}$ Answer: the Taylor (Maclaurin) series of $\sin{\left(x \right)}$ up to $n=5$ is $\sin{\left(x \right)}\approx P\left(x\right)=x- \frac{1}{6}x^{3}+\frac{1}{120}x^{5}$<\|endoftext\|>	4.40625
184	Here is the learning that can take place while a child does crosswords. Practice reading: the clues are out of context of a story. Sequencing: doing in order. Organisation: Doing the across clues first and then the down. Memory: Remembering which clue s/he is up to. Reinforcement of the size of letters: fitting them into the square. Having a space for each letter. Spelling: knowledge about how many letters are in a word. Seeing that sometimes some letters are already in place, so s/he has to allow for that. Develops flexible knowledge about letters in words. Expanding vocabulary: the example crossword is about rhyming words. Some are about a topic so may be introducing new words. They can be about words that sound the same but have different meanings, or about a words that has more than one meaning.<\|endoftext\|>	4.53125
2,010	a yoga & creative movement lesson for parents & teachers of children from preschool-5th grade Some Animals Migrate "Migrate" means to move somewhere warmer for the winter season, and then come back when it's spring. - Monarch Butterflies fly to Mexico and southern California to escape the cold winters of the north. For Butterfly Pose, sit on the floor with the bottoms of your feet touching, and knees bent. Hold on to your ankles or toes with your hands. Sit up tall and move your legs up and down like the flapping wings of a butterfly. - Snow Geese are large, white birds with black-tipped wings that take the Central Flyway route from Canada to the Gulf of Mexico, passing over lots of rich farmland where they can stop and eat grain. For our Goose Pose (Low Lunge), let's stand on our knees and put one foot on the floor in front of you, leaning forward. Open your arms to the sides and flap them up and down, slowly and then faster and faster. Try putting your other foot forward, and flap your wings again. Can you take off into the sky from this shape? (If you have enough students and enough space, you can make a "V" shape and try to move together like a flock of geese.) - Reindeer (Caribou) travel the farthest of all migrating animals, up to 5,000 miles, looking for forests where they can search under the snow for food. They are fast runners and can also swim across lakes and rivers to get where they are going. Stand up with your feet wide, like Warrior 1 Pose. Bend your front knee (toes forward) but keep the back leg straight (toes turn to side). Attach your thumbs to the sides of your head above your ears, fingers spread out like antlers. Lift your head and look up to the sky. Then lean forward, lowering head in front of knee to look for food under the snow. Then repeat with the other foot forward. - Humans! Lots of people like to leave the cold weather and travel somewhere warm for part of the winter, or all of the winter. How do they get to a warm place far away? Let's try flying in Airplane Pose. Balance on one foot with your other leg lifted behind you like the tail of a jet. Tip forward and reach your arms out to the sides like sturdy airplane wings. Try changing legs. "Adapt" means to change how you look or how you act. Let's talk about some animals who change when the winter comes! - Squirrels gather and store extra food for the long winter, when they might not be able to find much to eat. Usually, they don't like when other squirrels come into their favorite hiding spots, but in the winter they share their homes, huddling together to keep warm. Stand up with your feet apart, as wide as your body. Bend your knees like you're about to sit in a chair--Chair Pose. Bring your elbows close to your sides, lifting cupped hands up in front of your chest like squirrel's hands. Wiggle your back side as though you have a tall bushy tail to wave. If you have a group of students, try these Squirrel Games: Pass the Acorn: Stand in a circle, everyone in Squirrel Pose. Pass an "acorn" (any small object about the size of an orange) from squirrel to squirrel, holding it only with the inside edges of your hands (no fingers). Make sure everyone keeps their knees bent until the acorn goes all the way around the circle. Scatter and Huddle: Squirrels roam around the room in whatever ways the children imagine squirrels might--darting, shuffling, pausing to twitch their noses, until the teacher rings a bell or calls out "Huddle!" Move into the middle of the room to keep each other warm, until the bell rings again or the teacher calls out "Scatter!" - Fish in some cold places spend their winters being very active and moving around a lot so their bodies don't freeze. Lay down on your belly and lift your legs and arms. Swish your feet like a tail and paddle your hands like fins, twisting your body from side to side. Move so you don't freeze! - Snowshoe Rabbits are a rusty brown color... until it snows! Then their fur turns white so they match the snow and they can't be seen by animals who might try to eat them. Also, their feet are large and hairy so they don't sink down into the snowdrifts. For Rabbit Pose, sit on your shins and tip forward slowly till your head touches the mat in front of your knees. Reach your arms alongside your shins toward your ankles and start to lift your seat off your legs, keeping your head on the floor. Imagine you are hiding in the snow! - Humans! People adapt to cold winter weather by wearing lots of warm clothes and eating hot food and drinking hot drinks. Hot Chocolate Breath: Take a seat and hold your hands around an imaginary mug. Breathe in and smell your cocoa. Do you like marshmallows in it? Or some whip cream and sprinkles? Breathe out slowly through your mouth, so your breath warms your hands. Repeat a few a few more times, breathing in through your nose and breathing out warm breath through your mouth. If you want to add a movement game for a larger group, this would be a great time to play a Bundle-Up Relay Race, with two piles of oversized sweaters, hats, scarves, mittens, etc, at one end of the room. In two lines, a student from each team runs to the clothes, puts them all on and runs back, where they tag the next person in line to return to the former pile with them. The first player takes all the items off and goes to the end of their line while the next player puts them on. Continue till one team's last player is dressed in the warm clothes. Some Animals Hibernate "Hibernate" means to take a long nap during winter. - Grizzly Bears sleep from 5-7 months of the year. They eat a lot before they hibernate, and then they don't eat or go to the bathroom until winter is over. Let's Bear Walk, on hands and feet, around the room, getting slower and slower and more tired until finally, we curl up and sleep. - Box Turtles get so quiet during hibernation that they don't even breathe! Their skin takes in oxygen while they rest. Turtle Pose: Sit with your legs in front of you and the bottoms of your feet touching, so your legs form a diamond shape. Slide your hands, palms down, under your ankles and round your back, letting your forehead rest on your feet (or above your feet). It's nice and dark and quiet in your turtle shell. - Humans! Of course we don't sleep for months, but many people in cold places are less active in the winter, staying inside and maybe curling up with a blanket next to a fire. Let's stack our legs in Firelog Pose, sitting on the floor and crossing one shin on top of the other. Rub your hands together until they feel warm, and then give yourself a big warm hug. Make a fire with the other leg on top. "Dormant" is a way of hibernating, when the animal's body goes "on pause" for the winter. Part of their body may even freeze, but they wake up again when it gets warmer. - Frogs can go dormant in the mud at the bottom of a pond, or by burrowing into old leaves on the ground. Their body temperature drops and they get almost as cold as the earth around them. Frog Pose, squatting low with your sticky, webbed hands gripping the floor in front of you. Try a few hops and then stop moving and show a frozen frog shape! - Snakes in some places burrow into little tunnels underground, where the temperature is cold but not freezing. They do not move or eat for months. Laying on your belly with legs and feet touching, reach your arms by your sides as though you don't have arms at all, in Snake Pose. Lift your head and shoulders away from the floor and look from one side to the other, hissing. Wiggle into an imaginary burrow and show what a dormant snake might look like. - Insects, such as the Ladybird Beetle (Ladybug) find a sheltered spot to sleep in a dormant state till it warms up. We can do Flipped Bug Pose (a nicer way to say Dead Bug pose) by laying on our backs with our knees bent toward our armpits, bottoms of feet up to the ceiling. Now hold on to your feet with your hands and rock side to side a few times. Then find a paused shape, like a sleepy ladybug. - Humans! Though we don't hibernate or go dormant like some animals do, deep rest is very important to our health. Let's practice the best yoga pose of all, Final Resting Pose, or Savasana, for a few minutes and let our muscles, bones, heart and brain be soft and calm. Make yourself comfortable on your mat and feel very heavy and quiet. Listen to your breath. (Teachers/Parents, you may want to dim the lights, turn on soft music or count slowly to ten so the children know how long they will be resting for. After sitting up and stretching and yawning, talk about how good it would feel to wake up after months of sleep. Where would you go? What would you eat?<\|endoftext\|>	3.71875
1,332	Genetics is the science of studying the genes, the genetic variation and the inheritance of the characteristics from one generation to the other. Every living organism is defined by its genetics. Genetics is all around… Our knowledge on genetics first started during the pre-historic period, as humans domesticated animals and started breeding specific plant species, acknowledging that some characteristics were passed from the parents to the next generation. Nowadays, we know that all physical inborn characteristics that combine a human being, depend on genes…..Genes- this magic word- are inherited by the parents to their children and define all of their features. Eye and hair color, height, weight and many more characteristics are a result of the combination of many genes. Even mental skills and natural talents are affected by genes in close interaction with the environment. On the other hand, many disorders and diseases are caused by genes. Nowadays, we know that all the genetic information is enclosed in the inner of our cells, specifically in the cell nucleus. The genetic information could be compared to a book, whose half of the chapters derive from the mother and the other half from the father. This book contains sentences- the chromosomes, which consist of words- the genes. Human beings have 23 pairs of chromosomes in each somatic cell, which in total add up to 46 chromosomes. Half of these 46 chromosomes are of maternal origin and the other half are of paternal origin. Some chromosomes have a large number of genes, while other carry only a small number. Chromosomes consist of what we all have heard as “DNA”, which refers to the chemical structure of Deoxyribonucleic Acid. Chromosomes of the pairs 1-22 are called autosomes and are common for both sexes. The last pair consists of chromosomes X and Y, which are called sex chromosomes and are differently distributed between males and females: a male human has one Y and one X chromosome (symbolized XY), while a female human has two X chromosomes (symbolized XX). The chromosomes that form a pair (homologous chromosomes) are identical in terms of having the same kind of genes at the same location. However, genes of the same location in the two homologous chromosomes may differ on the nature of their genetic information. These genes are called allelic (there is a gene that defines a characteristic, the allelic gene A results in phenotype A and the allelic gene B results in phenotype B. Both genes are in the same location on the homologous chromosomes and control the same characteristic, but in a different way). Every human inherits one chromosome of each homologous pair from the mother and one from the father. Especially, regarding the sex chromosomes, all humans inherit one X chromosome from their mother, who has two X chromosomes (XX). The second sex chromosome is inherited by the father and could be either X or Y, as he has an X and an Y chromosome (XY). As a result, sex is defined by the Y chromosome and all females have as sex chromosomes a pair of XX, while all males have a pair of XY. Every word- gene- consists of letters- bases- which are adenine (A), thymine (T), guanine (G) and cytosine (C). These four bases can be combined in multifarious ways, defining the function of the body cells. Genes could be affected by age and may develop errors- mutations and malfunctions due to environmental factors and endogenous toxins. If a mutation occurs inside a gene, then alters the base sequence and could eventually change or destroy the genetic information for the synthesis of a protein. In such cases, depending on the importance of the non-functioning protein, various diseases could emerge. Genetic mutations can cause three different kind of disorders: 1) Chromosomal abnormalities 2) Multifactorial genetic diseases/ disorders 3) Monogenic diseases Monogenic diseases result from alterations in a single gene. We all have two copies of each gene (allelic genes); one comes from our mother and the other from our father. Monogenic diseases may result by a mutation in a single gene. A monogenic disorder example is Achondroplasia, which results in dwarfism. In other cases of monogenic diseases, mutations in both the allelic genes have to coexist for someone to be a patient. Such examples of monogenic diseases are Thalassemia a, Thalassemia b, or cystic fibrosis. If the mutation exists only in one of the two allelic genes, then this person is not considered patient, but a carrier of the disease. Carriers do not show symptoms of the disease. However, if both the partners are carriers of the same gene mutation, they have 25% chances of having a child suffering from the disease. Chromosomal abnormalities are cases of irregularities in the structure or the number of the chromosomes. Therefore, there are structural or numerical chromosomal abnormalities. These abnormal situations are present when someone has one or more chromosomes with detriment. A numerical chromosome abnormality could exist due to surplus chromosomes (trisomy or polysomy) or on the opposite, due to lack of some of them (monosomy). The most characteristic case of trisomy- existence of one surplus chromosome- is Down syndrome. People with Down syndrome have one extra 21 chromosome, instead of just a pair of chromosomes 21, and for this reason the syndrome is also called trisomy 21. Regarding monosomies, the most characteristic case is Turner syndrome or X0. People showing Turner syndrome are women, who lack one X chromosome. Multifactorial genetic diseases Multifactorial genetic diseases may arise from the interaction of genes with a combination of environmental factors and lifestyle. Usually, there are many implicating genes in the determination of a disease and not just one. Such examples are various cancer types, heart conditions, Alzheimer disease and diabetes. In these cases, a genetic analysis can relief, provide therapy or help in the prevention of a disease. Moreover, information of a genetic examination can help other family members as well, since there may be predisposition to a genetic disease. Such an example is breast cancer. Women with a family history of breast cancer are advised to check for BRCA1 and BRCA2 genes, in order to evaluate the chances of showing the disease. A positive result in a genetic analysis of this kind may seem portentous in the beginning, but the right consultation can lead to prevention and prompt therapy.<\|endoftext\|>	4
2,132	<meta http-equiv="refresh" content="1; url=/nojavascript/"> # Decimal Subtraction % Progress Practice Decimal Subtraction Progress % Decimal Subtraction After winning the Regional Championship, Ms. Sutter took the girls out for an ice cream sundae. While they were sitting at the table, the four girls began rethinking all of their times and talking about how much fun they had running. “It was great, but I was nervous too,” Uniqua said. “especially since I was the last runner.” “Yes, I can see that, but you did great,” Jessica complimented her. “It was a close race,” Tasha said. “Not that close,” Jessica said. “Sure it was,” Karin chimed in. “It was close between all three winning teams and it was a close call for us to beat last year’s time too.” “Are you sure?” Jessica questioned. “Sure I am, look at these times,” Karin said writing on a napkin. 53.87 Last year’s team 53.73 Our time 53.70 $2^{nd}$ place 53.68 $3^{rd}$ place “You can figure this out girls, by doing the math,” Ms. Sutter said smiling. Ms. Sutter is correct. Subtracting decimals is a way to figure out the difference between winning and coming in second or third. What is the difference between last year’s time and this year’s? What is the difference between second and first? What is the difference between first and third? The best way to figure out these problems is to subtract the decimals. This Concept is all about subtracting decimals. By the time you are finished, you will be able to figure out the differences as well. ### Guidance To compare two values, or find the difference in two values, or find out “how many more” we subtract. Anytime we want to find a “left over” value or a “less than” value we subtract. Anytime that you see these key words you can be sure that subtraction is necessary. Previously we worked on how to add decimals and estimate sums. Now it is time to learn how to subtract decimals and estimate differences. How do we subtract decimals? We subtract decimals the same way we subtract whole numbers—with special care to place value. For instance, when you subtract 571 from 2,462 you make sure to line up the different place values, so that the thousands are subtracted from thousands, hundreds are subtracted from hundreds, and so on. $& \ 2,462\\& \underline{- \ \ 571}\\& \quad 1891$ When you subtract two decimals, we do the same thing. We line up the numbers according to place value and use the decimal points as a starting point. If there are missing digits, we can add in zeros to help us keep our work straight. $& \ \ 18.98\\& \underline{- \ 4.50}\\& \ \ 14.48$ Notice that the decimal points are lined up and then each digit is matched according to place value. You can see that the zero in red was added to help us keep each digit in the correct location. When is it useful to use rounding? Rounding is useful when estimating or you could think of it as times when you need an approximate answer not an exact one. What kinds of situations are alright to round? One time that we can use rounding is if the decimals are too long for pencil calculations. An approximate answer is sufficient. Also, if the problem itself tells us that we don’t need an exact answer. The question might use words like close to or approximate. In these cases, we round the decimals before we subtract. Look for clues in the problem to tell you whether or not to round before subtracting. If you are asked to round, make sure you round to the right place! Round the numbers to the nearest tenth then find the difference, 72.953 - 52.418 This problem asks us to round each number to the tenth place before subtracting. Underline the number we are rounding to and bold or circle the number directly to the right of it. We are rounding to the tenth place, so round to the place directly to the right of the decimal place. The bolded number in the hundredths place, is the one to look at when deciding to round up or down. 72. 9 5 3 $\rightarrow$ rounded to the tenth place $\rightarrow$ 73.0 52. 4 1 8 $\rightarrow$ rounded to the tenth place $\rightarrow$ 52.4 Now that the numbers are rounded, we line up the decimal places, and subtract. $& \quad 73.0\\& \underline{- \; 52.4}\\& \quad 20.6$ While rounding doesn’t give us an exact answer, you can see that it simplifies the subtraction! Now it's your turn to try a few. Find each difference. #### Example A $5.674 - 2.5 = \underline{\;\;\;\;\;\;\;\;}$ Solution: $3.174$ #### Example B Take $5.67$ from $12.378$ Solution: $6.708$ #### Example C Round to the nearest tenth and then subtract, $8.356 - 1.258$ Solution: $8.4 - 1.3 = 7.1$ Now back to the original problem. After winning the Regional Championship, Ms. Sutter took the girls out for an ice cream sundae. While they were sitting at the table, the four girls began rethinking all of their times and talking about how much fun they had running. “It was great, but I was nervous too,” Uniqua said. “especially since I was the last runner.” “Yes, I can see that, but you did great,” Jessica complimented her. “It was a close race,” Tasha said. “Not that close,” Jessica said. “Sure it was,” Karin chimed in. “It was close between all three winning teams and it was a close call for us to beat last year’s time too.” “Are you sure?” Jessica questioned. “Sure I am, look at these times,” Karin said writing on a napkin. 53.87 Last year’s team 53.73 Our time 53.70 $2^{nd}$ place 53.68 $3^{rd}$ place “You can figure this out girls, by doing the math,” Ms. Sutter said smiling. Ms. Sutter is correct. Subtracting decimals is a way to figure out the difference between winning and coming in second or third. What is the difference between last year’s time and this year’s? What is the difference between second and first? What is the difference between first and third? First, we find the difference between last year’s time and this year’s time. We subtract one time from the other. We line up our decimal points and subtract. $& \quad 53.87\\& \underline{- \; 53.73}\\& \qquad .14$ The girls beat the previous year’s time by fourteen hundredths of a second. Wow, that is close! Next, we look at the times between first and second place. $& \quad 53.73\\& \underline{- \; 53.70}\\& \qquad .03$ The difference between first and second place was very close! The girls only won by three hundredths of a second. Finally, we can look at the difference between the first and third place finishes. $53.73 -53.68 = .05$ Wow! The first three teams had very close finishes! ### Guided Practice Here is one for you to try on your own. Subtract $4.56 - 2.37$ Answer To subtract, we simply line up the digits according to place value and subtract. $2.19$ This is our answer. ### Explore More Directions: Find each difference. 1. 6.57 - 5.75 2. .0826 - .044 3. 19.315 - 6.8116 4. 2056.04 - 2044.1 5. 303.45 - 112.05 6. 16.576 - 8.43 7. 199.2 - 123.45 8. 1.0009 - .234 9. 789.12 - .876 10. 102.03 - .27 Directions: Find the difference after rounding each decimal to the nearest hundredth. 11. 63.385 - 50.508 12. .535 - .361 13. 747.005 - 47.035 14. .882 - .546 15. .9887 - .0245 ### Vocabulary Language: English Decimal Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths). Decimal point Decimal point A decimal point is a period that separates the complete units from the fractional parts in a decimal number. Difference Difference The result of a subtraction operation is called a difference. Estimate Estimate To estimate is to find an approximate answer that is reasonable or makes sense given the problem. Magnitude Magnitude The magnitude of a number is the size of a number without respect to its sign. The number -35.6 has a magnitude of 35.6. Place Value Place Value The value of given a digit in a multi-digit number that is indicated by the place or position of the digit. Rounding Rounding Rounding is reducing the number of non-zero digits in a number while keeping the overall value of the number similar. ### Explore More Sign in to explore more, including practice questions and solutions for Decimal Subtraction. Please wait... Please wait...<\|endoftext\|>	4.65625

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