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317	# 2016 AMC 8 Problems/Problem 4 ## Problem When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy? $\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$ ## Solution When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes $= 3\times60 + 30$ minutes $= 210$ minutes, thus running $\frac{210}{15} = 14$ minutes per mile. Now that he is an old man, he can walk $10$ miles in $4$ hours $= 4 \times 60$ minutes $= 240$ minutes, thus walking $\frac{240}{10} = 24$ minutes per mile. Therefore, it takes him $\boxed{\textbf{(B)}\ 10}$ minutes longer to walk a mile now compared to when he was a boy. ~CHECKMATE2021 ## Video Solution (THINKING CREATIVELY!!!) ~Education, the Study of Everything ~savannahsolver<\|endoftext\|>	4.46875
428	What are work rights?: The parable is familiar: A poor woman says to two fishermen, “I’m hungry. Can you help?” The first gives her a fish. The second says, “I’ll teach you to fish.” Yet, if our poor woman is a refugee who fled her home in order to keep her family safe from war or persecution… she says, “I know how to fish. I’m just not allowed to.” Being able to fish, having the pertinent work permit, is only the tip of the iceberg when addressing refugee work rights. In addition to that, safety while fishing and fair share of the revenue from the fish market are two more conditions that fall under the umbrella of work rights. Refugee work rights are the laws and policies that protect refugees in entering and participating in the labor economy. Access to work complemented by labor rights protections decide whether refugees can thrive in the host country. Work rights allow refugees to: - Secure lawful work without discrimination on the basis of their refugee status; - Access labor protections that safeguard them from exploitation or wage theft; - Earn a fair wage - Work in a safe environment “Work is a start. It is a start to changing one’s life…Work gives a person the opportunity to educate their children, to feed them and to have a home.” – Alberto, Colombian Refugee Why work rights matter: Denying refugees the right to work forces them to sacrifice economic independence and rely on handouts for survival. As a result, refugees endure isolation, loss of confidence and erosion of skills. Without a job or income, refugees are forced to subsist as the underclass and stay dependent on social welfare, even though most are capable and eager to support themselves. In contrast, self-sufficient refugees who have the opportunity to work lawfully provide economic and social contributions to their host communities and countries. They foster the potential to rejuvenate communities, expand markets, import new skills and create new jobs to provide opportunities for others.<\|endoftext\|>	3.71875
1,027	Let’s talk about the beautiful Gulf of California, which exists in all of its beauty because of the flow of freshwater from the Colorado River. A flow that is reaching the Gulf less and less nowadays. You see, since the damming of the Colorado River the amount of water that reaches the mouth of the river in Mexico has dropped, and in certain years no water reaches the mouth. That’s a problem. The Gulf of California is important for a number of reasons. For 891 fish, 181 bird, 34 marine mammal, and seven marine reptile species this is where they live, reproduce, and feed making this one of the Western Hemisphere’s most diverse seas. Because of this it is also where over 60% of the fish and 90% of the shrimp caught by Mexico are taken in, but the number of fish caught every year is dwindling (this can also be attributed, in part, to overfishing). The Gulf is also very important to Mexico’s tourism industry with people going there to sport fish, scuba dive, and relax on the beautiful beaches. But all of this beauty and diversity is at risk because of what is going on upstream in the US. So what’s going on upstream? Dams. Dams hold back the two things that the Gulf needs to survive; water and sediment. Historically, the Colorado River deposited between 77 and 91 million tons of sediment into the Gulf every year. This sediment is the nourishment that is needed by plants and small animals to survive. But today most of the sediment is caught behind the walls of the Hoover Dam and the Glen Canyon Dam, never making it anywhere near the Gulf. The waters of the Colorado River are also held back by the dams. This by itself reduces the flow, but there are other factors such as evaporation of the lakes formed behind the dams that also lead to reduced flow. A report by the US Bureau of Reclamation says that the loss from evaporation alone is over 15%. The fact that water isn’t reaching the Gulf leads to another problem: salinity. If the system was working correctly the northern part of the Gulf would be less saline than the southern part due to the freshwater flowing in from the Colorado, but it’s not. Saline water has negatively impacted the habitats of many plant and animal species that thrived here for centuries while the flow was natural. Also, the salinity of the lower Colorado River is changing due to lack of flow. In its natural state the lower Colorado river had a salt content of 50 ppm, but by 1960 that had shot up to over 2,000 ppm. This is causing millions of dollars in damage to crops that use this water, both in the US and in Mexico. While there’s little that can be done upstream short of taking down the dams (which won’t happen) conservation efforts in the Gulf have been steadily increasing in recent years by organizations such as The Nature Conservatory and the World Wildlife Fund, but they’ve been faced with challenges. One is that the fishing industry doesn’t want to lose any money, and they fear conservation means they’ll be able to catch less. Conservation groups realize this and try to come up with economically viable solutions, but most of the time these solutions are still defeated by the fishing industry. What the industry doesn’t understand, or isn’t thinking about, is that if they don’t do something now they won’t have any fish to catch at all. Another problem that the Mexican government faces is how to enforce laws that are put into place to help with conservation. The coastline of the gulf is thousands of miles long, and it takes a lot of resources to monitor the area, and enforce any laws that are broken. Also, the Mexican government isn’t at a consensus as to how important conservation of this area is, and so conservation laws are often not passed. This doesn’t make any sense especially since the Mexican government and businesses want to develop the gulf as a tourist destination (something that I’m sure will also have negative impacts on the environment). So what happens now? Well, that’s a good question that I don’t have an answer to. The dams are staying, and as the climate warms I would bet that the amount of water released isn’t going up. People have thrown out ideas of using treated water from treatment plants to increase the flow of the river, but that idea hasn’t gained much ground because of a legitimate fear of what will still be in the treated water. Our best hope is that conservationist can gain ground with the people, government, and industry in Mexico and make some positive impact. But with the water becoming more and more saline I don’t know how much conservation is going to do. If the plants and animals can’t survive in the region because it’s too salty what are you conserving?<\|endoftext\|>	3.796875
639	Aim To develop information retrieval, listening & observing, scientific reading, data representation, scientific writing and knowledge presentation. Activities Projects inPhysical Science (chemistry, geology, astronomy and physics), Applied Science (Health & Engineering), Battle of the Scientists (technology & inventions). Earth and space science (ESS) connect systems. Earth and space science explores the interconnections between the land, ocean, atmosphere, and life of our planet. These include the cycles of water, carbon, rock, and other materials that continuously shape, influence, and sustain Earth and its inhabitants. Physical science is the study of non-living things including: chemistry, geology, astronomy and physics. An example of physical science is a course teaching the concepts of energy and gravity. Applied science is a discipline that is used to apply existing scientific knowledge to develop more practical applications, for example: technology or inventions. In natural science, basic science (or pure science) is used to develop information to explain phenomena in the natural world. Construction Blocks – Build It Up! Best suited for children between the ages of 2-12 Years. Blocks have been the most basic brain games for kids since the beginning of time and have remained a constant in the ‘toy-sphere’ — and there’s a good reason why. Expose your child to blocks of different colors and sizes — and that’s it! Let your child explore the blocks and let their imagination run wild. All aspects of your child’s development are exposed including shape/ color recognition, creativity, spatial awareness, and so much more. You can start off with basic color and shape blocks for younger children and then upgrade to Legos or abstract building blocks for older children. Create simple patterns with blocks, have your toddler try to copy the patterns. This is a simple way to help your child observe patterns. Battle of the Scientists This is where applied science the application of existing scientific knowledge to practical applications, like technology or inventions. Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatability of findings are used to try to ensure the validity of scientific advances. These topics range from the areas such as, Engineering, Aerospace, Agriculture, Civil Computer science, Healthcare cutting across from Medicine, Veterinary, Dentistry, Midwifery, Pharmacy and Nursing. Other projects include - Avoiding Disaster: The Right Bridge Design - Holding Power of Nails - How Does the Ratio of Sand to Cement Affect the Strength of Concrete? - Shapes with Straws - Solving a ‘Windy’ Problem - Set Your Table for a Sweet and Sticky Earthquake Shake - How Do You Make the ‘Best’ Cookie? - Converting Oil into Clean Fuel - Interpreting Area Data from Maps vs. Graphs: An Experiment in Visual Perception - Motion After-Effects in Vision - Testing the Accuracy of Eyewitness Testimony - Time is Money<\|endoftext\|>	3.796875
419	# How do you find an equation of variation: y varies inversely as the square of x, and y = 0.15 when x= 0.1? Apr 5, 2016 $y = \frac{0.0015}{x} ^ 2$ Or if you prefer $y = \frac{15}{10000 {x}^{2}}$ Or as scientific notation: $y = \frac{15}{{x}^{2}} \times {10}^{- 4}$ #### Explanation: Splitting the question down into its component parts: y varies inversely as: ->y=1/? the square of ->y=1/(?^2) $x \text{ } \to y = \frac{1}{{x}^{2}}$ But we need a constant of variation Let the constant of variation be $k$ then we have: $y = k \times \frac{1}{{x}^{2}}$ '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $\textcolor{b l u e}{\text{Determine the value of } k}$ Known condition: at $y = 0.15 \text{; } x = 0.1$ So by substitution we have $0.15 = k \times \frac{1}{{\left(0.1\right)}^{2}}$ $\implies k = 0.15 \times {\left(0.1\right)}^{2}$ $k = 0.15 \times 0.01$ To calculate this directly think of $0.01 \text{ as } \frac{1}{100}$ Then we have: $k = \frac{0.15}{100} = 0.0015$ So the equation becomes: $y = \frac{0.0015}{x} ^ 2$ Or if you prefer $y = \frac{15}{10000 {x}^{2}}$ '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\|endoftext\|>	4.53125
333	Pupils work out how adventurers can maximise the area in which they can dig for gold given a fixed perimeter. - National Curriculum levels 4 to 6 (can be extended up to level 8) - 30 minutes to 1 hour - Ruler, pencil, calculator and 1 cm squared paper. It may also be helpful to make available some graph paper. Key Processes involved - Draw various rectangles with a given perimeters. - Find areas of rectangles. - Interpreting and evaluating - Consider different rectangles while trying to maximise the area. Find that a square is the best. Go on to find that, if the adventurers work together, they can get a bigger area each. - Explain how they know this is the best arrangement and that working together is best. Check that pupils understand the context, for example with questions such as: - How do you think the adventurers would mark out their plot of land? - How should they share a plot that two of them had marked out? - The adventurers need to understand your thinking, so show your findings in an organised way. Pupils can tackle this task in different ways, but they might be expected to: - draw rectangles and find their areas - collect results in a table or ordered list - find a pattern in their data The work could be extended to: - finding patterns in their results and finding an nth term - drawing a graph to clarify their results - finding the areas of other shapes This would take the content to National Curriculum levels 6 - 8.<\|endoftext\|>	4.40625
2,147	Categories: Class 10 # What is Quadrilateral? Definition,Types, Properties, Examples A polygon with four sides, four angles, and four vertices is called a quadrilateral. As suggested by the name, it always has four sides, four angles, and four vertices. The quadrilateral has angles at each of its four vertices, or corners. The sum of 4 angles is always 360 degrees. In the below figure.ABCD is a quadrilateral. • The quadrilateral ABCD’s four angles are ∠A, ∠B, ∠C, and ∠D • The four sides of the quadrilateral ABCD are AB, BC, CD, and DA. • The quadrilateral ABCD’s four vertices are A, B, C, and D. • The quadrilateral ABCD’s two diagonals are AC and BD. A quadrilateral is a two-dimensional shape that has four sides, four angles, and four vertices.A quadrilateral does not always have to have equal lengths on each of its four sides. As a result, depending on the sides and angles, we have several types of quadrilaterals. There are four different types of quadrilaterals: There are many different types of quadrilaterals, such as 1. Trapezium 2. Parallelogram 3. Squares 4. Rectangle 5. Rhombus 6. Kite The following quadrilaterals each have unique characteristics. Nevertheless, all quadrilaterals have a few characteristics. They are listed below. 2. They have four vertices. 3. There are two diagonals in each 4. They have four interior angles. 5. The Sum of all interior angles of a quadrilateral is 360° The basic classification of the quadrilaterals is between convex and concave quadrilaterals. Types of quadrilaterals are separated into different groups based on their properties. These quadrilaterals that are concave and convex can be divided further into their subgroups. Concave Quadrilaterals: Concave quadrilaterals are polygons with four sides and an interior angle larger than 180. A convex quadrilateral’s two diagonals are included in the enclosed figure.A Dart is an example of a concave quadrilateral. Convex Quadrilaterals: A four-sided polygon with inner angles that are less than 180 degrees each is referred to as a convex quadrilateral. Squares, rectangles, parallelograms, rhombuses, kites, and trapezoids are examples of convex quadrilaterals (trapezium). ## Types of Quadrilaterals and its Properties Let’s take a look at the various types of quadrilateral and their properties with the figure. ### Parallelogram A quadrilateral with two sets of parallel sides is known as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Properties of Parallelogram: • The opposing sides are equal and parallel. • Angles on either side are equal. • The two diagonals of the parallelogram are split in two. • The parallelogram is divided into two congruent triangles by each diagonal. • A parallelogram’s total square sum and the total square sum of its diagonals are equivalent. Also known as the parallelogram law. ### Rhombus A rhombus is a quadrilateral with opposite sides that are parallel to one another and equal opposite angles. A rhombus has equal lengths on all sides, making it an equilateral quadrilateral. Properties of Rhombus: • A rhombus has an equal number of sides. • Every Rhombus has two diagonals that join the opposite vertices in pairs. • A rhombus’ diagonals will always cut each other at a 90-degree angle. • A rhombus is divided into four right-angled congruent triangles by its two diagonals. • Any two parallel or following angles added together equal 180 degrees. ### Trapezium The trapezium is a quadrilateral in two dimensions with a single pair of parallel opposite sides. The bases and legs of a trapezium are its parallel and non-parallel sides, respectively. Properties of Trapezium • A trapezium has four sides, four vertices, and four angles. • In A trapezium two parallel sides are known as bases and the other two non-parallel sides are called legs. • Trapezium has one set of opposite sides which is parallel to the other. • In a trapezium, the sum of all four internal angles is always 360°. ### Rectangle A quadrilateral with equal angles and parallel opposite sides is known as a rectangle.Each rectangle shape has two dimensions: length and width. All of a rectangle’s angles are equal to 90 degrees, and its opposite sides are equal and parallel to one another. Properties of Rectangle: • Each of a rectangle’s opposing sides is equal and parallel to the other. • At each vertex, a rectangle’s inner angle is 90°. • The sum of all interior angles is 360°. • Each of the diagonals is divided in half. • The diagonals are equal in Length ### Square A quadrilateral with four equal sides is called a square. Equal sides and internal angles that are both 90 degrees distinguish each square shape. Properties of Square: • A Square has four sides and four vertices. • The square’s four sides are equal to one another. • Each vertex of a square has a 90° internal angle. • The Sum of all interior angles is 360°. • A square’s diagonals are parallel to one another. • The diagonals are all the same length. ### Kite A kite is a quadrilateral that has two sets of adjacent, equal-length sides. A kite has four vertices, four sides, and four angles. Properties of Kite • A kite has two adjacent pairs of equal sides. • It has a single pair of equal, obtuse, opposite angles. • In the intersection of the two angles, the two sides are equally sized. • It has two diagonals that right-angle intersect with one another. • The diagonals are parallel with one another. • The longer diagonal splits the two opposing angles in half. • A kite has symmetry along its main diagonal. There are two fundamental quadrilateral formulas, which are as follows: The area of the Quadrilateral refers to the total amount of space that is enclosed by a quadrilateral. Below is a list of the different types of quadrilaterals area formulas: Types of Quadrilateral Area Formula Parallelogram Base x Height Rectangle Length x Width Square Side x Side Rhombus (1/2) x Diagonal 1 x Diagonal 2 Trapezium ½ (Sum of parallel sides)(Distance between parallel sides) Kite 1/2 x Diagonal 1 x Diagonal 2 In order to determine the path that encircles a quadrilateral, we must add the lengths and widths of each of its four sides, with lengths being the longest and widths being the shortest. Perimeter is calculated using the formula Perimeter = Length + Length + Width + Width. Here is a list of Perimeter formulas for different types of quadrilaterals Types of Quadrilateral Formula of Perimeter Square 4 x Side Rectangle 2(Length + Breadth) Parallelogram 2(Base + Side) Rhombus 4 x Side Trapezium Sum of all four sides Kite 2 (a + b), a and b are adjacent pairs A polygon with four sides, four angles, and four vertices is called a quadrilateral. As suggested by the name, it always has four sides, four angles, and four vertices. A polygon with four sides, four angles, and four vertices is called a quadrilateral. The Latin words Quadri, which means four, and latus, which means side, were combined to create the English word quadrilateral. The first mathematician to present the formula for the area of a cyclic quadrilateral was Brahmagupta (ad 628) His explanation of how to locate a cyclic quadrilateral with rational sides is the first of its kind. 2. They have four vertices. 3. There are two diagonals in each ## FAQs A polygon with four sides, four angles, and four vertices is called a quadrilateral. As suggested by the name, it always has four sides, four angles, and four vertices. ### Why are they called quadrilaterals? A polygon with four sides, four angles, and four vertices is called a quadrilateral. The Latin words Quadri, which means four, and latus, which means side, were combined to create the English word quadrilateral. The first mathematician to present the formula for the area of a cyclic quadrilateral was Brahmagupta (ad 628) His explanation of how to locate a cyclic quadrilateral with rational sides is the first of its kind. They have four vertices. There are two diagonals in each Share ## Science Sample Paper Class 10 2022- 2023 with Solutions, PDF Science Sample Paper Class 10 2023 CBSE Class 10 Science Sample Question Paper 2023 is… 14 hours ago ## CBSE Class 10 Science Additional Practice Question Paper 2023 Class 10 Science Additional Practice Question Paper 2023 The Central Board Board of Secondary Education… 15 hours ago ## JEE Main Result 2023, Session 1 Cut Off Marks, Scorecard, Rank List JEE Main Result 2023 As per the reports, National Testing Agency will soon release the… 20 hours ago ## JEE Main Answer key 2023 for Session 1, PDF Out at jeemain.nta.nic.in JEE Main Answer Key 2023 The National Testing Agency has released the Provisional JEE Main… 1 day ago ## CBSE Result 2023 Live: 10th & 12th Score Card Updates CBSE Result 2023 CBSE Result 2023:The Central Board of Secondary Education will release the CBSE… 1 day ago ## JEE Mains 2023 Exam Dates, Notification, Fees For Jan & June Online Applications for JEE Main 2023 are Started, The form's eligibility can be verified on… 1 day ago<\|endoftext\|>	4.625
1,478	A brief history of The Fukushima PrefecturePrior to the 4th century AD, the islands that are now Japan were controlled by individual clans -close-knit groups of people related to one another biologically or by marriage. Around the 4th century A. D, Japan’s first imperial government, The Yamato Court, started in western Japan, and by the mid-7th century, all of Japan was under control of the national government. However the clans were still striving for power The Tokugawa Shogunate Clan, who were still trying to rule the whole country from Edo (now Tokyo), placed 2 related clans in the Fukushima area in an effort to maintain control. But the Tokugawa Shogunate’s power declined, and, by the mid-9th century, Japan was in a severe political and economic crisis. In the mid-15th century through the mid-16th century, the central military government collapsed during what became known as The Warring States Period. The clan known as The Date Clan (based in the Fukushima Basin) and the clan known as The Ashina Clan (based in the Aizu Basin(, rose up as strong powers in the northeastern part of Japan. Both basins are now part of the Fukushima Prefecture. In the late 19th century, Japan finally unified under an imperial government, and in 1871 created a prefectural system, which replaced the clan system. A prefecture is the highest administrative division below the Japanese federal government – similar to our states. Originally, the Fukushima area were divided into three prefectures, which were soon merged to form today’s Fukushima Prefecture. Culturally during this period, the Fukushima area was recognized as one of the centers of Buddhist culture in northeastern Japan from the 9th century through the l2th century. After the formation of the Japanese impreial government, Japan began to industrialize, and the Fukushima Prefecture flourished – particularly thanks to their exports of raw silk and coal. No history of Japan would be complete without a brief mention of the Samurai. The Samurai we think of today, the brave warrior – adept with his sword, an eminent horseman, and a master in martial fighting techniques – emerged around the 10th century. Before that, the term “Samurai” actually meant a mid-to-low-ranking court administrator. From the 10th century to the 19th century, the Samurai served both as interior military “police” and as defenders against foreign invaders. They were known and admired (or feared, depending on whose side you were on) for their loyalty to their chosen ruler and for their sense of honor and duty. Being family members of various clans, the Samurai were often warring against other Samurai. When Japan united as an empire, the Samurai became imperial warriors with a social status on par with the aristocracy. By the mid-19th century, as Western influences began to prevail, the Japanese started to modernize their armed forces. The Samurai, while still the ruling class of Japan, became modern soldiers – involved in administrative government positions, both local and federal. Still, the samurai ideals of honor, loyalty and patriotism live on in modern Japanese culture, and the renowned martial fighting techniques of the Samurai live on throughout the world as the sports of martial arts. Modern Fukushima: Prefecture and CityThe city of Fukushima is the capital city of the Fukushima Prefecture. As the third largest of Japan’s 47 prefectures, Fukushima Prefecture stretches nearly 100 miles from the Pacific ocean through the ancient mountains in the northeastern part of Honshu, the main Japanese island. Fukushima was not well known to the outside world before the March 11, 2011, however the earthquake and tsunami that caused the subsequent nuclear melt down at the Fukushima Daiichi Nuclear Power plant most empathically put Fukushima on the world-map. The prefecture of Fukushima is divided up into three regions: Hama-dori by the east coast, Naka-dori in the middle, and Aizu to the west. The Fukushima Daiichi Nuclear Power plant is located in Hama-dori. After the disaster, restricted zones were established, based on levels of radioactivity. The main restricted zone around the Daiichi Nuclear Power plant is still (as of 2014) off limits. A good 90% of the Fukushima Prefecture has beem deemed safe, both to live in and to visit. The restricted zone around the Fukushima Daiichi Nuclear Power plant makes up less than 10 % of the Fukushima Prefecture. Fukushima city itself is nearly 40 miles from the restricted zone. Attractions and Culture:Within Japan, the Fukushima Prefecture has long been very well known, and not just for it’s nuclear power plants. Activities and spectacular sights in the Fukushima Prefecture are many and varied. For example there is the more-than-one-thousand year old Enzoji Temple, the traditional Japanese Paper Museum, where you can see artisans making hand-made paper, the Sukagawa Shakadogawa Fireworks Festival which includes the hanabi-e-maki performance (a contest which combines music and fireworks), and some of Japan’s most beautiful places for the annual viewing of cherry blossoms in mid to late April. There are also many camping sites, hiking trails, hot springs, ski & snowboard parks, golf courses, and race courses. All of it surrounded in amazing nature, including majestic landscapes shaped by ancient volcanic activity. The many restaurants in Fukushima city range from fine dining over more casual to fast food. (Yes, they have that, too!) Fukushima city is well known as a hotspot for delicious ramen noodles. At the yearly Fukushima Ramen Show (early to mid May), you can buy real Ramen, original Japanese style, and watch it being cooked. Fukushima Tourism – Post-DisasterRight after the March 2011 nuclear disaster, Japan’s tourist industry declined by more than 50%. All of Japan saw this drastic decline, not just the Fukushima Prefecture.It was more than a year before tourism began to pick up again, and then it was mostly Japanese tourists. In an attempt to revive the industry, Japan’s tourist industry began to offer numerous discounts, including reduced airfare and reduced fees at some attractions. All the while keeping visitors informed with reports of falling radiation levels in the different restricted zones and reports of normal radiation levels in most of Japan While tourism in the Fukushima Prefecture is still lagging, the safe areas are beginning to attract tourists again. Foreign visitors have slowly started to come back, and now Japan’s tourist industry is finally on the way back to pre-disaster levels. At least outside of the main restricted zone. All activity around the disaster area, including all tourist activities, remain at a perfect stand-still. As of now, the towns, villages, and farms are still empty of people. Nobody lives there – even the looters have fled. For the Japanese tourist industry, this means a continued loss of many previously-popular tourist attractions. The restricted zone has many gorgeous beaches, historical buildings, and sacred shrines – none of which are accessible now.<\|endoftext\|>	3.71875
725	Please get in touch with us if you: 1. Have any suggestions 2. Have any questions 3. Have found an error/bug 4. Anything else ... We can be reached at # Fourteen% off twenty three Dollars How to calculate fourteen % off \$twenty three. How to figure out percentages off a price. Using this calculator you will find that the amount after the discount is \$19.78. ### Inputs Original Price of the Item: \$ Discount Percent (% off): % ### Results Amount Saved (Discount): \$ Sale / Discounted Price: \$ Using this calculator you can find the discount value and the discounted price of an item. It is helpfull to answer questions like: • 1) What is fourteen percent (%) off \$twenty three? • 2) How much will you pay for an item where the original price before discount is \$ twenty three when discounted fourteen percent (%)? What is the final or sale price? • 3) \$3.22 is what percent off \$23? • See how to solve these questions just after the Percent-off Calculator (or Discount) below. ## how to work out discounts - Step by Step To calculate discount it is ease by using the following equations: • Amount Saved = Orig. Price x Discount % / 100 (a) • • Sale Price = Orig. Price - Amount Saved (b) Now, let's solve the questions stated above: ### 1) What is 14 percent off \$23? Find the amount of discount. Suppose you have a Kohls coupon of \$23 and you want to know how much you will save for an item if the discount is 14. Solution: Replacing the given values in formula (a) we have: Amount Saved = Original Price x Discount in Percent / 100. So, Amount Saved = 23 x 14 / 100 Amount Saved = 322 / 100 In other words, a 14% discount for a item with original price of \$23 is equal to \$3.22 (Amount Saved). Note that to find the amount saved, just multiply it by the percentage and divide by 100. ### 2) How much to pay for an item of \$23 when discounted 14 percent (%)? What is item's sale price? Suppose you have a L.L. Bean coupon of \$23 and you want to know the final or sale price if the discount is 14 percent. Using the formula (b) and replacing the given values: Sale Price = Original Price - Amount Saved. So, Sale Price = 23 - 3.22 This means the cost of the item to you is \$19.78. You will pay \$19.78 for a item with original price of \$23 when discounted 14%. In this example, if you buy an item at \$23 with 14% discount, you will pay 23 - 3.22 = \$19.78. ### 3) 3.22 is what percent off \$23? Using the formula (b) and replacing given values: Amount Saved = Original Price x Discount in Percent /100. So, 3.22 = 23 x Discount in Percent / 100 3.22 / 23 = Discount in Percent /100 100 x 3.22 / 23 = Discount in Percent 322 / 23 = Discount in Percent, or Discount in Percent = 14 (answer). To find more examples, just choose one at the bottom of this page.<\|endoftext\|>	4.4375
724	# Verifying trig identities help Verifying trig identities help can support pupils to understand the material and improve their grades. So let's get started! ## The Best Verifying trig identities help Verifying trig identities help can be a helpful tool for these students. In other words, x would be equal to two (2). However, if x represented one third of a cup of coffee, then solving for x would mean finding the value of the whole cup. In this case, x would be equal to three (3). The key is to remember that, no matter what the size of the fraction, solving for x always means finding the value of the whole. With a little practice, solving for x with fractions can become second nature. Pythagoras’ theorem states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of an adjacent side. The Pythagorean theorem solver can be used to find this value by calculating the length of one leg in terms of the lengths of the other two. The Pythagorean theorem solver can be used to solve simple right triangles with legs and hypotenuse lengths as well as right triangles whose sides are not all equal (other than their length). It can even be used to find values for right triangles whose shape has been distorted, such as when one side has been extended or shortened. The Pythagorean theorem solver can also measure angles from which it can be determined whether or not a given triangle is a right triangle. When inputting values into the Pythagorean theorem solver, it is important to take into account any non-right triangle factors (such as non-integer sides or non-perfect squares) that may affect your results. Values for these factors should be added to your final answer before proceeding. Any math student worth their salt knows that equations can be a real pain to solve, especially when they involve more than one variable. Thankfully, there's a tool that can help: the variable equation solver. This online tool allows users to input an equation and see the results in real-time. Plus, it can handle equations with multiple variables, making it a real lifesaver for students who are struggling with algebra. So next time you're stuck on a math problem, be sure to give the variable equation solver a try. You might just be surprised at how helpful it can be. Hard math equations with answers are difficult to find. However, there are a few websites that have a compilation of hard math equations with answers. These websites have a variety of equations, ranging from algebra to calculus. In addition, the answers are provided for each equation. This is extremely helpful for students who are struggling with a particular equation. Hard math equations with answers can be very challenging, but by using these websites, students can get the help they need to succeed. ## We cover all types of math problems Better than math way. It actually shows me how to solve the problems and helps me learn, and I cannot express how much I love that. It was annoying when I had my unit of math that it didn't support and I had to use math way and figure out how to get the right answers. But I know their working on expanding the app. This app is great I highly recommend. Edith Wilson Ok so it's the most beautiful app, considering the fact that you don't have to pay anything. It's in any language, and so easy to use. And there is also a calculator with many other keys and letters, and I love it. Totally recommend Tabitha Flores<\|endoftext\|>	4.6875
870	# Proving/disproving that √7 - √2 is irrational [duplicate] It's been proven that √7 and √2 are irrational. However, I am not sure how to go about proving that √7 - √2. Is it an acceptable proof to just solve the equation which would prove/disprove the equation or as should the proof be done as a contrapositive, similar to how √7 and √2 are proven to be irrational. What would a valid proof/disproof of irrationality look like in this case? • One way is to find the equation with integer coefficients that $\,\sqrt{7}-\sqrt{2}\,$ satisfies, then prove that it has no rational roots. – dxiv Feb 8, 2018 at 5:42 • not quite sure what you mean by integer coefficients, could you provide an example? Feb 8, 2018 at 5:44 • Please use MathJax to format your posts. Feb 8, 2018 at 6:15 Suppose $\sqrt{7}-\sqrt{2}$ were rational; that is, suppose $$\sqrt{7}-\sqrt{2}=\frac{a}{b},$$ where $\text{gcd}(a,b)=1$. Multiply both sides of the equation by $\sqrt{7}+\sqrt{2}$ to obtain $$5=7-2=(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2}) = \frac{a}{b}(\sqrt{7}+\sqrt{2}).$$ Since $\frac{5b}{a}\in\mathbb{Q}$, $\sqrt{7}+\sqrt{2}$ is also a rational number. Since the sum of two rational numbers is rational, $$(\sqrt{7}-\sqrt{2}) + (\sqrt{7}+\sqrt{2}) = 2\sqrt{7}$$ is rational. So $\sqrt{7}$ is rational. This is a contradiction. Hint: suppose $\,\sqrt{7}-\sqrt{2}\,$ were rational, then so would be $\,\dfrac{5}{\sqrt{7}-\sqrt{2}}=\sqrt{7}+\sqrt{2}\,$, then so would be their difference $\,2 \sqrt{2}\,$. [ EDIT ] Following up on the previous comment: let $\,x=\sqrt{7}-\sqrt{2}\,$, then $\,x^2=9-2\sqrt{14}\,$, then $(x^2-9)^2=4 \cdot 14 \iff x^4 - 18 x^2 + 25 = 0\,$. But the latter equation has no rational roots, since by the rational root theorem the only such roots could be $\,\pm1, \pm5, \pm25\,$ which none work. • Not really a hint :) – user223391 Feb 8, 2018 at 5:46 • Your alt hint is not really a hint either :) – user223391 Feb 8, 2018 at 5:49 • @ZacharySelk Right about the alt, just dropped the "hint" misnomer. The first one still leaves at least something to work out, though ;-) – dxiv Feb 8, 2018 at 5:51 Suppose $\sqrt{a}-\sqrt{b} = r$ is rational. Squaring this, $a+b-2\sqrt{ab} = r^2$, so $\sqrt{ab}$ is rational. If $\sqrt{ab}$ is irrational, this can not hold. Therefore, if $\sqrt{ab}$ is irrational, so is $\sqrt{a}-\sqrt{b}$. Since $\sqrt{14}$ is irrational, so is $\sqrt{7}-\sqrt{2}$. Note that this works for $\sqrt{a}+\sqrt{b}$ also. Note 2: There are many proofs here that if $n$ is not a perfect square then $\sqrt{n}$ is irrational.<\|endoftext\|>	4.40625
91	General music students take a journey through music history. Beginning with the Middle Ages and continuing through the twenty-first century, students learn about the composers and instruments that help create a musical culture. Students study the music of different cultures and how these genres evolved into what we listen to today. Students also learn the basics of music theory. The construction of simple melodies and rhythms are explored each year, adding skills and vocabulary as students move through middle school.<\|endoftext\|>	3.765625
874	A while back we posted a holiday recipe for eggnog that explained how 7,500 years or so ago, humans in the region between the central Balkans and central Europe developed “lactase persistence.” According to a study by Professor Mark Thomas of University College London (UCL) Genetics, Evolution and Environment, “Most adults worldwide do not produce the enzyme lactase and so are unable to digest the milk sugar lactose. However, most Europeans continue to produce lactase throughout their life, a characteristic known as lactase persistence. In Europe, a single genetic change (13,910*T) is strongly associated with lactase persistence and appears to have given people with it a big survival advantage.”1 Domestication of animals and a rise in farming spurred the evolution of milk products, and humans adapted accordingly. Early protein remnants in clay vessels have been found in present-day Romania and Hungary dating back more than 7,000 years and attesting to the presence of dairy farming. Farms in England of 6,000 years ago give evidence of yogurt, butter and cheese production. Romans used goat and sheep milk for cheese, and Germanic and Celtic tribes drank abundant quantities of fresh milk from cattle. As populations migrated, this genetic trait became more widespread.2 Justin Cook, assistant professor of economics at the University of California-Merced, furthers the elaboration of the benefits of dairy by correlating lactose persistence with economic development, and by extension, the rise in later colonial explorations. Cook says the lactase persistent allele, or genetic variant, evolved along with the growth of dairy production, which conferred upon humans three benefits related to economic development: - Dairying represented a technological advance in “fixed resources,” that is, land and animals provided reliable resources to enable continued and increased sustenance. - The fats, proteins and other nutrients in milk were consistently available to farmers, improving overall health and resistance to illness, which in turn led to increased production and economic growth. - Milk production could have had the effect of increasing fertility, offering women another milk source for their infants and thus re-starting their fertility cycle that would have not been active while lactating.3 Thus, according to Cook, “A statistically strong and robust relationship is found between the fraction of a country’s population that is lactase persistent, or able to consume milk, and economic development in 1500 C.E., a period representative of the precolonization era. And given the high frequency of lactose tolerance associated with European countries, milk consumption may have contributed to Europe’s colonization of most of the world starting in the late 15th century.”4 Got milk I mean, that? So enjoy your ice cream, cheeses, yogurt, milk and all their variations. And be sure to give a nod to our ancestors’ quirk in genetics that thousands of years ago paved the way. Shrimp Avocado Salad Recipe courtesy of iFoodReal Prep Time: 15 minutes Total Time: 15 minutes Yield: 7 servings Method: No Cook Cuisine: American Ukrainian - 3/4 cup regular or Greek plain yogurt, 2+% fat - 2 tsp any light colour vinegar - 1 tsp garlic powder - 1/2 tsp salt - Ground black pepper, to taste Shrimp Avocado Salad: - 1 lb cooked frozen shrimp, thawed & drained - 1 pint grape tomatoes, cut in halves - 2 large bell peppers, chopped - 3 medium avocados, chopped - 1 (1 lb) long English cucumber, chopped - 1/2 cup cilantro, finely chopped - In a small bowl, add yogurt, vinegar, garlic powder, salt and pepper. Whisk with a fork and set aside. - Chop vegetables and add them to a large salad bowl. - Pour dressing on top and mix gently to combine. Serve chilled. Store: Refrigerate covered for up to 2 days (dressed is OK). If using thick Greek yogurt, thin it out with a few tbsp of water. Smaller size shrimp is great for this salad as it’s cheaper. If using large shrimp, cut in half.<\|endoftext\|>	3.921875
2,272	# The beta distribution as order statistics The beta distribution is mathematically defined using the beta function as discussed in the previous post. The beta distribution can also arise naturally from random sampling from the uniform distribution. This natural generation of the beta distribution leads to an interesting discussion of order statistics and non-parametric inference. _______________________________________________________________________________________________ Sampling from the Uniform Distribution Suppose $X_1,X_2,\cdots,X_n$ is a random sample from a continuous distribution. Rank the sample in ascending order: $Y_1. The statistic $Y_1$ is the least sample item (the minimum statistic). The statistic $Y_2$ is the second smallest sample item (the second order statistic), and so on. Of course, $Y_n$ is the maximum statistic. Since we are sampling from a continuous distribution, assume that there is no chance for a tie among the sample items $X_i$ or the order statistics $Y_i$. Let $F(x)$ and $f(x)$ be the cumulative distribution function and the density function of the continuous distribution from which the random sample is drawn. The following is the density function of the order statistic $Y_i$ where $1 \le i \le n$. $\displaystyle f_{Y_i}(y)=\frac{n!}{(i-1)! \ (n-i)!} \ F(y)^{i-1} \ f(y) \ [1-F(y)]^{n-i} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ To see how $(1)$ is derived, see the discussion here and here. When sampling from the uniform distribution on $(0,1)$, the order statistics have the beta distribution. Suppose that the distribution from which the random sample is drawn is the uniform distribution on the unit interval $(0,1)$. Then $F(x)=x$ and $f(x)=1$ for all $0. Then the density function in $(1)$ becomes the following: $\displaystyle f_{Y_i}(y)=\frac{n!}{(i-1)! \ (n-i)!} \ y^{i-1} \ [1-y]^{n-i} \ \ \ \ \ \ 0 The density in $(2)$ is a beta density function with $a=i$ and $b=n-i+1$. The following is the mean of the beta distribution described in $(2)$. $\displaystyle E(Y_i)=\frac{a}{a+b}=\frac{i}{n+1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$ To summarize, in a random sample of size $n$ drawn from a uniform distribution on $(0,1)$, the $i$th order statistic $Y_i$ has a beta distribution with parameters $a=i$ and $b=n-i+1$. The following table shows the information when $n=7$. $\displaystyle \begin{array}{ccccccc} \text{ } &\text{ } & \text{beta parameter } a & \text{ } & \text{beta parameter } b& \text{ } & E(Y_i) \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_1 &\text{ } & 1 & \text{ } & 7 & \text{ } & \displaystyle \frac{1}{8}\\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_2 &\text{ } & 2 & \text{ } & 6 & \text{ } & \displaystyle \frac{2}{8} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_3 &\text{ } & 3 & \text{ } & 5 & \text{ } & \displaystyle \frac{3}{8} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_4 &\text{ } & 4 & \text{ } & 4 & \text{ } & \displaystyle \frac{4}{8} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_5 &\text{ } & 5 & \text{ } & 3 & \text{ } & \displaystyle \frac{5}{8} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_6 &\text{ } & 6 & \text{ } & 2 & \text{ } & \displaystyle \frac{6}{8} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ Y_7 &\text{ } & 7 & \text{ } & 1 & \text{ } & \displaystyle \frac{7}{8} \\ \end{array}$ See here for a fuller discussion on the beta distribution. _______________________________________________________________________________________________ The Non-Parametric Angle In descriptive statistics, the sample statistics are used as point estimates for population parameters. For example, the sample mean can be used as an estimate for the population mean and the sample median is used as an estimate of the population median and so on. Such techniques are part of non-parametric statistics since there are no assumptions made about the probability distributions of the variables being assessed. Let’s focus on estimation of percentiles. We do not need to assume a population probability distribution. Simply generate the sample data from the population. Then rank the sample data from the smallest to the largest. Use the sample item in the “middle” to estimate the population median. Likewise, the population 75th percentile is estimated by the sample item that ranks higher than approximately 75% of the sample items and ranks below 25% of the sample items. And so on. In other words, order statistics can be used as estimates of population percentiles. This makes intuitive sense. The estimate will always be correct from the perspective of the sample. For example, the estimate of the 75th percentile is chosen to rank approximately above 75% of the sample. But will the estimate chosen this way rank above 75% of the population? The remainder of the post shows that the answer is yes. On average the estimate will rank above the appropriate percentage of the population. Thus the non-parametric approach of using order statistics as estimate of population percentiles makes mathematical sense as well. The estimate is “unbiased” in the sense that it is expected to be rank correctly among the population values. As before, $X_1,X_2,\cdots,X_n$ is the random sample and $Y_1 is the resulting ordered sample. We do not know the distribution from which the sample is obtained. To make the argument clear, let $f(x)$ be the density function and $F(x)$ be the CDF of the unknown population, respectively. We show that regardless of the probability distribution from which the sample is generated, the expected area under the density curve $f(x)$ and to the left of $Y_i$ is $\displaystyle \frac{i}{n+1}$. Thus the order statistic $Y_i$ is expected to be greater than $\displaystyle \biggl(100 \times \frac{i}{n+1}\biggr)$% of the population. This shows that it is mathematically justified to use the order statistics $Y_1 as estimates of the population percentiles. For example, if the sample size $n$ is 11, then the middle sample item is $Y_6$. Then the area under the density curve of the unknown population distribution and to the left of $Y_6$ is expected to be 6/12 = 0.5. So $Y_6$ is expected to be greater than 50% of the distribution, even though the form of the distribution is unknown. Note that $F(X)$ has a uniform distribution on the interval $(0,1)$ for any continuous random variable $X$. So $F(X_1),F(X_2),\cdots,F(X_n)$ is like a random sample drawn from the uniform distribution. Furthermore $F(Y_1) is an order sample from the uniform distribution since $F(x)$ as a CDF is an increasing function. Thus each item $F(Y_i)$ is an order statistic. Recall the result described in $(2)$. The order statistic $F(Y_i)$ has a beta distribution with $a=i$ and $b=n-i+1$. Then the mean of $F(Y_i)$ is $\displaystyle E[F(Y_i)]=\frac{a}{a+b}=\frac{i}{n+1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$ , which is the same as the mean in $(3)$. Note that the area under the density curve $f(x)$ and to the left of $Y_i$ can be conceptually expressed as $F(Y_i)$. The expected value of this area is the ratio indicated in $(4)$. This concludes the argument that when the order statistic $Y_i$ is used as an estimate of a certain population percentile, it is on average a correct estimate in that it is expected to rank above the correct percentage of the population. _______________________________________________________________________________________________ Remarks A subclass of the beta distribution can be naturally generated from random sampling of the uniform distribution, as described in $(2)$. The non-parametric approach is not only for producing point estimates . It can be used as an inference procedure as well. For example, a wildlife biologist may be interested in estimating the median weight of black bear in a certain region in Alaska. The procedure is to capture a sample of bears, sedate the bears and then take the weight measurements. Rank the sample. The middle sample item is then estimate of the population median weight of black bears. The ordered sample can also be used to form a confidence interval for the median bear weight. For an explanation on how to form such distribution-free confidence intervals, see here. _______________________________________________________________________________________________ $\copyright \ 2016 - \text{Dan Ma}$<\|endoftext\|>	4.46875
1,325	# How to Solve 'And' & 'Or' Compound Inequalities An error occurred trying to load this video. Try refreshing the page, or contact customer support. Coming up next: How to Solve and Graph One-Variable Inequalities ### You're on a roll. Keep up the good work! Replay Your next lesson will play in 10 seconds • 0:05 What Are Inequalities? • 2:21 How to Solve a… • 4:44 Important Notes • 6:35 Lesson Summary Want to watch this again later? Timeline Autoplay Autoplay Speed #### Recommended Lessons and Courses for You Lesson Transcript Instructor: Sarah Spitzig Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree. In this lesson, we will learn the difference between a conjunction and a disjunction and how to solve and graph 'and' and 'or' inequalities on the number line. ## What Are Inequalities? Sydney has an important math test coming up on Friday. She wants to study for her test for at least 30 minutes. Sydney's parents told her she needs to leave for swimming practice in 60 minutes. Therefore, she can study for no less than 30 minutes and no more than 60 minutes. This scenario can be written as a compound inequality. But what exactly does this mean? You can think of an inequality as an equation, except that the equals sign is replaced with a less than or greater than sign. We still need to solve the inequality just like you would an equation. The only difference is that instead of one answer that makes the equation true, like x = 3, there are many answers that make an inequality true, like x < 5. In this case, all numbers less than five would make the inequality true. • 2x + 5 = 7 is an equation because it has an equals sign. • 2x + 5 < 7 is an inequality because it has an inequality sign. A compound inequality is just more than one inequality that we want to solve at the same time. We can either use the word 'and' or 'or' to indicate if we are looking at the solution to both inequalities (and), or if we are looking at the solution to either one of the inequalities (or). x < 7 and x > -3, which can also be written as -3 < x < 7, is a compound inequality because it is two inequalities connected by the word 'and'. This is also known as a conjunction. In this case, we are looking for the solution to both inequalities. In other words, this solution satisfies both inequalities. x > 7 or x < -3 is a compound inequality, also known as a disjunction, because it is two inequalities connected by the word 'or'. In this case, we are looking for a solution to either one of the equations. Let's check back in with Sydney. We know she needs to study for at least 30 minutes, but less than 60. If we set this up as a compound inequality, it looks like this: x > 30 and x < 60, also written as 30 < x < 60. ## How to Solve a Compound Inequality #### Example 1 Let's take a look at the inequality 2 + x < 5 and -1 < 2 + x, which can also be written as -1 < 2 + x < 5. This is a compound inequality because it uses the word 'and.' Now let's go ahead and solve it. 1) Solve each part of the inequality separately. 2 + x < 5 and -1 < 2 + x In the first equation, 2 + x < 5, we need to subtract 2 from each side to get the variable by itself. We then get x < 3. In the second equation, -1 < 2 + x, we again subtract 2 from both sides. This gives us -3 < x. Our solution, then, is x < 3 and -3 < x, or -3 < x < 3. 2) Graph on the number line. Since this is a conjunction, the space between -3 and 3 is where the answer lies. In other words, any value between -3 and 3 satisfies this compound inequality. Remember Sydney? If we were to display her inequality on a number line, it would show that all numbers between 30 and 60 would be possible solutions. Meaning, she could study for 35, minutes, 42 minutes and so on. #### Example 2 But what if we are solving a disjunction? Let's take a look at the following inequality: 7 > 2x + 5 or 7 < 5x - 3. This time the word 'or' is used instead of the word 'and'. How do we solve this? 1) Solve each inequality: For 7 > 2x + 5, we subtract 5 from each side to get 2 > 2x. Divide each side by 2 and we get 1 > x. For 7 < 5x - 3, we add 3 to each side and get 10 < 5x. Divide each side by 5 and we have 2 < x. 2) Graph on the number line. Since this is a disjunction, any value greater than 2 and less than 1 is where the answer lies. All real numbers satisfy this compound inequality. ## Important Notes There are a few important points to keep in mind as you solve compound inequalities. To unlock this lesson you must be a Study.com Member. ### Register to view this lesson Are you a student or a teacher? ### Unlock Your Education #### See for yourself why 30 million people use Study.com ##### Become a Study.com member and start learning now. Back What teachers are saying about Study.com ### Earning College Credit Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.<\|endoftext\|>	4.59375
2,741	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. ## Algebra (all content) ### Course: Algebra (all content)>Unit 10 Lesson 1: Intro to polynomials # The parts of polynomial expressions Discover the magic of polynomials! Learn to identify terms, coefficients, and exponents in a polynomial. Understand that terms are the parts being added, coefficients are the numbers multiplying the powers of x, and exponents are the powers to which x is raised. Dive into the world of polynomials and make math fun! Created by Sal Khan and Monterey Institute for Technology and Education. ## Want to join the conversation? • Could someone explain to me what concept a polynomial represents? Why is it special? What can i do with it that I can't do with some other concept? What are the limits of what can be done with a polynomial, i.e. the constraints? Until I know what a polynomial actually is I can't move on to learning about its constituent parts. I can't find this information anywhere. It is almost as if nobody actually knows what a polynomial really is. Jumping in and telling me about its constituent parts or simply defining it with regards to these constituent parts is not what I'm looking for. A referential analogy would be great. For instance: a sine wave is nothing more and nothing less than a means of describing fluid oscillating motion as it appears in nature. Light, sound and electricity are all connected by the fluid oscillating nature of a sine wave. Since the sine wave is a fluid and non-linear type of motion it is a necessary component in the creation of and understanding of curved geometric shapes in the real or modeled world, i.e one complete oscillation of the sine wave is a sufficient component to help describe and/or create two-dimensional curved geometric shapes and two complete oscillations are a necessary component to help describe and/or create 3 dimensional curved geometric shapes. I understand that the above analogy may not be perfect. My mathematical skills are poorly developed at this point. I will give you that BUT it does help me to understand a fundamental principal and move forward having received an answer. Kind regards and thanks, Tony • Wow! What a fantastic question! Polynomials have been around for a long time, but the name polynomial has only been in use since around the 17th century. Polynomials evolved directly from word problems. Way, way back, farmers, economists and kings (that is to say, anyone involved in business) used to describe and solve their problems with words. So back then, what we would now call a polynomial equation would be written out using words, for example, “2 plots of carrots, 3 plots of peas and one plot of cabbage are sold for 50 pieces of silver.” Today, we would write that with the polynomial 2x + 3y + z = 50. As the transition from writing words to symbols progressed, mathematicians of the day began investigating the properties of these expressions and began to develop better and better ways to solve them leading to the theories and methods we have today. Now, perhaps you understand why we put so much emphasis on word problems. This method of codifying problems described by words into a system that permits the easy solution of the problems is so very, very useful. To make this connection, you are asked to translate word problems into math, just as has been done for 100s of years. For example, “I need to build the largest enclosed area I can for my cattle, but I only have 300 meters of fencing material. What should the length of each side be to make this area as big as possible?” I hope this helps you start doing the math! Keep Studying and Keep Asking Questions! • I get the whole concept in the video really well, however, I had one question: In the video, the question states "In the following polynomial", meaning, that there is polynomial involved. 3x^2 - 8x + 7 I thought that polynomials were 4+ terms, hence, this will be a trinomial (since it has 3 terms). Can anyone explain? :) Thank you so much! • A monomial, binomial, and trinomial are all also polynomials, they just have unique names. So you could call this either a trinomial (which gives you more information because you know it has 3 terms) or a polynomial. • How should I learn Algebra 2 to get ahead for next year? • Hey! I asked this exact question too before I started. Some things that really helped me were adding Khan's "Algebra 2" course and doing a portion of it everyday throughout the summer. This will keep your mind sharp and teach you the basics of what you'll be doing next year. You can take notes from the videos that you can use later when you are taking the actual class. Make sure to take the mastery quizzes because they are extremely helpful to test your knowledge and teach you what you get wrong if you answer incorrectly. I hope this helps! You got this! Honestly, confidence is everything! • What is coefficient and constant? I get mixed up • The coefficient is the number that is being multiplied by a variable. If you see 12x, the x is the variable and 12 is the coefficient. The prefix co- in front of coefficient means "together". Another word that has co- as a prefix is cooperation. You cooperate when you are working "together" with something or someone else to complete a goal. So just think of coefficient as a number that is cooperating with the variable through multiplication. With 12x, the coefficient 12 is cooperating with variable x. A constant is a quantity that does not change it's value. What does constant literally mean? Constant means "remaining the same over time". It doesn't change. It always has the same value. Take the number 6, for example. It's always going to be the value of 6. 6 will never equal 7 or 8 or 9 etc. It has always and will always equal 6. The value is constant, so it is a constant! This is true for any number not connected to a variable. If you look at a variable such as x, it's not a constant, it's a variable. Variables are the opposite of a constant. Variables vary in its value. Variables change depending on the equation and can equal any number. Hopefully that clears everything up. Good question. • Why does a constant have a degree of 0? • You can rewrite a constant c as c * x^0, since anything to the power of 0 is 1. Let's define that 0^0 is also 1 in this context. You can see that the highest power of x, which is called the degree, is 0. • I have to factor (2t^3)-(14t^2)+(24t) using multiplication tables, how do you do that? -even my sister who is smarter than me cant figure it out- • ((2 • (t3)) - (2•7t2)) + 24t >>> (2t3 - (2•7t2)) + 24t >>> Pull out like factors : 2t3 - 14t2 + 24t = 2t • (t2 - 7t + 12)>>> Factoring t2 - 7t + 12 The first term is, t2 its coefficient is 1 . The middle term is, -7t its coefficient is -7 . The last term, "the constant", is +12 Step-1 : Multiply the coefficient of the first term by the constant 1 • 12 = 12 Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is -7 . -12 + -1 = -13 -6 + -2 = -8 -4 + -3 = -7 That's it Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -3 t2 - 4t - 3t - 12 Step-4 : Add up the first 2 terms, pulling out like factors : t • (t-4) Add up the last 2 terms, pulling out common factors : 3 • (t-4) Step-5 : Add up the four terms of step 4 : (t-3) • (t-4) Which is the desired factorization Final Answer :- 2t • (t - 3) • (t - 4) • Does the following equation have 2 terms or 3 terms? (2x + 3y) - 7z Do the parenthesis around the 2x + 3y make it a single term or is it considered still 2 terms? • This has 3 terms. The parenthesis doesn't combine terms, unlike other BEDMAS. Terms are always divided by an operation. Vote me up if this helped :) • Hi, so my question is if a polynomial has more than 3 terms do you still call it a trinomial or is there a different name for it? or can polynomials not have more than 3 terms? thanks! • Yes - A polynomial can have more than 3 terms. A trinomial always has 3 terms. A polynomial with more than 3 terms is just a polynomial. If the polynomial with 4 terms, it could be called a quadnomial. But that terminology is rarely used. • Would an expression like 9x^2+3x^0 be a polynomial if x=0? • Yes, because 9(0)^2+3(0)^0 is 0, and 0 is a monomial. And a monomial is a polynomial. #2020gang (1 vote) • am currently learning about polynomials what are the rules in adding/subtracting polynomials? • Basic ± Rules for polynomials are that you may only add and subtract terms of the same degree and variable types. `x^3 + x^3 = 2•x^3`, `x^3 + x^2` cannot be added together. `x^2•y^3 + x^2•y^3 = 2•x^2•y^3`, `x^2•y^3 + x^3•y^2` cannot be added together. ## Video transcript In the following polynomial, identify the terms along with the coefficient and exponent of each term. So the terms are just the things being added up in this polynomial. So the terms here-- let me write the terms here. The first term is 3x squared. The second term it's being added to negative 8x. You might say, hey wait, isn't it minus 8x? And you could just view that as it's being added to negative 8x. So negative 8x is the second term. And then the third term here is 7. It's called a polynomial. Poly, it has many terms. Or you could view each term as a monomial, as a polynomial with only one term in it. So those are the terms. Now let's think about the coefficients of each of the terms. The coefficient is what's multiplying the power of x or what's multiplying in the x part of the term. So over here, the x part is x squared. That's being multiplied by 3. So 3 is the coefficient on the first term. On the second term, we have negative 8 multiplying x. And we want to be clear, the coefficient isn't just 8. It's a negative 8. It's negative 8 that's multiplying x. So that's the coefficient right over here. And here you might say, hey wait, nothing is multiplying x here. I just have a 7. There is no x. 7 isn't being multiplied by x. But you can think of this as 7 being multiplied by x to the 0 because we know that x to the zeroth power is equal to 1. So we would even call this constant, the 7, this would be the coefficient on 7x to the 0. So you could view this as a coefficient. So this is also a coefficient. So let me make it clear, these three things are coefficients. Now the last part, they want us to identify the exponent of each term. So the exponent of this first term is 2. It's being raised to the second power. The exponent of the second term, remember, negative 8x, x is the same thing as x to the first power. So the exponent here is 1. And then on this last term, we already said, 7 is the same thing as 7x to the 0. So the exponent here on the constant term on 7 is 0. So these things right over here, those are our exponents. And we are done.<\|endoftext\|>	4.40625
579	# Examples of mixed fractions Perhaps it is best to address the various expressions that can serve as an example for mixed fractions, it is to briefly review some definitions, which will allow each of these cases to be understood within its precise mathematical context. ## Fundamental definitions It may therefore also be useful to delimit this theoretical review to three specific notions: the concepts of Fractions, Natural Numbers and post-position that of the Mixed Fraction itself, as it is these definitions that will allow us to understand the the nature of the mathematical expressions subsequently exposed. Here’s each one: ## Fractions First, it must then be said that Mathematics has explained fractions as one of the two types of mathematical expression with which fractional numbers count, that is, that serve to account for non-whole or non-exact amounts. Likewise, the mathematical discipline has pointed out that fractions can be considered to consist of two elements: • Numerator: which will be located at the top of the expression, indicating how many parts of the whole the fraction refers to. • Denominator: For its part, the denominator will be located at the bottom of the mathematical expression, indicating in how many parts the whole has been divided. ## Integers As for integers, Mathematics has chosen to explain them as those mathematical expressions, consisting of a positive number, a negative number or even the zero, which are used to express whole or exact amounts, as well as lack of a specific quantity, or even the total absence of quantity. These numbers are also considered conforming to the Z-set. ## Mixed fractions Given these two definitions, it may be much easier to understand the concept of Mixed Fractions, which will be a mathematical expression that will consist of an integer and a fraction, and serve to account for amounts that where it has taken a whole part and some parts, then counting on the following form: Likewise, Mathematics has pointed out that the use of Mixed Fractions are often much more common in the everyday realm, than in Mathematics, since it is more common for people to make use of expressions such as “I ate 1 1/2 pizza” to use the form that this cant would have as an improper fraction, referring “I ate 3/2 pizza”. Consequently, it can also be inferred from this reflection that any mixed fraction can be expressed as an improper fraction, and vice versa, a situation that is achieved by following a series of steps and operations that allow the conversion to be done. However, these two types of expressions are equivalent, as they can account for the same fractional amount. ## Examples of mixed fractions However, in addition to the exposure of these definitions, it may be necessary to closely review some mathematical expressions that can be understood as concrete examples of mixed fractions, in order to see in practice how this type of Fractions. Here are some of them:<\|endoftext\|>	4.40625
2,939	Equivalent Fractions Math Lesson Plan for Kids \| 3rd, 4th & 5th Grade \| Generation Genius 1% It was processed successfully! It was processed successfully! WHAT ARE EQUIVALENT FRACTIONS? Equivalent fractions are fractions that represent the same part of a whole. You can use multiplication and division to make equivalent fractions. To better understand equivalent fractions… WHAT ARE EQUIVALENT FRACTIONS?. Equivalent fractions are fractions that represent the same part of a whole. You can use multiplication and division to make equivalent fractions. To better understand equivalent fractions… ## LET’S BREAK IT DOWN! ### Pizza April, Adesina, and Marcos have leftover pizza. Everyone's pizza started out the same size and shape. April’s pizza was cut into 2 equal pieces, and she has one piece left. This is represented with [ggfrac]1/2[/ggfrac]. Adesina’s pizza was cut into 4 equal pieces and she has 2 pieces left. This is represented with [ggfrac]2/4[/ggfrac]. Marcos’ pizza was cut into 8 equal pieces and he has 4 pieces left. This is represented with [ggfrac]4/8[/ggfrac]. Who has more pizza left? Visually, it looks like all three have the same amount of pizza, but what’s the difference? Even though the fractions have different numerators and denominators, they represent the same amount of a whole pizza. They are called equivalent fractions. We prove this using a pizza cutter: if we double the number of slices by cutting April’s slice in half, we turn [ggfrac]1/2[/ggfrac] into [ggfrac]2/4[/ggfrac]. If we again double the number of slices by cutting each [ggfrac]1/4[/ggfrac] in half, we turn [ggfrac]2/4[/ggfrac] into [ggfrac]4/8[/ggfrac]. Now you try: Draw a pizza that has [ggfrac]2/3[/ggfrac] left. Draw lines to show that this is the same amount as [ggfrac]4/6[/ggfrac]. Pizza April, Adesina, and Marcos have leftover pizza. Everyone's pizza started out the same size and shape. April’s pizza was cut into 2 equal pieces, and she has one piece left. This is represented with [ggfrac]1/2[/ggfrac]. Adesina’s pizza was cut into 4 equal pieces and she has 2 pieces left. This is represented with [ggfrac]2/4[/ggfrac]. Marcos’ pizza was cut into 8 equal pieces and he has 4 pieces left. This is represented with [ggfrac]4/8[/ggfrac]. Who has more pizza left? Visually, it looks like all three have the same amount of pizza, but what’s the difference? Even though the fractions have different numerators and denominators, they represent the same amount of a whole pizza. They are called equivalent fractions. We prove this using a pizza cutter: if we double the number of slices by cutting April’s slice in half, we turn [ggfrac]1/2[/ggfrac] into [ggfrac]2/4[/ggfrac]. If we again double the number of slices by cutting each [ggfrac]1/4[/ggfrac] in half, we turn [ggfrac]2/4[/ggfrac] into [ggfrac]4/8[/ggfrac]. Now you try: Draw a pizza that has [ggfrac]2/3[/ggfrac] left. Draw lines to show that this is the same amount as [ggfrac]4/6[/ggfrac]. ### Dividing Clay April and Marcos have a block of clay to use for an art project. April wants to use [ggfrac]1/2[/ggfrac] of the block, while Marcos wants to use [ggfrac]8/16[/ggfrac] of the block. How can we find out if these amounts are the same? We start by cutting the block into two halves. Now we can show what [ggfrac]1/2[/ggfrac] looks like. Next, cut each of the pieces in half to show what [ggfrac]2/4[/ggfrac] looks like. If we repeat this step, we can show what [ggfrac]4/8[/ggfrac] looks like, and finally, what [ggfrac]8/16[/ggfrac] looks like. We can represent all of these pieces using the same amount of clay, so [ggfrac]1/2[/ggfrac], [ggfrac]2/4[/ggfrac], [ggfrac]4/8[/ggfrac], and [ggfrac]8/16[/ggfrac] are all equivalent fractions. To make equivalent fractions using math, we multiply both the denominator and the numerator by 2 each time. For [ggfrac]1/2[/ggfrac], we multiply 1 × 2 = 2, and 2 × 2 = 4 to get [ggfrac]2/4[/ggfrac]. They are equivalent. Each time we double the numerator, we must double the denominator as well to get an equivalent fraction. In fact, we can find an equivalent fraction by multiplying the numerator and denominator by any number, as long as it is the same number. Now you try: Find 3 fractions that are equivalent to [ggfrac]1/3[/ggfrac]. Dividing Clay April and Marcos have a block of clay to use for an art project. April wants to use [ggfrac]1/2[/ggfrac] of the block, while Marcos wants to use [ggfrac]8/16[/ggfrac] of the block. How can we find out if these amounts are the same? We start by cutting the block into two halves. Now we can show what [ggfrac]1/2[/ggfrac] looks like. Next, cut each of the pieces in half to show what [ggfrac]2/4[/ggfrac] looks like. If we repeat this step, we can show what [ggfrac]4/8[/ggfrac] looks like, and finally, what [ggfrac]8/16[/ggfrac] looks like. We can represent all of these pieces using the same amount of clay, so [ggfrac]1/2[/ggfrac], [ggfrac]2/4[/ggfrac], [ggfrac]4/8[/ggfrac], and [ggfrac]8/16[/ggfrac] are all equivalent fractions. To make equivalent fractions using math, we multiply both the denominator and the numerator by 2 each time. For [ggfrac]1/2[/ggfrac], we multiply 1 × 2 = 2, and 2 × 2 = 4 to get [ggfrac]2/4[/ggfrac]. They are equivalent. Each time we double the numerator, we must double the denominator as well to get an equivalent fraction. In fact, we can find an equivalent fraction by multiplying the numerator and denominator by any number, as long as it is the same number. Now you try: Find 3 fractions that are equivalent to [ggfrac]1/3[/ggfrac]. ### Sandwich Deli Adesina shows April and Marcos how comparing equivalent fractions helps us serve lunch. How can we check if [ggfrac]2/3[/ggfrac] is equal to [ggfrac]6/9[/ggfrac] or [ggfrac]10/15[/ggfrac]? Start with [ggfrac]2/3[/ggfrac]. If we multiply the numerator and denominator by 2, we get [ggfrac]4/6[/ggfrac], so it is an equivalent fraction. If we instead multiply the numerator and denominator by 3, we get [ggfrac]6/9[/ggfrac]. [ggfrac]6/9[/ggfrac] is an equivalent fraction. Multiplying by 4, we get [ggfrac]8/12[/ggfrac], and multiplying by 5, we get [ggfrac]10/15[/ggfrac]. All of these fractions are equivalent, and we can find an infinite number of equivalent fractions, multiplying both the numerator and denominator by the same number. Now you try: Is [ggfrac]2/5[/ggfrac] equivalent to [ggfrac]14/35[/ggfrac]? How do you know? Sandwich Deli Adesina shows April and Marcos how comparing equivalent fractions helps us serve lunch. How can we check if [ggfrac]2/3[/ggfrac] is equal to [ggfrac]6/9[/ggfrac] or [ggfrac]10/15[/ggfrac]? Start with [ggfrac]2/3[/ggfrac]. If we multiply the numerator and denominator by 2, we get [ggfrac]4/6[/ggfrac], so it is an equivalent fraction. If we instead multiply the numerator and denominator by 3, we get [ggfrac]6/9[/ggfrac]. [ggfrac]6/9[/ggfrac] is an equivalent fraction. Multiplying by 4, we get [ggfrac]8/12[/ggfrac], and multiplying by 5, we get [ggfrac]10/15[/ggfrac]. All of these fractions are equivalent, and we can find an infinite number of equivalent fractions, multiplying both the numerator and denominator by the same number. Now you try: Is [ggfrac]2/5[/ggfrac] equivalent to [ggfrac]14/35[/ggfrac]? How do you know? ### Chocolate Bars Chocolate Bars Marcos feeds his dog [ggfrac]10/15[/ggfrac] of a treat each day. He can chop the treat into 15 pieces and give the dog 10 of the pieces. This is a lot of work, and Marcos wants to find an easier way. Let’s see if there is an equivalent fraction that would let him cut the dog treat into fewer pieces. If we divide both the numerator and denominator in 1[ggfrac]10/15[/ggfrac] by 5, we get [ggfrac]2/3[/ggfrac]. So, [ggfrac]2/3[/ggfrac] is equivalent to [ggfrac]10/15[/ggfrac]. This saves Marcos a lot of effort because now he only has to cut the dog treat into 3 pieces. Now you try: A recipe calls for [ggfrac]12/16[/ggfrac] of a chocolate bar. What are some equivalent fractions that would make it more efficient to cut up the chocolate? ## EQUIVALENT FRACTIONS VOCABULARY Numerator The number above the line in a fraction, which tells us how many pieces we have. Denominator The number below the line in a fraction, which tells us the size of our unit, or how many equal-sized parts are in the whole. Equivalent Equal. Equivalent fractions Fractions that represent the same amount of the whole. Double Multiplied by 2. Half Divided by 2. ## EQUIVALENT FRACTIONS DISCUSSION QUESTIONS ### What should we do to turn [ggfrac]1/6[/ggfrac] of a pie into [ggfrac]2/12[/ggfrac]? Does the amount of pie change? We should cut each slice in half. Now there are twice as many slices and they are twice as small. The amount of pie is the same. ### What are some fractions that are equivalent to [ggfrac]2/3[/ggfrac]? [ggfrac]4/6[/ggfrac], [ggfrac]6/9[/ggfrac], [ggfrac]8/12[/ggfrac], [ggfrac]10/15[/ggfrac], [ggfrac]12/18[/ggfrac], ... ### How can we check if [ggfrac]5/6[/ggfrac] and [ggfrac]15/18[/ggfrac] are equivalent fractions? If we multiply 5 by 3, we get 15. If we multiply 6 by 3 we get 18. Both the numerator and denominator are multiplied by 3, so the fractions are equivalent. ### Why do we multiply the numerator and denominator by the same number to find equivalent fractions? When we cut more slices out of the same amount of a pie, we make more slices (the numerator grows) and at the same time, each slice gets smaller (the denominator grows). Notice that when the numerator and denominator are the same in a fraction, that is the same as 1 whole. Whenever we multiply something by 1, that means we do not change the amount, we just express it in a different way. ### How can we check if [ggfrac]2/5[/ggfrac] and [ggfrac]12/15[/ggfrac] are equivalent fractions? We can see that 5 × 3 = 15. Then if these are equivalent fractions, we multiply 2 by 3 as well. But 2 × 3 = 6, so [ggfrac]2/5[/ggfrac] and [ggfrac]12/15[/ggfrac] are not equivalent. Or we can see that 2 × 6 = 12. Then we multiply 5 by 6 as well. But 5 × 6 = 30, so [ggfrac]12/15[/ggfrac] is not equivalent to [ggfrac]2/5[/ggfrac]. We could also check using division. X Success We’ve sent you an email with instructions how to reset your password. Ok x Choose Your Free Trial Period 3 Days Continue to Lessons 30 Days Get 30 days free by inviting other teachers to try it too. Share with Teachers Get 30 Days Free By inviting 4 other teachers to try it too. 4 required *only school emails accepted. Skip, I will use a 3 day free trial Thank You! Enjoy your free 30 days trial<\|endoftext\|>	4.75
529	Wednesday, October 25, 2006 Serious Substitution. Topic: Calculus. Problem: Use the substitution $z = \sqrt{x}$ to solve the differential equation $4x \frac{d^2y}{dx^2}+2 \frac{dy}{dx}+y = 0$. Solution: Well, let's do what it says. From the chain rule, we have $\displaystyle \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz} = 2z \frac{dy}{dx}$. Then we also have by the product rule and chain rule again $\displaystyle \frac{d^2y}{dz^2} = 2 \frac{dy}{dx}+2z \frac{d^2y}{dx^2} \cdot \frac{dx}{dz} = \frac{1}{z} \cdot \frac{dy}{dz}+4z^2 \frac{d^2y}{dx^2}$. So we can make the substitutions $\displaystyle 4x \frac{d^2y}{dx^2} = 4z^2 \frac{d^2y}{dx^2} = \frac{d^2y}{dz^2}-\frac{1}{z} \cdot \frac{dy}{dz}$ and $\displaystyle \frac{dy}{dx} = \frac{1}{2z} \cdot \frac{dy}{dz}$ to obtain the differential equation $\frac{d^2y}{dz^2}-\frac{1}{z} \cdot \frac{dy}{dz} + 2 \cdot \frac{1}{2z} \cdot \frac{dy}{dz}+y = \frac{d^2y}{dz^2}+y = 0$. But we know the solution to this is $y = C_1 \cos{z}+C_2 \sin{z}$ so our final solution is $y = C_1 \cos{\sqrt{x}}+C_2 \sin{\sqrt{x}}$. QED. -------------------- Comment: This substitution is, of course, not really natural but was actually found after solving the ODE in another way. Fortunately, it simplifies the problem rather greatly and seems to be a useful technique to look out for. -------------------- Practice Problem: Find another way to solve the differential equation.<\|endoftext\|>	4.53125
716	From the CORDIS News service: Predicting hot days in Europe ‘Red sky at night, shepherd’s delight, red sky in morning, fisherman’s warning.’ This saying is one of Europe’s earliest rhymes that were used to predict weather for the following day. With advances in modern technology, from radar to satellite imagery, we can now predict weather well beyond the following day. European scientists have gone one step further, however, by predicting weather, not just days and weeks in advance, but a whole season ahead. Seasonal prediction can help us prepare against adverse weather conditions in the areas of agriculture, health and other industries. The findings were published in the journal Nature Climate Change. Seasonal predictions are the next hurdle faced by meteorologists but they remain a significant scientific challenge due to flow instability and nonlinearity, which occurs mostly in the mid-latitudes. In this latest study, researchers focused on whether preceding seasons rainfall allow scientists to predict the frequency of forthcoming summer hot days and physical causes of such a predictability. Led by the Laboratoire des Sciences du Climat et de l’Environnement (LSCE) and the Swiss Federal Institute of Technology Zurich (ETH Zurich), researchers in France and Switzerland observed that summer heat in Europe rarely develops after rainy winter and spring seasons over southern Europe. But they discovered that dry seasons are either followed by hot or cold summers. What this means is that the predictability of summer heat is asymmetric and that climate projections indicate a drying of southern Europe. The results suggest that the asymmetry that exists should create a favourable situation for the development of more summer heat waves with a modified seasonal predictability from winter and spring rainfall. The researchers noted that over the past decade Europe saw a number of exceptional summer heat waves with important impacts on society. The 2003 heat wave was the hottest summer on record since 1540 and led to a major health crisis and crop shortfall. Meteorologists were stunned when the record was shattered by the 2010 heat wave just a few years later. Maximum temperature measurements averaged over 7 days exceeding the average for this figure from 1871-2010 by 13.3 degrees Celsius. According to the researchers, extreme summers such as these could be considered as prototypes of summers of future warmer climate. But our ability to anticipate such events one or several months in advance, thereby giving us the chance to prepare ourselves, remains poor. The team analysed precipitation and temperature observations made in 200 European meteorological stations over a period of time stretching more than 60 years. From this they made some generalisations for the region of southeastern Europe, including that rainy winters and springs inhibit the development of hot summer days for the following summer season, while dry or normal rainfalls allow either a large or a weak number of hot temperature days. After dry months, a strong solar energy, associated with anticyclonic conditions, is transferred to the atmosphere through heat fluxes, amplifying drought and heat with a positive feedback. After rainy months, solar energy is largely used for evapo-transpiration instead, limiting the amplification of heat. Even after very dry winter and spring seasons, early summer heavy precipitations can annihilate the potential to develop extreme temperatures, which may have been the case during the 2011 summer, which followed an exceptional spring drought. For more information, please visit: Laboratoire des Sciences du Climat et de l’Environnement (LSCE): h/t to Dr. Leif Svalgaard<\|endoftext\|>	3.796875
1,999	# Converting Expressions Into Radical Form Writing expressions in radical form can seem daunting at first, but with practice and understanding the basics, you’ll be able to tackle even the most complex equations. Radicals are a type of mathematical notation that involve roots and exponents. They can be found in various forms of algebraic equations and are important for solving problems that involve square roots or cube roots. Understanding how to write expressions in radical form will not only improve your math skills but also help you solve real-world problems like calculating distances or volumes. So let’s dive into the basics of radicals and get started on our journey towards mastering this essential skill. ## Key Takeaways • Simplifying radicals involves finding the largest perfect square factor of a number and factoring it out. • Rationalizing denominators involves getting rid of square roots in the denominator of a fraction using conjugates. • Multiplying or dividing radicals involves using the properties of square roots to break them down into factors that can be multiplied or divided. ## Understand the Basics of Radicals Let’s dive into the basics of radicals so you can start understanding how to write expressions in radical form! Radicals are mathematical symbols that represent a root or roots of a number. They’re used to simplify complex numbers and make them easier to work with. Simplifying radicals involves finding the largest perfect square factor of a given number and factoring it out. Radicals have many real-life applications, such as in construction and engineering. For example, architects use radicals to calculate the length of diagonal braces needed for support beams. In medical imaging, radiologists use radical notation to measure the size of tumors or other abnormalities in organs. Understanding how to simplify radicals is essential when working with algebraic expressions involving square roots. It allows you to combine like terms and solve equations more efficiently. Moreover, simplifying radicals helps reduce errors when dealing with large numbers or complicated calculations. So take your time getting comfortable with this fundamental concept – it’ll serve you well in your math journey! ## Rationalizing Denominators To make our math look neater, we can simplify fractions by getting rid of square roots in the denominator. This process is called rationalizing denominators. Simplifying radicals involves finding a perfect square factor of the number under the radical sign and taking its root outside of the radical. However, when dealing with fractions, sometimes there’s a square root left in the denominator that cannot be simplified any further. In this case, we can use conjugates to rationalize the denominator. A conjugate is formed by changing the sign between two terms in an expression. For example, if we have (a + b), then its conjugate would be (a – b). To rationalize a denominator using conjugates, we simply multiply both numerator and denominator by the conjugate of the denominator. This eliminates any square roots from the denominator since multiplying two conjugates results in a difference of squares. Let’s look at an example: suppose we want to simplify 2/(sqrt(3) + 1). We can use its conjugate as follows: 2/(sqrt(3) + 1) * (sqrt(3) – 1)/(sqrt(3) – 1) = [2(sqrt(3) – 1)] / [(sqrt(3))^2 – (1)^2] = [2(sqrt(3)-1)] / [2] = sqrt(3)-1 By multiplying both numerator and denominator by (sqrt(3)-1), we were able to eliminate the square root from the denominator and simplify our fraction into radical form. Remember that when simplifying radicals or rationalizing denominators, it’s important to always check for extraneous solutions and only keep valid answers that satisfy all conditions given in your problem. If they’re not the same, you can use a process called simplifying radicals to make them match. Simplifying square roots involves breaking down a number into its prime factors and pairing up any repeated factors under one radical. Once you have simplified the radicals, you can add or subtract them just like any other algebraic term. For example, if you have √5 + 2√5, you can combine these like terms to get 3√5. On the other hand, if you have √3 – √6, you cannot combine these since they do not have matching radicands. It’s important to note that when adding or subtracting square roots with different radicands, there is no way to simplify further and obtain an exact answer. Instead, we leave our answer in radical form since it is more precise than a decimal approximation. By understanding this concept of adding and subtracting similar expressions with radicals in their simplest form, we can solve more complex problems involving higher degree expressions with ease! Get ready to simplify and conquer! When multiplying and dividing radicals, you’ll be amazed at how easy it is to streamline complex expressions into their most simplified forms. Simplifying radicals involves using the properties of square roots to break them down into factors that can be multiplied or divided. To multiply two or more radicals, simply multiply the coefficients together and then multiply the radicands together. For example, √3 * √5 = √15. If there are variables present in the radicand, you can only multiply them if they have the same base and exponent. For instance, √(x^2) * √(y^2) = xy. Dividing radicals works similarly – divide the coefficients first and then divide the radicands. To divide two square roots with different bases, we need to rationalize the denominator by multiplying both numerator and denominator by a factor that will eliminate any radical from the denominator. This is done by multiplying both sides of a fraction by its conjugate – a binomial that has opposite signs in between its terms. With these simple rules for simplifying square roots when multiplying or dividing them, you can now tackle even more complex expressions with ease! ## Practice Exercises Simplifying square roots can be a breeze if you remember to use the right simplification techniques. When multiplying two radicals, multiply their coefficients and radicands separately. On the other hand, when dividing radicals, divide their coefficients and radicands separately. Don’t forget to rationalize the denominator by multiplying both numerator and denominator by its conjugate. To simplify square roots even further, here are some additional tips that could come in handy: 1. Simplify as much as possible before multiplying or dividing. 2. Break down large numbers into smaller ones. 3. Look for perfect squares or cubes within the expression. By following these guidelines, simplifying expressions in radical form should become second nature to you. Real-life applications of simplifying radicals include solving problems involving geometry, physics and engineering. For example, architects need to calculate areas of triangles with irrational side lengths using Pythagoras’ theorem which requires finding square roots of numbers. Similarly, physicists may encounter equations involving irrational values such as Planck’s constant or Avogadro’s number which require simplification using radical forms. These principles are also important in everyday life when calculating distances between locations or splitting bills amongst friends equally, so it’s essential to understand how to simplify expressions in radical form for practical purposes! ### What is the history of the use of radical form in mathematical expressions? Have you ever wondered about the origins and evolution of radical form in mathematical expressions? This cultural significance dates back to ancient times, with roots in Babylonian and Egyptian mathematics. It has since evolved into a powerful tool for simplifying complex equations. ### How can I apply radical form to real-life situations or problems? Real world examples of radical form include calculating the length of a diagonal in a square, finding the distance between two points on a coordinate plane, and determining the amount of medication needed based on body weight. Practical applications abound in fields like architecture, engineering, and medicine. ### Are there any limitations or restrictions on the use of radical form in mathematical expressions? As you use radical form in mathematical expressions, it’s important to keep in mind the limitations. Not all expressions can be simplified using radicals, so sometimes you may need to use alternative methods for simplification. ### Can I simplify expressions in radical form further, and if so, how? You can simplify expressions in radical form by using simplification techniques such as factoring and combining like terms. For example, √18 can be simplified to 3√2. Complex radical expressions may require additional steps for simplification. Keep exploring new methods! ### What are some common mistakes to avoid when working with expressions in radical form? You can avoid common errors in simplifying expressions in radical form by checking for perfect squares, using the distributive property, and rationalizing denominators. Practice these simplification techniques for efficient and innovative problem-solving. ## Conclusion Congratulations! You’ve mastered the basics of writing expressions in radical form. From understanding what radicals are to knowing how to rationalize denominators, add, subtract, multiply, and divide radicals, you’ve gained valuable knowledge that will serve you well in your academic pursuits. But don’t stop here! To truly master this topic, it’s important to practice regularly. Take advantage of the practice exercises provided in this article and seek out additional resources to challenge yourself further. With dedication and perseverance, you can become a confident expert in expressing radicals in various forms. In conclusion, by following the steps outlined in this article and putting in consistent effort towards practicing, you can confidently write expressions using radical notation with ease. Remember that mastery takes time, but with persistence and a positive attitude, anything is possible! Author Michael Michael is a passionate writer and dedicated typist with a flair for helping others excel in the world of online typing. With years of experience in remote work and a deep understanding of the challenges and opportunities it presents, Michael is committed to sharing valuable insights, practical tips, and expert advice on typing online from home.<\|endoftext\|>	4.40625
11,780	Battle of Bunker Hill \|Battle of Bunker Hill\| \|Part of the American Revolutionary War\| Death of General Warren at the Battle of Bunker Hill by John Trumbull \|Commanders and leaders\| Joseph Warren † Sir Robert Pigot James Abercrombie † John Pitcairn † \|Casualties and losses\| 30 captured (20 POWs died) 19 officers killed\| 62 officers wounded 207 soldiers killed 766 soldiers wounded The Battle of Bunker Hill was fought on June 17, 1775, during the Siege of Boston in the early stages of the American Revolutionary War. The battle is named after Bunker Hill in Charlestown, Massachusetts, which was peripherally involved in the battle. It was the original objective of both the colonial and British troops, though the majority of combat took place on the adjacent hill which later became known as Breed's Hill. On June 13, 1775, the leaders of the colonial forces besieging Boston learned that the British were planning to send troops out from the city to fortify the unoccupied hills surrounding the city, which would give them control of Boston Harbor. In response, 1,200 colonial troops under the command of William Prescott stealthily occupied Bunker Hill and Breed's Hill. During the night, the colonists constructed a strong redoubt on Breed's Hill, as well as smaller fortified lines across the Charlestown Peninsula. By daybreak of June 17, the British became aware of the presence of colonial forces on the Peninsula and mounted an attack against them that day. Two assaults on the colonial positions were repulsed with significant British casualties; the third and final attack carried the redoubt after the defenders ran out of ammunition. The colonists retreated to Cambridge over Bunker Hill, leaving the British in control of the Peninsula. The battle was a tactical, though somewhat Pyrrhic victory for the British, as it proved to be a sobering experience for them, involving many more casualties than the Americans had incurred, including a large number of officers. The battle had demonstrated that inexperienced militia were able to stand up to regular army troops in battle. Subsequently, the battle discouraged the British from any further frontal attacks against well defended front lines. American casualties were comparatively much fewer, although their losses included General Joseph Warren and Major Andrew McClary, the final casualty of the battle. The battle led the British to adopt a more cautious planning and maneuver execution in future engagements, which was evident in the subsequent New York and New Jersey campaign, and arguably helped rather than hindered the American forces. Their new approach to battle was actually giving the Americans greater opportunity to retreat if defeat was imminent. The costly engagement also convinced the British of the need to hire substantial numbers of foreign mercenaries to bolster their strength in the face of the new and formidable Continental Army. - 1 Geography - 2 British planning - 3 Prelude to battle - 4 British assault - 5 Aftermath - 6 Political consequences - 7 Analysis - 8 "The whites of their eyes" - 9 Notable participants - 10 Commemorations - 11 See also - 12 Notes - 13 References - 14 Bibliography - 15 Further reading - 16 External links Boston, situated on a peninsula, was largely protected from close approach by the expanses of water surrounding it, which were dominated by British warships. In the aftermath of the battles of Lexington and Concord on April 19, 1775, the colonial militia, a force of about 15,000 men, had surrounded the town, and effectively besieged it. Under the command of Artemas Ward, they controlled the only land access to Boston itself (the Roxbury Neck), but, lacking a navy, were unable to even contest British domination of the waters of the harbor. The British troops, a force of about 6,000 under the command of General Thomas Gage, occupied the city, and were able to be resupplied and reinforced by sea. In theory, they were thus able to remain in Boston indefinitely. However, the land across the water from Boston contained a number of hills, which could be used to advantage. If the militia could obtain enough artillery pieces, these could be placed on the hills and used to bombard the city until the occupying army evacuated it or surrendered. It was with this in mind that the Knox Expedition, led by Henry Knox, later transported cannon from Fort Ticonderoga to the Boston area. The Charlestown Peninsula, lying to the north of Boston, started from a short, narrow isthmus (known as the Charlestown Neck) at its northwest and extended about 1 mile (1.6 km) southeastward into Boston Harbor. Bunker Hill, with an elevation of 110 feet (34 m), lay at the northern end of the peninsula. Breed's Hill, at a height of 62 feet (19 m), was more southerly and nearer to Boston. The town of Charlestown occupied flats at the southern end of the peninsula. At its closest approach, less than 1,000 feet (305 m) separated the Charlestown Peninsula from the Boston Peninsula, where Copp's Hill was at about the same height as Breed's Hill. While the British retreat from Concord had ended in Charlestown, General Gage, rather than immediately fortifying the hills on the peninsula, had withdrawn those troops to Boston the day after that battle, turning the entire Charlestown Peninsula into a no man's land. Throughout May, in response to orders from Gage requesting support, the British received reinforcements, until they reached a strength of about 6,000 men. On May 25, three generals arrived on HMS Cerberus: William Howe, John Burgoyne, and Henry Clinton. Gage began planning with them to break out of the city, finalizing a plan on June 12. This plan began with the taking of the Dorchester Neck, fortifying the Dorchester Heights, and then marching on the colonial forces stationed in Roxbury. Once the southern flank had been secured, the Charlestown heights would be taken, and the forces in Cambridge driven away. The attack was set for June 18. On June 13, the Massachusetts Provincial Congress was notified, by express messenger from the Committee of Safety in Exeter, New Hampshire, that a New Hampshire gentleman "of undoubted veracity" had, while visiting Boston, overheard the British commanders making plans to capture Dorchester and Charlestown. On June 15, the Massachusetts Committee of Safety decided that additional defenses needed to be erected. General Ward directed General Israel Putnam to set up defenses on the Charlestown Peninsula, specifically on Bunker Hill. Prelude to battle Fortification of Breed's Hill On the night of June 16, colonial Colonel William Prescott led about 1,200 men onto the peninsula in order to set up positions from which artillery fire could be directed into Boston. This force was made up of men from the regiments of Prescott, Putnam (the unit was commanded by Thomas Knowlton), James Frye, and Ebenezer Bridge. At first, Putnam, Prescott, and their engineer, Captain Richard Gridley, disagreed as to where they should locate their defense. Some work was performed on Bunker Hill, but Breed's Hill was closer to Boston and viewed as being more defensible. Arguably against orders, they decided to build their primary redoubt there. Prescott and his men, using Gridley's outline, began digging a square fortification about 130 feet (40 m) on a side with ditches and earthen walls. The walls of the redoubt were about 6 feet (1.8 m) high, with a wooden platform inside on which men could stand and fire over the walls. The works on Breed's Hill did not go unnoticed by the British. General Clinton, out on reconnaissance that night, was aware of them, and tried to convince Gage and Howe that they needed to prepare to attack the position at daylight. British sentries were also aware of the activity, but most apparently did not think it cause for alarm. Then, in the early predawn, around 4 a.m., a sentry on board HMS Lively spotted the new fortification, and notified her captain. Lively opened fire, temporarily halting the colonists' work. Aboard his flagship HMS Somerset, Admiral Samuel Graves awoke, irritated by the gunfire that he had not ordered. He stopped it, only to have General Gage countermand his decision when he became fully aware of the situation in the morning. He ordered all 128 guns in the harbor, as well as batteries atop Copp's Hill in Boston, to fire on the colonial position, which had relatively little effect. The rising sun also alerted Prescott to a significant problem with the location of the redoubt – it could easily be flanked on either side. He promptly ordered his men to begin constructing a breastwork running down the hill to the east, deciding he did not have the manpower to also build additional defenses to the west of the redoubt. When the British generals met to discuss their options, General Clinton, who had urged an attack as early as possible, preferred an attack beginning from the Charlestown Neck that would cut off the colonists' retreat, reducing the process of capturing the new redoubt to one of starving out its occupants. However, he was outvoted by the other three generals. Howe, who was the senior officer present and would lead the assault, was of the opinion that the hill was "open and easy of ascent and in short would be easily carried." General Burgoyne concurred, arguing that the "untrained rabble" would be no match for their "trained troops". Orders were then issued to prepare the expedition. When General Gage surveyed the works from Boston with his staff, Loyalist Abijah Willard recognized his brother-in-law Colonel Prescott. "Will he fight?" asked Gage. "[A]s to his men, I cannot answer for them;" replied Willard, "but Colonel Prescott will fight you to the gates of hell." Prescott lived up to Willard's word, but his men were not so resolute. When the colonists suffered their first casualty, Asa Pollard of Billerica, a young private killed by cannon fire, Prescott gave orders to bury the man quickly and quietly, but a large group of men gave him a solemn funeral instead, with several deserting shortly thereafter. It took six hours for the British to organize an infantry force and to gather up and inspect the men on parade. General Howe was to lead the major assault, drive around the colonial left flank, and take them from the rear. Brigadier General Robert Pigot on the British left flank would lead the direct assault on the redoubt, and Major John Pitcairn led the flank or reserve force. It took several trips in longboats to transport Howe's initial forces (consisting of about 1,500 men) to the eastern corner of the peninsula, known as Moulton's Point. By 2 p.m., Howe's chosen force had landed. However, while crossing the river, Howe noted the large number of colonial troops on top of Bunker Hill. Believing these to be reinforcements, he immediately sent a message to Gage, requesting additional troops. He then ordered some of the light infantry to take a forward position along the eastern side of the peninsula, alerting the colonists to his intended course of action. The troops then sat down to eat while they waited for the reinforcements. Colonists reinforce their positions Prescott, seeing the British preparations, called for reinforcements. Among the reinforcements were Joseph Warren, the popular young leader of the Massachusetts Committee of Safety, and Seth Pomeroy, an aging Massachusetts militia leader. Both of these men held commissions of rank, but chose to serve as infantry. Prescott ordered the Connecticut men under Captain Knowlton to defend the left flank, where they used a crude dirt wall as a breastwork, and topped it with fence rails and hay. They also constructed three small v-shaped trenches between this dirt wall and Prescott's breastwork. Troops that arrived to reinforce this flank position included about 200 men from the 1st and 3rd New Hampshire regiments, under Colonels John Stark and James Reed. Stark's men, who did not arrive until after Howe landed his forces (and thus filled a gap in the defense that Howe could have taken advantage of, had he pressed his attack sooner), took positions along the breastwork on the northern end of the colonial position. When low tide opened a gap along the Mystic River to the north, they quickly extended the fence with a short stone wall to the water's edge. Colonel Stark placed a stake about 100 feet (30 m) in front of the fence and ordered that no one fire until the regulars passed it. Just prior to the action, further reinforcements arrived, including portions of Massachusetts regiments of Colonels Brewer, Nixon, Woodbridge, Little, and Major Moore, as well as Callender's company of artillery. Behind the colonial lines, confusion reigned. Many units sent toward the action stopped before crossing the Charlestown Neck from Cambridge, which was under constant fire from gun batteries to the south. Others reached Bunker Hill, but then, uncertain about where to go from there, milled around. One commentator wrote of the scene that "it appears to me there never was more confusion and less command." While General Putnam was on the scene attempting to direct affairs, unit commanders often misunderstood or disobeyed orders. By 3 p.m., the British reinforcements, which included the 47th Foot and the 1st Marines, had arrived, and the British were ready to march. Brigadier General Pigot's force, gathering just south of Charlestown village, were taking casualties from sniper fire, and Howe asked Admiral Graves for assistance in clearing out the snipers. Graves, who had planned for such a possibility, ordered incendiary shot fired into the village, and then sent a landing party to set fire to the town. The smoke billowing from Charlestown lent an almost surreal backdrop to the fighting, as the winds were such that the smoke was kept from the field of battle. Pigot, commanding the 5th, 38th, 43rd, 47th, and 52nd regiments, as well as Major Pitcairn's Marines, were to feint an assault on the redoubt. However, they continued to be harried by snipers in Charlestown, and Pigot, when he saw what happened to Howe's advance, ordered a retreat. General Howe led the light infantry companies and grenadiers in the assault on the American left flank, expecting an easy effort against Stark's recently arrived troops. His light infantry were set along the narrow beach, in column, in order to turn the far left flank of the colonial position. The grenadiers were deployed in the middle. They lined up four deep and several hundred across. As the regulars closed, John Simpson, a New Hampshire man, prematurely fired, drawing an ineffective volley of return fire from the regulars. When the regulars finally closed within range, both sides opened fire. The colonists inflicted heavy casualties on the regulars, using the fence to steady and aim their muskets, and benefit from a modicum of cover. With this devastating barrage of musket fire, the regulars retreated in disarray, and the militia held their ground. The regulars reformed on the field and marched out again. This time, Pigot was not to feint; he was to assault the redoubt, possibly without the assistance of Howe's force. Howe, instead of marching against Stark's position along the beach, marched instead against Knowlton's position along the rail fence. The outcome of the second attack was much the same as the first. One British observer wrote, "Most of our Grenadiers and Light-infantry, the moment of presenting themselves lost three-fourths, and many nine-tenths, of their men. Some had only eight or nine men a company left ..." Pigot did not fare any better in his attack on the redoubt, and again ordered a retreat. Meanwhile, in the rear of the colonial forces, confusion continued to reign. General Putnam tried, with only limited success, to send additional troops from Bunker Hill to Breed's Hill to support the men in the redoubt and along the defensive lines. The British rear was also in some disarray. Wounded soldiers that were mobile had made their way to the landing areas, and were being ferried back to Boston, and the wounded lying on the field of battle were the source of moans and cries of pain. General Howe, deciding that he would try again, sent word to General Clinton in Boston for additional troops. Clinton, who had watched the first two attacks, sent about 400 men from the 2nd Marines and the 63rd Foot, and then followed himself to help rally the troops. In addition to the new reserves, he also convinced about 200 of the wounded to form up for the third attack. During the interval between the second and third assaults, General Putnam continued trying to direct troops toward the action. Some companies, and leaderless groups of men, moved toward the action; others retreated. John Chester, a Connecticut captain, seeing an entire company in retreat, ordered his company to aim muskets at that company to halt its retreat; they turned about and headed back to the battlefield. The third assault, concentrated on the redoubt (with only a feint on the colonists' flank), was successful, although the colonists again poured musket fire into the British ranks, and it cost the life of Major Pitcairn. The defenders had run out of ammunition, reducing the battle to close combat. The British had the advantage once they entered the redoubt, as their troops were equipped with bayonets on their muskets while most of the colonists were not. Colonel Prescott, one of the last colonists to leave the redoubt, parried bayonet thrusts with his normally ceremonial sabre. It is during the retreat from the redoubt that Joseph Warren was killed. The retreat of much of the colonial forces from the peninsula was made possible in part by the controlled retreat of the forces along the rail fence, led by John Stark and Thomas Knowlton, which prevented the encirclement of the hill. Their disciplined retreat, described by Burgoyne as "no flight; it was even covered with bravery and military skill", was so effective that most of the wounded were saved; most of the prisoners taken by the British were mortally wounded. General Putnam attempted to reform the troops on Bunker Hill; however the flight of the colonial forces was so rapid that artillery pieces and entrenching tools had to be abandoned. The colonists suffered most of their casualties during the retreat on Bunker Hill. By 5 p.m., the colonists had retreated over the Charlestown Neck to fortified positions in Cambridge, and the British were in control of the peninsula. The British had taken the ground but at a great loss; they had suffered 1,054 casualties (226 dead and 828 wounded), with a disproportionate number of these officers. The casualty count was the highest suffered by the British in any single encounter during the entire war. General Clinton, echoing Pyrrhus of Epirus, remarked in his diary that "A few more such victories would have shortly put an end to British dominion in America." British dead and wounded included 100 commissioned officers, a significant portion of the British officer corps in North America. Much of General Howe's field staff was among the casualties. Major John Pitcairn who was physically in charge of the Regulars in Lexington and Concord had been injured, along with Lieutenant Colonel James Abercrombie. They both died hours later in Military Hospitals. General Gage, in his report after the battle, reported the following officer casualties (listing lieutenants and above by name): - 1 lieutenant colonel killed - 2 majors killed, 3 wounded - 7 captains killed, 27 wounded - 9 lieutenants killed, 32 wounded - 15 sergeants killed, 42 wounded - 1 drummer killed, 12 wounded The colonial losses were about 450, of whom 140 were killed. Most of the colonial losses came during the withdrawal. Major Andrew McClary was technically the highest ranking colonial officer to die in the battle; he was hit by cannon fire on Charlestown Neck, the last person to be killed in the battle. He was later commemorated by the dedication of Fort McClary in Kittery, Maine. A serious loss to the Patriot cause, however, was the death of Dr. Joseph Warren. He was the President of Massachusetts' Provincial Congress, and he had been appointed a Major General on June 14. His commission had not yet taken effect when he served as a volunteer private three days later at Bunker Hill. Only thirty men were captured by the British, most of them with grievous wounds; twenty died while held prisoner. The colonials also lost numerous shovels and other entrenching tools, as well as five out of the six cannon they had brought to the peninsula. When news of the battle spread through the colonies, it was reported as a colonial loss, as the ground had been taken by the enemy, and significant casualties were incurred. George Washington, who was on his way to Boston as the new commander of the Continental Army, received news of the battle while in New York City. The report, which included casualty figures that were somewhat inaccurate, gave Washington hope that his army might prevail in the conflict. A British officer in Boston, after the battle The Massachusetts Committee of Safety, seeking to repeat the sort of propaganda victory it won following the battles at Lexington and Concord, commissioned a report of the battle to send to England. Their report, however, did not reach England before Gage's official account arrived on July 20. His report unsurprisingly caused friction and argument between the Tories and the Whigs, but the casualty counts alarmed the military establishment, and forced many to rethink their views of colonial military capability. King George's attitude toward the colonies hardened, and the news may have contributed to his rejection of the Continental Congress' Olive Branch Petition, the last substantive political attempt at reconciliation. Sir James Adolphus Oughton, part of the Tory majority, wrote to Lord Dartmouth of the colonies, "the sooner they are made to Taste Distress the sooner will [Crown control over them] be produced, and the Effusion of Blood be put a stop to." About a month after receiving Gage's report the Proclamation of Rebellion would be issued in response; this hardening of the British position would also lead to a hardening of previously weak support for the rebellion, especially in the southern colonies, in favor of independence. Gage's report had a more direct effect on his own career. His dismissal from office was decided just three days after his report was received, although General Howe did not replace him until October 1775. Gage wrote another report to the British Cabinet, in which he repeated earlier warnings that "a large army must at length be employed to reduce these people", that would require "the hiring of foreign troops". Much has been written in the wake of this battle over how it was conducted. Both sides made strategic and tactical missteps which could have altered the outcome of the battle. While hindsight often gives a biased view, some things seem to be apparent after the battle that might reasonably have been within the reach of the command of the day. Years after the battle, and after Israel Putnam was dead, General Dearborn published an account of the battle in Port Folio magazine, accusing General Putnam of inaction, cowardly leadership and failing to supply reinforcements during the battle, which subsequently sparked a long lasting and major controversy among veterans of the war, various friends, family members and historians.[a] People were shocked by the rancor of the attack, and this prompted a forceful response from defenders of Putnam, including such notables as John and Abigail Adams. Historian Harold Murdock wrote that Dearborn's account "abounds in absurd misstatements and amazing flights of imagination." The Dearborn attack received considerable attention because at the time he was in the middle of considerable controversy himself. He had been relieved of one of the top commands in the War of 1812 due to his mistakes. He had also been nominated to serve as Secretary of War by President Monroe, but was rejected by the United States Senate (which was the first time that the Senate had voted against confirming a presidential cabinet choice). Disposition of Colonial forces The colonial forces, while nominally under the overall command of General Ward, with General Putnam and Colonel Prescott leading in the field, often acted quite independently. This was evident in the opening stages of the battle, when a tactical decision was made that had strategic implications. After deliberating with General Putnam and Colonel Gridley, Colonel Prescott and his staff, apparently in contravention of orders, decided to fortify Breed's Hill rather than Bunker Hill. The fortification of Breed's Hill was more provocative; it would have put offensive artillery closer to Boston. It also exposed the forces there to the possibility of being trapped, as they probably could not properly defend against attempts by the British to land troops and take control of Charlestown Neck. If the British had taken that step, they might have had a victory with many fewer casualties. While the front lines of the colonial forces were generally well managed, the scene behind them, especially once the action began, was significantly disorganized, due at least in part to a poor chain of command. Only some of the militias operated directly under Ward's and Putnam's authority, and some commanders also disobeyed orders, staying at Bunker Hill rather than joining in the defense on the third British assault. Several officers were subjected to court martial and cashiered. Colonel Prescott was of the opinion that the third assault would have been repulsed, had his forces in the redoubt been reinforced with either more men, or more supplies of ammunition and powder. Disposition of British forces The British leadership, for its part, acted slowly once the works on Breed's Hill were spotted. It was 2 p.m. when the troops were ready for the assault, roughly ten hours after the Lively first opened fire. This leisurely pace gave the colonial forces time to reinforce the flanking positions that had been poorly defended. Gage and Howe decided that a frontal assault on the works would be a simple matter, when an encircling move (gaining control of Charlestown Neck), would have given them a more resounding victory. (This move would not have been without risks of its own, as the colonists could have made holding the Neck expensive with fire from the high ground in Cambridge.) But the British leadership was excessively optimistic, believing that "two regiments were sufficient to beat the strength of the province". Once in the field, Howe, rather than focusing on the redoubt, opted (twice) to dilute the force attacking the redoubt with a flanking maneuver against the colonial left. It was only with the third attack, when the flank attack was merely a feint, and the main force (now also reinforced with additional reserves) squarely targeted the redoubt, that the attack succeeded. Following the taking of the peninsula, the British arguably had a tactical advantage that they could have used to press into Cambridge. General Clinton proposed this to Howe; having just led three assaults with grievous casualties, he declined the idea. The colonial military leaders eventually recognized Howe as a tentative decision-maker, to his detriment; in the aftermath of the Battle of Long Island (1776), he again had tactical advantages that might have delivered Washington's army into his hands, but again refused to act. Historian John Ferling maintains that had General Gage used the Royal Navy to secure the narrow neck to the Charleston peninsula, cutting the Americans off from the mainland, he could have achieved a far less costly victory, but he was motivated by revenge over patriot resistance at the Battles of Lexington and Concord and relatively heavy British losses, and also felt that the colonial militia were completely untrained and could be overtaken with little effort, opting for a frontal assault. "The whites of their eyes" The famous order "Don't fire until you see the whites of their eyes" was popularized in stories about the battle of Bunker Hill. It is uncertain as to who said it there, since various histories, including eyewitness accounts, attribute it to Putnam, Stark, Prescott, or Gridley, and it may have been said first by one, and repeated by the others. It was also not an original statement. The idea dates originally to the general-king Gustavus Adolphus (1594–1632) who gave standing orders to his musketeers: "never to give fire, till they could see their own image in the pupil of their enemy's eye". Gustavus Adolphus's military teachings were widely admired and imitated and caused this saying to be often repeated. It was used by General James Wolfe on the Plains of Abraham, when his troops defeated Montcalm's army on September 13, 1759. The earliest similar quote came from the Battle of Dettingen on June 27, 1743, where Lieutenant-Colonel Sir Andrew Agnew of Lochnaw warned his Regiment, the Royal Scots Fusiliers, not to fire until they could "see the white of their e'en." The phrase was also used by Prince Charles of Prussia in 1745, and repeated in 1755 by Frederick the Great, and may have been mentioned in histories the colonial military leaders were familiar with. Whether or not it was actually said in this battle, it was clear that the colonial military leadership were regularly reminding their troops to hold their fire until the moment when it would have the greatest effect, especially in situations where their ammunition would be limited. A significant number of notable American patriots fought in this battle. Henry Dearborn and William Eustis, for example, went on to distinguished military and political careers; both served in Congress, the Cabinet, and in diplomatic posts. Others, like John Brooks, Henry Burbeck, Christian Febiger, Thomas Knowlton, and John Stark, became well known for later actions in the war. Stark became known as the "Hero of Bennington" for his role in the 1777 Battle of Bennington. Free African-Americans also fought in the battle; notable examples include Barzillai Lew, Salem Poor, and Peter Salem. Another notable participant was Daniel Shays, who later became famous for his army of protest in Shays' Rebellion. Israel Potter was immortalized in Israel Potter: His Fifty Years of Exile, a novel by Herman Melville. Colonel John Paterson commanded the Massachusetts First Militia, served in Shays' Rebellion, and became a congressman from New York. Lt. Col. Seth Read, who served under John Paterson at Bunker Hill, went on to settle Geneva, New York and Erie, Pennsylvania, and was said to have been instrumental in the phrase E pluribus unum being added to U.S. coins. George Claghorn of the Massachusetts militia was shot in the knee at Bunker Hill and went on after the war to become the master builder of the USS Constitution, a.k.a. "Old Ironsides", which is the oldest naval vessel in the world that is still commissioned and afloat. John Trumbull's painting, The Death of General Warren at the Battle of Bunker Hill (displayed in lede), was created as an allegorical depiction of the battle and Warren's death, not as an actual pictorial recording of the event. The painting shows a number of participants in the battle including a British officer, John Small, among those who stormed the redoubt, yet came to be the one holding the mortally wounded Warren and preventing a fellow redcoat from bayoneting him. He was friends of Putnam and Trumbull. Other central figures include Andrew McClary who was the last man to fall in the battle. The Bunker Hill Monument is an obelisk that stands 221 feet (67 m) high on Breed's Hill. On June 17, 1825, the fiftieth anniversary of the battle, the cornerstone of the monument was laid by the Marquis de Lafayette and an address delivered by Daniel Webster. (When Lafayette died, he was buried next to his wife, Adrienne de La Fayette, at the Cimetière de Picpus under soil from Bunker Hill, which his son Georges sprinkled over him.) The Leonard P. Zakim Bunker Hill Memorial Bridge was specifically designed to evoke this monument. There is also a statue of William Prescott showing him calming his men down. The National Park Service operates a museum dedicated to the battle near the monument, which is part of the Boston National Historical Park. A cyclorama of the battle was added in 2007 when the museum was renovated. In nearby Cambridge, a small granite monument just north of Harvard Yard bears this inscription: "Here assembled on the night of June 16, 1775, 1200 Continental troops under command of Colonel Prescott. After prayer by President Langdon, they marched to Bunker Hill." See footnote for picture. (Samuel Langdon, a Congregational minister, was Harvard's 11th president.) Another small monument nearby marks the location of the Committee of Safety, which had become the Patriots' provisional government as Tories left Cambridge. These monuments are on the lawn to the west of Harvard's Littaeur Center, which is itself the west of Harvard's huge Science Center. See footnote for map. Bunker Hill Day, observed every June 17, is a legal holiday in Suffolk County, Massachusetts (which includes the city of Boston), as well as Somerville in Middlesex County. Prospect Hill, site of colonial fortifications overlooking the Charlestown Neck, is now in Somerville, which was previously part of Charlestown. State institutions in Massachusetts (such as public institutions of higher education) in Boston also celebrate the holiday. However, the state's FY2011 budget requires that all state and municipal offices in Suffolk County be open on Bunker Hill Day and Evacuation Day. On June 16 and 17, 1875, the centennial of the battle was celebrated with a military parade and a reception featuring notable speakers, among them General William Tecumseh Sherman and Vice President Henry Wilson. It was attended by dignitaries from across the country. Celebratory events also marked the sesquicentennial (150th anniversary) in 1925 and the bicentennial in 1975. Over the years the Battle of Bunker Hill has been commemorated on four U.S. Postage stamps. - List of American Revolutionary War battles - List of Continental Forces in the American Revolutionary War - List of British Forces in the American Revolutionary War - Dr. John Hart, Regimental Surgeon of Col Prescott's Regiment who treated the wounded at Bunker Hill - Royal Welch Fusiliers - USS Bunker Hill - Chidsey p. 122 counts 1,400 in the night-time fortification work. Frothingham is unclear on the number of reinforcements arriving just before the battle breaks out. In a footnote on p. 136, as well as on p. 190, he elaborates the difficulty in getting an accurate count. - Chidsey p. 90 says the initial force requested was 1,550, but Howe requested and received reinforcements before the battle began. Frothingham p. 137 puts the total British contingent likely to be over 3,000. Furthermore, according to Frothingham p. 148, additional reinforcements arrived from Boston after the second attack was repulsed. Frothingham, p. 191 notes the difficulty in attaining an accurate count of British troops involved. - Chidsey, p. 104 - Frothingham pp. 191, 194. - Borneman, Walter R. American Spring: Lexington, Concord, and the Road to Revolution, p. 350, Little, Brown and Company, New York, Boston, London, 2014. ISBN 978-0-316-22102-3. - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, p. 85, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 85–87, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 87–95, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Clinton, p. 19. General Clinton's remark is an echoing of Pyrrhus of Epirus's original sentiment after the Battle of Heraclea, "one more such victory and the cause is lost". - "Battle of Bunker Hill". Encyclopædia Britannica. Encyclopædia Britannica, Inc. December 8, 2016. Retrieved January 25, 2016. Although the British eventually won the battle, it was a Pyrrhic victory that lent considerable encouragement to the revolutionary cause. - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 94–95, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - 18th century Boston was a peninsula. Primarily in the 19th century, much land around the peninsula was filled, giving the modern city its present geography. See the history of Boston for details. - Chidsey, p. 72 New Hampshire 1,200, Rhode Island 1,000, Connecticut 2,300, Massachusetts 11,500 - Alden, p. 178 - Visitors to Boston, upon seeing the nearby hills, may conclude that they are too low. The hills were once higher, but were lowered by excavations to obtain landfill used to expand Boston in the 19th century. - Martin, James Kirby (1997). Benedict Arnold: Revolutionary Hero. New York: New York University Press. p. 73. ISBN 978-0-8147-5560-0. OCLC 36343341. - Chidsey p. 91 has an historic map showing elevations. - French, p. 220 - French, p. 249 - Brooks, p. 119 - Ketchum, pp. 45–46 - Ketchum, p. 47 - Ketchum, pp. 74–75 - French, p. 255 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, p. 84, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Frothingham, pp. 122–123 - Ketchum, pp. 102, 245 - Frothingham, pp. 123–124 - Frothingham, p. 135 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 87–88, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Ketchum, p. 115 - Frothingham, p. 125 - Brooks, p. 127 - Ketchum, p. 117 - Ketchum, pp. 120–121 - Wood, p. 54 - Ketchum, p. 122 - Graydon, p. 424 - Chidsey, p. 84 - Frothingham, p. 133 - Ketchum, p. 139 - Ketchum, p 143 - Chidsey p. 93 - Chidsey p. 96 - Frothingham, p. 136 - Ketchum, p. 147 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 92–93, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Ketchum, pp. 152–153 - Ketchum, pp. 151–152 - Frothingham, pp. 144–145 - Ketchum, p. 160 - Ketchum, p. 152 - Fusillers, Mark Urban p38 - Frothingham, pp. 141–142 - Ketchum, p. 161 - Ketchum, p. 162 - Frothingham, p. 146 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, p. 92, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Ketchum, p. 163 - Ketchum, p. 164 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 92–95, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Ketchum, pp. 165–166 - Chidsey p. 99 - Frothingham, p. 150 - Frothingham, p. 151 - Ketchum, p. 181 - Frothingham, pp. 151–152 - Brooks, p. 237 - Brooks, pp. 183–184 - Frothingham, pp. 145, 196 - Frothingham, pp. 387–389 lists the officer casualties by name, as well as this summary - Bardwell, p. 76 - Ketchum, p. 150 - Ketchum, p. 255 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 94–96, McFarland & Company, Inc., Jefferson, North Carolina, 2017. ISBN 978-1-4766-6453-8. - Ketchum, pp. 207–208 - Ketchum, p. 209 - Ketchum, pp. 208–209 - Ketchum, p. 211 - Ketchum, p. 213 - Scheer, p. 64 - Cray, 2001 - Purcell, 2010, pp.164-168 - Ketchum, Richard M. The Battle for Bunker Hill, p. 178, The Cresset Press, London, England, 1963. - Murdock, Harold. Bunker Hill, Notes and Queries on a Famous Battle, Kessinger Publishing, LLC, 2010. ISBN 1163174912, - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, pp. 191–92, McFarland & Company, Inc., Jefferson, North Carolina, 2017 ISBN 978-1-4766-6453-8. - Frothingham, p. 131 - Frothingham, p. 19 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, p. 87, McFarland & Company, Inc., Jefferson, North Carolina, 2017 ISBN 978-1-4766-6453-8. - Frothingham, p. 155 - Frothingham, pp. 158–159 - French, pp. 274–276 - Frothingham, p. 153 - French, pp. 263–265 - Frothingham, p. 156 - French, p. 277 - Frothingham, p. 148 - Frothingham pp. 152–153 - Jackson, p. 20 - Ferling, 2015, p. 127-129 - Lewis, John E., ed. The Mammoth Book of How it Happened. London: Robinson, 1998. Print. P. 179 - Hubbard, Robert Ernest. Major General Israel Putnam: Hero of the American Revolution, p. 97, McFarland & Company, Inc., Jefferson, North Carolina, 2017 ISBN 978-1-4766-6453-8. - Joannis Schefferi, "Memorabilium Sueticae Gentis Exemplorum Liber Singularis" (1671) p. 42 - R. Reilly, The Rest to Fortune: The Life of Major-General James Wolfe (1960), p. 324 - Anderson, p. 679 - Winsor, p. 85 - French, pp. 269–270 - Abbatt, p. 252 - Ketchum, pp. 132,165 - Woodson, p. 204 - Ketchum, p. 260 - Richards, p. 95 - Ketchum, p. 257 - Biographical Directory of the United States - Buford, 1895, Preface - Marvin, p. 425, 436 - "Massachusetts Coppers 1787–1788: Introduction". University of Notre Dame. Archived from the original on November 8, 2007. Retrieved October 9, 2007. - "e pluribus unum FAQ #7". www.treas.gov. Retrieved September 29, 2007.[permanent dead link] - Wheeler, O. Keith (January 30, 2002). "Individual Summary for COL. GEORGE CLAGHORN". Retrieved October 10, 2012. - HMS Victory is the oldest commissioned vessel by three decades; however, Victory has been in dry dock since 1922. "HMS Victory Service Life". HMS Victory website. Archived from the original on August 28, 2012. - Bunce, p. 336 - Hayward, p. 322 - MTA Bridges - Bunker Hill Museum - "Album Archive". Picasaweb.google.com. Retrieved November 19, 2017. - "Archived copy". Archived from the original on January 1, 2013. Retrieved 2013-12-31.CS1 maint: Archived copy as title (link) - Committee of Safety (American Revolution) - "Harvard University Campus Map". Map.harvard.edu. Archived from the original on January 6, 2013. Retrieved November 19, 2017. - MA List of legal holidays - Somerville Environmental Services Guide - University of Massachusetts, Boston, observed holidays - Bunker Hill Day closings - "Commonwealth of Massachusetts FY2011 Budget, Outside Section 5". Mass.gov. July 14, 2010. Retrieved August 6, 2010. - See the Centennial Book for a complete description of the events. - Sesquicentennial celebration - New York Times, June 15, 1975 - Scotts 2008 United States stamp catalogue Major sources Most of the information about the battle itself in this article comes from the following sources. - Brooks, Victor (1999). The Boston Campaign. Conshohocken, PA: Combined Publishing. ISBN 1-58097-007-9. OCLC 42581510. - Chidsey, Donald Barr (1966). The Siege of Boston. Boston, MA: Crown. OCLC 890813. - Frothingham, Jr, Richard (1851). History of the Siege of Boston and of the Battles of Lexington, Concord, and Bunker Hill, Second Edition. Boston, MA: Charles C. Little and James Brown. OCLC 2138693. - French, Allen (1911). The Siege of Boston. New York: McMillan. OCLC 3927532. - Ketchum, Richard (1999). Decisive Day: The Battle of Bunker Hill. New York: Owl Books. ISBN 0-385-41897-3. OCLC 24147566. (Paperback: ISBN 0-8050-6099-5) - Philbrick, Nathaniel. Bunker Hill: A City, a Siege, a Revolution (New York: Viking, 2013) Minor sources Specific facts not necessarily covered by the major sources come from the following sources. - Buford, Mary Hunter (1895). Seth Read, Lieut.-Col. Continental Army; Pioneer at Geneva, New York, 1787, and at Erie, Penn., June, 1795. His Ancestors and Descendants. Boston, Mass. pp. 167 Pages on CD in PDF Format. - Bunce, Oliver Bell (1870). The romance of the revolution: being true stories of the adventures, romantic incidents, hairbreath escapes, and heroic exploits of the days of '76. Philadelphia: Porter & Coates. OCLC 3714510. - Abbatt, William (ed) (1883). The Magazine of American History with Notes and Queries, volume 8. A.S. Barnes. OCLC 1590082.CS1 maint: Extra text: authors list (link) - Alden, John R (1989). A History of the American Revolution. Da Capo. ISBN 0-306-80366-6. - Anderson, William (1863). The Scottish Nation: Or, The Surnames, Families, Literature, Honours, and Biographical History of the People of Scotland, volume 2. Fullarton. OCLC 1290413. - Bardwell, John D (2005). Old Kittery. Arcadia Publishing. ISBN 978-0-7385-2476-4. - Clinton, Henry (1954). Willcox, William B. (ed.). The American Rebellion: Sir Henry Clinton's Narrative of His Campaigns, 1775–1782. Yale University Press. OCLC 1305132. - Graydon, Alexander (1846). Littell, John Stockton (ed.). Memoirs of His Own Time: With Reminiscences of the Men and Events of the Revolution. Philadelphia: Lindsay & Blakiston. OCLC 1557096. - Ferling, John (2015). Whirlwind, The American Revolution and the War That Won it. Bloomsbury Press, New York, London. - Hayward, John (1854). A Gazetteer of the United States of America. self published. OCLC 68756962. - Jackson, Kenneth T; Dunbar, David S (2005). Empire City: New York Through the Centuries. Columbia University Press. ISBN 978-0-231-10909-3. - Melville, Herman (1855). Israel Potter: his fifty years of exile. G. Routledge. OCLC 13065897. - Richards, Leonard L (2003). Shays's Rebellion: The American Revolution's Final Battle. University of Pennsylvania Press. ISBN 978-0-8122-1870-1. - Scheer, George F; Rankin, Hugh F (1987). Rebels and Redcoats: The American Revolution Through the Eyes of Those Who Fought and Lived It. Da Capo Press. ISBN 978-0-306-80307-9. - Winsor, Justin; Jewett, Clarence F (1882). The Memorial History of Boston: Including Suffolk County, Massachusetts, 1630–1880, Volume 3. James R. Osgood. OCLC 4952179. - Wood, Gordon S. (2002). The American Revolution: A History. Modern Library. ISBN 0-8129-7041-1. - Woodson, Carter Godwin; Logan, Rayford Whittingham (1917). The Journal of Negro History, Volume 2. Association for the Study of Negro Life and History. OCLC 1782257. - Cray, Robert E. (2001). Bunker Hill Refought: Memory Wars and Partisan Conflicts, 1775-1825 (PDF). Historical Journal of Massachusetts. - "Congressional bio of John Patterson". Biographical Directory of the United States. Commemorations Various commemorations of the battle are described in the following sources. - "Charles River Bridges". Massachusetts Turnpike Authority. Archived from the original on September 28, 2007. Retrieved November 26, 2008. - "Massachusetts List of Legal Holidays". Massachusetts Secretary of State. Retrieved December 16, 2008. - "Environmental Guide 2008" (PDF). City of Somerville, Massachusetts. Archived from the original (pdf) on March 4, 2009. Retrieved February 26, 2009. - "UMass Boston Holidays observed". University of Massachusetts, Boston. Archived from the original on February 25, 2009. Retrieved March 16, 2009. - "Bunker Hill Day Closings". Boston Globe. June 18, 2007. Retrieved March 16, 2009. - Winsor, Justin (1875). Celebration of the centennial anniversary of the battle of Bunker Hill. Boston, MA: Boston City Council. OCLC 2776599. - Celebration of the sesquicentennial anniversary of the Battle of Bunker Hill, June 17, 1925. Boston, MA: City of Boston. 1925. OCLC 235594934. - "Bunker Hill Museum". National Park Service. Archived from the original on April 3, 2009. Retrieved March 17, 2009. - Clary, David (2007). Adopted Son: Washington, Lafayette, and the Friendship that Saved the Revolution. New York City: Bantam Books. pp. 443–448. ISBN 978-0-553-80435-5. OCLC 70407848. - Kifner, John (July 15, 1975). "Not Unusual Occurrence: British Take Bunker Hill". New York Times. Retrieved March 17, 2009. (ProQuest document number: 118450359) - McKenna, Kathleen (June 10, 2007). "On Bunker Hill, a boost in La Fayette profile". Boston Globe. Archived from the original on February 20, 2009. Retrieved March 17, 2009. - Axelrod, Alan (2007). The real History of the American Revolution. Sterling Publishing Company, New York. - Beck, Derek W. (2016). The War Before Independence: 1775-1776. Sourcebooks, Inc., 480 pages - Doyle, Peter (1998). Bunker Hill. Charlottesville, VA: Providence Foundation. ISBN 1-887456-08-2. OCLC 42421560. - Drake, Samuel Adams (1875). Bunker Hill: the story told in letters from the battle field by British Officers Engaged. Boston: Nichols and Hall. - Elting, John R. (1975). The Battle of Bunker's Hill. Monmouth Beach, NJ: Phillip Freneau Press. ISBN 0-912480-11-4. OCLC 2867199. - Ferling, John (2007). Almost a Miracle. Oxford University Press. - Fast, Howard (2001). Bunker Hill. New York: ibooks inc. ISBN 0-7434-2384-4. OCLC 248511443. - Lanning, Michael Lee (2008). The American Revolution 100. Source Books, Naperville, Illinois. - O'Brien, Michael J. (1968). The Irish at Bunker Hill: Evidence of Irish Participation in the Battle of 17 June 1775. Irish University Press. - Ristow, W. Walter (1979). Cartography of the Battle of Bunker Hill. - Swett, S (1826). History of Bunker Hill Battle, With a Plan, Second Edition. Boston, MA: Munroe and Francis. OCLC 3554078. This book contains printings of both Gage's official account and that of the Massachusetts Congress. - Commager, Henry Steele; Morris, Richard B. (1958). The Spirit of 76. Harper & Row Publishers, New York, London., eBook, Vol 2 \|Wikisource has the text of the 1911 Encyclopædia Britannica article Bunker Hill.\| \|Wikimedia Commons has media related to Battle of Bunker Hill.\| About the battle - Library of Congress page about the battle - Bunker Hill Web Exhibit of the Massachusetts Historical Society - SAR Sons of Liberty Chapter list of colonial fallen at Bunker Hill - SAR Sons of Liberty Chapter description of the battle - The Battle of Bunker Hill: Now We Are at War, a National Park Service Teaching with Historic Places (TwHP) lesson plan - TheAmericanRevolution.org description of the battle - BritishBattles.com description of the battle - Animated History of the Battle of Bunker Hill - The American Cyclopædia. 1879. . About people in the battle<\|endoftext\|>	3.765625
124	Brown dwarfs started out the same as ordinary stars, collapsing from giant nebulas of dust and gas. Most brown dwarfs are not quite massive enough to sustain a nuclear fusion reaction at their cores. Brown dwarfs, therefore, are transitional objects, standing between stars and giant gas planets. The mass of a brown dwarf can range from 13 to 90 times the mass of the planet Jupiter, or up to about a tenth the mass of the sun. (Image: © by Karl Tate, Infographics Artist) Have a news tip, correction or comment? Let us know at [email protected].<\|endoftext\|>	3.75
1,227	# RD Sharma Solutions Class 9 Chapter 7 Introduction To Euclids Geometry Read RD Sharma Solutions Class 9 Chapter 7 Introduction To Euclids Geometry below, students should study RD Sharma class 9 Mathematics available on Studiestoday.com with solved questions and answers. These chapter wise answers for class 9 Mathematics have been prepared by teacher of Grade 9. These RD Sharma class 9 Solutions have been designed as per the latest NCERT syllabus for class 9 and if practiced thoroughly can help you to score good marks in standard 9 Mathematics class tests and examinations Question 1. Define the following terms: (i) Line segment (ii) Collinear points (iii) Parallel lines (iv) Intersecting lines (v) Concurrent lines (vi) Ray (vii) Half-line Solution 1. (i) Line segment:- The line which has two end points it is called line segment. Here (ab) is a line segment. (ii) Collinear points:- When two or more point lie on a same line. Here point a, b and c are collinear points. (iii) Parallel lines:- Two lines having no common points are parallel lines. Two or more than two lines are parallel if they have no intersecting point. Both lines have common distance from each other. For example green and orange lines are parallel to each other. (iv) Intersecting lines:- Two lines are intersecting line if they have a common point. The common point is called point of intersection. Here point p is a intersecting point. (v) Concurrent lines:- Three or more than three lines are said to ne concurrent if there is a point which lies on all of them. (vi) Ray:- A line which one end point is fixed and the part can be extended endlessly. In other words A ray is defined as the part of the line with one end point such that it can be extended infinitely in the other direction. For example here (ab). (vii) Half-line:- If A, B, C be the points on a line, such that b lies between a and c, and if we remove the point b from line, the other two parts of that remain are each corner called half-line. It is different from ray as end point is not included in the half-line. Question 2. (i) How many lines can pass through a given point? (ii) In how many points can two distinct lines at the most intersect? Solution 2. (i) Infinitely many lines can pass through a given point. (ii) Two lines can intersect from only one point. Question 3. (i) Given two points P and Question , find how many line segments do they deter-mine. (ii) Name the line segments determined by the three collinear points P, Q and R. Solution 3. (i) One line segment we can determine. Question 4. Write the turth value (T/F) of each of the following statements: (i) Two lines intersect in a point. (ii) Two lines may intersect in two points. (iii) A segment has no length. (iv) Two distinct points always determine a line. (v) Every ray has a finite length. (vi) A ray has one end-point only. (vii) A segment has one end-point only. (viii) The ray AB is same as ray BA. (ix) Only a single line may pass through a given point. (x) Two lines are coincident if they have only one point in common. Solution 4. (i) False, it is not compulsory that the two lines always intersect each other. For example parallel lines not intersect. (ii) False, as there is only one common point between two intersecting lines. (iii) False, a segment is a part of a line so it is always of a definite length. (iv) True, a line can be determined by minimum of two distinct points. (v) False, a ray has not a finite length as it can be stretched infinitely. (vi) True, there is only a one end point in a ray other end will stretched infinitely. (vii) False, a line segment is a defined part and it has a definite length. (viii) False, as only one point is defined in a ray other is stretched infinitely so we can’t denote the ray with reversing name. (ix) False, many lines can pass through a given point. (x) False, two lines can said coincident if they lie exactly on each other. Question 5. In Fig. 7.17, name the following: (i) five line segments. (ii) five rays (iii) four collinear points (iv) two pairs of non-intersecting line segments. Solution 5. (iii) Four collinear points are in the given figure:- C,D,Q,S (iv) Two pairs of non-intersecting line segments are:- AB and CD, AB and LS Question 6. Fill in the blanks so as to make the following statements true: (i) Two distinct points in a plane determine a _________line. (ii) Two distinct _________ in a plane cannot have more than one point in common. (iii) Given a line and a point, not on the line, there is one and only _________line which passes through the given point and is _________to the given line. (iv) A line separates a plane into________parts namely the ________and the________itself. Solution 6. (i) Two distinct points in a plane determine a unique line. (ii) Two distinct lines in a plane cannot have more than one point in common. (iii) Given a line and a point, not on the line, there is one and only perpendicular line which passes through the given point and is perpendicular to the given line. (iv) A line separates a plane into three parts namely the two half plane and the line itself.<\|endoftext\|>	4.75
2,205	Bipedalism defines a method of locomotion by which organisms maneuver in their environment on two feet, and includes actions such as running, hopping, and walking. Organisms that habitually walk on two feet are called habitual bipeds and inhabit terrestrial environments. Organisms that occasionally support their weight on two hind legs, such as when fighting, foraging, copulating, or eating, are said to exhibit limited bipedalism. Organisms whose only method of locomotion on land involves two feet are called exclusive bipeds. Other terms used to label types of bipedal movement include facultative and obligate bipedalism; however, the distinction between organisms that do and don’t use bipedalism isn’t so clear. Bipedal behaviors are found on a spectrum upon which animals can be on the facultative end, the obligate end, or somewhere in between. Skeletal Changes for Bipedalism in Humans In humans, the foramen magnum—the hole in the skull through which the spinal cord leaves the head—is positioned more directly under the skull compared to quadrupeds, allowing bipeds to hold their heads erect when walking upright. The chest of a human is flatter (dorsal to ventral) than that of a quadruped. This keeps most of the weight of the chest near the spine and above one’s center of gravity, effectively helping to increase balance and stop us from falling forward. The spine of humans has a characteristic S-shaped curvature. The concave curve of the S positions the chest directly above where the spine and pelvis meet, and puts the weight of the chest, again, at the center of gravity. Aside from offering better balance, the S-shaped spine is also good for absorbing the mechanical shock that comes from walking. The pelvis of humans is wide and short. This squat shape provides increased stability to hold up an erect torso and to transfer much of the mechanical stress of weight bearing to the two lower limbs. Humans have legs longer than their arms, while quadrupeds have arms longer than their legs. The femur of humans is longer, straighter, and thinner than that of their quadruped counterparts. The longer bones allow for a bigger stride. The straighter shape ensures that the weight is evenly distributed down the length of the bone. In addition, the thinness of a biped femur makes for a lighter structure. The valgus angle (the angle at which the femur descends from the pelvis) in a biped is bigger than the angle in a quadruped. In a quadruped, the angle is zero; the femur descends with no slant, providing a wider stance. In a human, the femur angles inwards to bring the knees together, providing support at the center of gravity when walking upright. The feet of humans are specialized for walking only. Human feet are arched and have lost the ability to grasp objects. The arch acts like a spring that absorbs shock, and allows the weight of the body to be transferred from the heel to the ball of the foot as we stride. The toes are positioned to provide a thrusting motion against the ground to push the body forward. Bipedalism Origin Theories There are several highly contested theories surrounding the evolution of bipedalism in hominines. Some have already been debunked, and others are still viable contenders. Most of these hypotheses, however, emphasize some sort of environmentally-based pressure that favored the bipedal gait. Others focus on how bipedalism facilitates food acquisition, predator avoidance, and reproductive success. The Savanna-based Theory Widely debunked by modern-day scientists, this theory posits that bipedalism first arose when hominines migrated out of shaded forests and into the dry heat of the grasslands because of receding forests during a time of climate change. With tall grasses that obscured vision, and no trees to shelter our early ancestors from the sun, this theory suggests that bipedalism was advantageous in that it allowed early hominines to see farther into the distance by providing the height needed to peer over the grass. It was also believed that the bipedal posture acted as a thermoregulatory mechanism that decreased the surface area of skin exposed to an overhead sun (thermoregulatory model). Other advantages that pushed the evolution of bipedalism in the savanna, according to this theory, were that the upright posture made it easier to acquire food in trees that would have otherwise been out of reach, and walking on two legs was more energy efficient. We know now, however, that hominines had already acquired the ability to walk upright while still living the aeries of the forests, before moving out into the savannah. Hence, the challenges of the savannah and the advantages of bipedalism within it are irrelevant to the rise of bipedalism in human evolution. Aquatic Ape Hypothesis More popular among the general public than with scientists themselves, this hypothesis suggests our early ancestors took to a more aquatic lifestyle after being outcompeted and forced from the forest trees. This band of apes, then, depending heavily on aquatic food sources, acquired bipedalism to allow them to wade into deeper waters for improved food acquisition. Supporting this hypothesis, the wading model is based on the fact that great apes and other large primates will wade into waters looking for food and begin walking upright when they’ve waded in waist-deep to keep their heads above water. While recent studies have found some supportive evidence for this hypothesis, there is still not enough to allow us to conclusively accept or reject it. Still, most scientists do not consider this hypothesis seriously, especially because most primates will avoid going into the water unless absolutely necessary. The waters are inhabited by creatures deadly to apes and humans, such as crocodiles and hippopotami. The Postural Feeding Hypothesis One of the more likely scenarios of the evolution of bipedalism is the postural feeding hypothesis. This hypothesis is based on the observation that chimpanzees, our closest living relatives, use bipedalism only while eating. Being able to stand on two hind feet allows chimps on the ground to stand and reach for low-hanging fruit, as well as allowing chimps in the trees to stand and reach for a higher branch. This hypothesis suggests that these occasional bipedal actions eventually became habitual actions because of how advantageous they were in acquiring food. Further evidence supporting the idea that bipedalism arose from the need to maneuver better in the treetops comes from the observation of orangutans using their hands for stability when the branches they were moving through were unsteady. In other words, orangutans were relying on their bipedal actions to move through the trees, only using their hands for extra support when needed. This adaptation would have been very useful while the forests and their trees were thinning. Early Bipedalism in Homininae Model An interesting idea surrounding the evolution of bipedalism is that it was a trait in all early hominids that was either lost or retained in varying lineages. The idea came about when a 4.4-million year old fossil of A. ramidus was found. The fossil structures suggested that the organism was bipedal, and this gave birth to the idea that chimpanzees and gorillas both started out with a bipedal gait, but that each evolved more specialized means of locomotion for their different environments. Some chimps, this model posits, lost the ability to walk upright when they settled into arboreal habitats. Some of these arboreal chimps later acquired a particular type of locomotion called knuckle-walking, as seen in gorillas. Other chimps acquired a different type of locomotion that made it easier to walk and run on the ground, as seen in humans. The Threat Model A recent suggestion posits that bipedalism was used by early hominines to make themselves look bigger and more threatening to prevent predators from seeing them as easy prey. Many non-bipeds will do this when threatened; however, this idea states that early hominines used bipedalism for as long, and as often, as they could, whether or not they were currently being threatened, until it eventually became habitual. The (Male) Provisioning Model This theory on the evolution of bipedalism links bipedalism to the practice of monogamy. It suggests that, in order to improve the chances that an offspring will survive, early hominines entered into a binding paired relationship in which the male would forage for provisions and the female, in return for these provisions, would reserve herself for her partner and care for their offspring. The provisions-providing males supposedly walked upright to free their arms to carry food and resources back to their families. Furthermore, as bipedalism emerged and physiological changes arose to accommodate the new lifestyle, newborn babies would experience increased difficulty in independently hanging onto the mother, requiring the mother to use her arms to carry the child and forcing the use of a bipedal gait. 1. Which is not a driving force of bipedalism mentioned by the many proposed hypotheses? A. Bipedalism allows for the acquisition of food sources at higher places. B. Bipedalism makes the organism look big and scary. C. Bipedalism frees up the hands for uses other than locomotion. D. Bipedalism facilitates stealthy movement. 2. When walking on the ground, a bird uses its legs to hop around. What type of bipedalism is this? A. Obligate bipedalism B. Facultative bipedalism C. Limited bipedalism D. Habitual bipedalism - Aquatic Ape Theory (n.d.). Retrieved June 13, 2017, from http://www.primitivism.com/aquatic-ape.htm - Human skeletal changes due to bipedalism. (2017, June 08). Retrieved June 13, 2017, from https://en.wikipedia.org/wiki/Human_skeletal_changes_due_to_bipedalism - Ko, K. H. (n.d.). Origins of Bipedalism. Retrieved June 13, 2017, from http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1516-89132015000600929<\|endoftext\|>	4.15625
795	One of the simplest and highly conserved bacterial shapes is a rod. In order to grow, rod-shaped bacteria restrict their growth so that they maintain a constant width as they elongate along their length. However, it is not understood how bacteria form into a rod shape, much less how they maintain rod shape as they grow. One key protein, MreB, plays an important role in this process. Bacteria that lose MreB become spherical. MreB polymerizes into short, curved membrane-bound filaments, interacting with the enzymes that build the cell wall. These MreB filament/enzyme complexes, as they insert new material into the growing cell wall, move in circle around the cell width, and perpendicularly to the cell length. It is thought that this circumferential motion causes the insertion of “hoops” of new material around the rod width, an organized arrangement that helps the rod shape resist the high pressures within the cell. Saman Hussain and Carl Wivagg sought to understand how this organized movement arises: how do these short, independent filaments know how to move around the rod width, instead of moving in any other direction? To study this, they examined MreB motion in cells of different shapes. When cells were rods of any width, they could see that MreB moved circumferentially. But when cells became round, they were surprised to see that MreB motion became random, with filaments moving in all directions. If they confined these round cells into rod shapes in microfluidic molds, the motion of the filaments became oriented again, indicating MreB filaments can sense the local cell shape to move around the cell. To test if the MreB filaments were moving along tracks of old material in the cell wall, they removed the cell wall. In round wall-less cells, filaments were oriented in all directions, but when they confined the wall-less cells into rods, MreB filaments aligned around the rod width. To test if purified MreB filaments alone were able to orient around tubes, they examined (in collaboration with Piotr Szwedziak in the Löwe lab at Cambridge University) the structure of pure MreB assembled inside liposomes. MreB filaments deformed the liposomes into rod-like tubes, and oriented along the circumference, just like in cells. Putting all of this data together, Carl and Saman realized that these inwardly curved MreB filament orient so they point along the most inwardly curved surface thus maximizing membrane interaction. In round cells, there is no difference in the two curvatures causing MreB filaments to have no preference in their orientation, resulting in a random distribution of orientations. In collaboration with Felix Wong in the Amir lab at Harvard, they showed by mathematical modelling that circumferential membrane binding is the most energetically favorable conformation for highly curved filaments, a behavior predicted to be robust no matter how wide the cell, or pressurized it is. Thus, by sensing the existing cell shape and orienting new cell wall synthesis to reinforce that shape, curved MreB filaments create a local feedback loop allowing rod shape to be robustly maintained. This local feedback of sensing then reinforcing local shape may also help bacteria to form into rod shapes in the first place. When round cells were forced to reform into rods, they found that rods form not from the sphere thinning down to make a rod, but instead they suddenly emerge from small local outward bulged on the sphere surface. These bulges create small rod like regions where MreB filaments could orient within, allowing them to quickly elongate as growing rods. Therefore, the curvature-orienting properties of MreB not only help cells maintain rod shape, they also help cells form into rods in the first place.<\|endoftext\|>	3.8125
926	# How to Calculate and Solve for Reaction: Lift Moves Down \| Motion The image above represents reaction: lift moves down. To calculate reaction: lift moves down, three essential parameters are needed and these parameters are mass (m), acceleration (a)Â andÂ acceleration due to gravity (g). The formula for calculating reaction: lift moves down: R = m(g – a) Where; R = Reaction m = Mass g = acceleration due to gravity a = Acceleration Let’s solve an example; Find the reaction when the mass is 28, acceleration is 9 and acceleration due to gravity is 9.8. This implies that; m = Mass = 28 g = acceleration due to gravity = 9.8 a = Acceleration = 9 R = m(g – a) So, R = 28(9.8 – 9) R = 28(0.80) R = 22.4 Therefore, theÂ reactionÂ is 22.4 N. ### Calculating the Mass when the Reaction, the Acceleration and the Acceleration due to Gravity are Given m = R / gÂ – a Where; m = Mass R = Reaction g = acceleration due to gravity a = Acceleration Let’s solve an example; Find the mass when the reaction is 42, the acceleration is 8 and the acceleration due to gravity is 9.8 ### Master Every Calculation Instantly Unlock solutions for every math, physics, engineering, and chemistry problem with step-by-step clarity. No internet required. Just knowledge at your fingertips, anytime, anywhere. This implies that; R = Reaction = 42 g = acceleration due to gravity = 9.8 a = Acceleration = 8 m = R / g – a Then, m = 42 / 9.8 + 8 m = 42 / 17.8 m = 2.359 Therefore, theÂ massÂ isÂ 2.359. ### Calculating the Acceleration when the Reaction, the Mass and the Acceleration due to Gravity are Given a = g – (R / m) Where; a = Acceleration R = Reaction m = Mass g = acceleration due to gravity Let’s solve an example; Find the acceleration when the reaction is 35, mass is 19 and acceleration due to gravity is 9.8. This implies that; R = Reaction = 35 m = Mass = 19 g = acceleration due to gravity = 9.8 a = g – (R / m) So, a = g – (35 / 19) a = 9.8 – 1.84 a = 7.96 Therefore, theÂ accelerationÂ isÂ 7.96. ### Calculating the Acceleration when the Reaction, the Mass and the Acceleration are Given g = (R / m) + a Where; g = acceleration due to gravity R = Reaction m = Mass a = Acceleration Let’s solve an example; Find the acceleration due to gravity when the reaction is 40, the mass is 4 and the acceleration is 7. This implies that; R = Reaction = 40 m = Mass = 4 a = Acceleration = 7 g = (R / m) + a Then, g = (40 / 4) + 7 g = 10 + 7 g = 17 Therefore, theÂ acceleration due to gravityÂ isÂ 17. ### How to Calculate and Solve for Reaction: Lift Moves Down Using Nickzom Calculator Nickzom Calculator â€“Â The Calculator Encyclopedia is capable of calculating the reaction: lift moves down. To get the answer and workings of the reaction: lift moves down using the Nickzom Calculator â€“ The Calculator Encyclopedia.Â First, you need to obtain the app. You can get this app via any of these means: To get access to theÂ professionalÂ version via web, you need toÂ registerÂ andÂ subscribeÂ forÂ NGN 1,500Â perÂ annumÂ to have utter access to all functionalities. You can also try theÂ demoÂ version viaÂ https://www.nickzom.org/calculator<\|endoftext\|>	4.40625
1,367	Question Video: Finding the Area of a Lamina given the Expression of Its Rate of Change by Using Indefinite Integration \| Nagwa Question Video: Finding the Area of a Lamina given the Expression of Its Rate of Change by Using Indefinite Integration \| Nagwa # Question Video: Finding the Area of a Lamina given the Expression of Its Rate of Change by Using Indefinite Integration Mathematics • Third Year of Secondary School ## Join Nagwa Classes Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher! The area A of a lamina is changing at the rate d๐ด/d๐ก = ๐^(โ0.7๐ก) cmยฒ/s, starting from an area of 60 cmยฒ. Give an exact expression for the area of the lamina after 30 seconds. 04:13 ### Video Transcript The area ๐ด of a lamina is changing at the rate d๐ด by d๐ก equals ๐ to the power of negative 0.7๐ก square centimeters per second, starting from an area of 60 square centimeters. Give an exact expression for the area of the lamina after 30 seconds. The key to answering this question is to spot that weโve been given information about the rate of change of the area. Thatโs d๐ด by d๐ก, in other words, the derivative of ๐ด with respect to ๐ก. Now, we know that integration and differentiation are the reverse of one another. So we can find an expression for ๐ด by integrating d๐ด by d๐ก with respect to ๐ก. Now, what will happen is this will give us a general solution. And weโll need to use the fact that the starting area is 60 square centimeters to find a particular solution to this equation. But to begin, weโll simply integrate the expression for d๐ด by d๐ก with respect to ๐ก. Thatโs the indefinite integral of ๐ to the power of negative 0.7๐ก with respect to ๐ก. Now, here we can recall the general result for the integral of ๐ to the power of ๐๐ฅ with respect to ๐ฅ for real constant values of ๐ด. Itโs one over ๐ times ๐ to the power of ๐๐ฅ plus a constant of integration ๐ถ. Now, in our example, we can see we can let ๐ be equal to negative 0.7 or negative seven-tenths. This means when we integrate ๐ to the power of negative 0.7๐ก, we get one over negative seven-tenths times ๐ to the power of negative 0.7๐ก. And of course we need that constant of integration ๐ถ. And we recall that to divide by a fraction, we simply multiply by the reciprocal of that fraction. Now, letโs think of negative seven over 10 as negative seven-tenths. And we see that this is equal to one times negative 10 over seven, which is simply negative 10 over seven. And so weโve found a general equation for ๐ด. Itโs ๐ด is equal to negative 10 over seven times ๐ to the power of negative 0.7๐ก plus our constant of integration ๐ถ. Now, recall we actually want to find the area of the lamina after 30 seconds, in other words, when ๐ก is equal to 30. So weโll begin by finding the value of our constant. And weโll use the fact that the starting area was 60 square centimeters. In other words, when ๐ก is equal to zero, ๐ด is equal to 60. Substituting these values into our equation, and we get 60 equals negative ten-sevenths times ๐ to the power of zero plus ๐ถ. But of course ๐ to the power zero is one. So we have 60 equals negative ten-sevenths plus ๐ถ. Letโs clear some space and solve our equation for ๐ถ. Weโre going to rewrite 60 as 420 over seven. Now, that comes from the fact that we can write 60 as 60 over one and then multiply the numerator and the denominator by seven. Then weโre easily able to add ten-sevenths to both sides of our equation to solve for ๐ถ. So we find ๐ถ is equal to four hundred and thirty sevenths. And so weโve found a particular equation for the area, given this information about its starting area. ๐ด is equal to negative ten-sevenths times ๐ to the power of negative 0.7๐ก plus 430 over seven. Now, remember, we want an exact expression for the area of the lamina after 30 seconds. So we substitute ๐ก equals 30 into this equation. Weโre not going to type this into our calculator. Remember, weโre looking to find an exact expression. So instead, weโll simply evaluate negative 0.7 times 30. Negative 0.7 times 30 is negative 21. And of course weโre working in square centimeters. So we can say that the exact expression for the area of the lamina after 30 seconds is negative ten-sevenths ๐ to the power of negative 21 plus 430 over seven square centimeters. ## Join Nagwa Classes Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher! • Interactive Sessions • Chat & Messaging • Realistic Exam Questions Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy<\|endoftext\|>	4.5625
1,282	Multiplication Table of 5 Repeated addition by 5’s means the multiplication table of 5. (i) When 6 bowls of five fruits each. By repeated addition we can show 5 + 5 + 5 + 5 + 5 + 5 = 30 Then, five 6 times or 6 fives 6 × 5 = 30 Therefore, there are 30 fruits. (ii) When 9 baskets each having 5 shirts. By repeated addition we can show 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 45 Then, five 9 times or 9 fives 9 × 5 = 45 Therefore, there are 45 shirts. We will learn how to use the number line for counting the multiplication table of 5. (i) Start at 0. Hop 5, four times. Stop at 20. 4 fives are 20 4 × 5 = 20 (ii) Start at 0. Hop 5, seven times. Stop at ____. Thus, it will be 35 7 fives are 35 7 × 5 = 35 (iii) Start at 0. Hop 5, twelve times. Stop at ____. Thus, it will be 60 12 fives are 60 12 × 5 = 60 How to read and write the table of 5? The above chart will help us to read and write the 5 times table. Read 1 five is 5 2 fives are 10 3 fives are 15 4 fives are 20 5 fives are 25 6 fives are 30 7 fives are 35 8 fives are 40 9 fives are 45 10 fives are 50 11 fives are 55 12 fives are 60 Write 1 × 5 = 5 2 × 5 = 10 3 × 5 = 15 4 × 5 = 20 5 × 5 = 25 6 × 5 = 30 7 × 5 = 35 8 × 5 = 40 9 × 5 = 45 10 × 5 = 50 11 × 5 = 55 12 × 5 = 60 Now we will learn how to do forward counting and backward counting by 5’s. Forward counting by 5’s: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, …… Backward counting by 5’s: ……, 120, 115, 110, 105, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 0. Multiplication Table Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. Recent Articles 1. Fraction in Lowest Terms \|Reducing Fractions\|Fraction in Simplest Form Feb 28, 24 04:07 PM There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1… 2. Equivalent Fractions \| Fractions \|Reduced to the Lowest Term \|Examples Feb 28, 24 01:43 PM The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole. We can see the shade portion with re… 3. Fraction as a Part of Collection \| Pictures of Fraction \| Fractional Feb 27, 24 02:43 PM How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i… 4. Fraction of a Whole Numbers \| Fractional Number \|Examples with Picture Feb 24, 24 04:11 PM Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One… 5. Identification of the Parts of a Fraction \| Fractional Numbers \| Parts Feb 24, 24 04:10 PM We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been Worksheet on Multiplication Table of 0 Worksheet on Multiplication Table of 2 Worksheet on Multiplication Table of 3 Worksheet on Multiplication Table of 4 Worksheet on Multiplication Table of 5 Worksheet on Multiplication Table of 6 Worksheet on Multiplication Table of 7 Worksheet on Multiplication Table of 8 Worksheet on Multiplication Table of 10 Worksheet on Multiplication Table of 11 Worksheet on Multiplication Table of 12 Worksheet on Multiplication Table of 13 Worksheet on Multiplication Table of 14 Worksheet on Multiplication Table of 17 Worksheet on Multiplication Table of 18 Worksheet on Multiplication Table of 19 Worksheet on Multiplication Table of 20 Worksheet on Multiplication Table of 21 Worksheet on Multiplication Table of 23 Worksheet on Multiplication Table of Worksheet on Multiplication Table of 25<\|endoftext\|>	4.84375
755	US Constitution Article 1 Section 5 Clause 2: Rules Each House may determine the Rules of its Proceedings,... This includes how discussions are held and how votes are brought to the floor. President of the Senate Unlike in the House of Representatives, where the Speaker is both the presiding officer and the leader of the majority party, the presiding officer in the Senate (whether the Vice President of the US, the President Pro Tempore or a junior senator who is just filling in) has little political power according to Senate rules. The main powers are to cast tie-breaking votes and to make judgments about parliamentary rules (which can be appealed to the whole chamber - incidentally, this is how the "nuclear option" was invoked). Otherwise, the majority leader, committee chairs and even individual senators share the power, and the presiding officer must mostly act as they wish. If that means the majority leader (backed by the majority party) do not want to hold a vote, then the presiding officer must respect that under the rules. Even if the presiding officer was particularly brazen and did not respect the rules, the senators in the majority could begin a quick rebellion using tactics like e.g. fleeing to eliminate a quorum (see below). At some point, the executive branch needs the Senate to accomplish its goals (e.g. passing appropriations), and so fomenting a rebellion among senators would be counterproductive. Why can't the rules change? The Senate is a permanent body since only 1/3 of senators' terms expire at the end of a given Congress, unlike the House of Representatives, where all members' terms expire at the end of every Congress. Since the Senate is permanent, its rules live on between Congresses. Rule changes require either a supermajority or the politically-fraught "nuclear option." The Presiding Officer may at any time lay, and it shall be in order at any time for a Senator to move to lay, before the Senate, any bill or other matter sent to the Senate by the President or the House of Representatives for appropriate action allowed under the rules and any question pending at that time shall be suspended for this purpose. Any motion so made shall be determined without debate. This rule allows the presiding officer to supersede pending business with two major constraints: the superseding matter must be from either the President or the House or Representatives and whatever action must be an "appropriate action allowed under the rules." This rule is used for things like announcing that the president has made a nomination, that the president has vetoed a Senate bill or that the House has passed a bill, all three of which require action by the Senate. Other rules (both precedents and explicit rules in the Standing Rules) govern what the appropriate actions are (in general, a first reading and then referral to committee). The Constitution requires that each house can only conduct business when a quorum is present, and defines as quorum as a majority of members of each respective house. This means that if a minority party (or a presiding officer aligned with them) attempted to take command, the majority party could simply flee en masse, leaving only one member to raise points of order about the absence of a quorum and so on. The Senate rules allow a minority of the Senate to empower the Sergeant-at-Arms to arrest vacant members and bring them back to the Senate, but it would still be very hard to do this on a mass scale. (Something similar happened several years ago in Wisconsin, where a slim Republican majority in the state legislature were trying to pass legislation that would weaken unions, and Democrats in the state senate fled Wisconsin. Wisconsin requires a quorum of 60% of members, so in that case the minority party had an effective veto.)<\|endoftext\|>	3.9375
625	# The Commutative and Associative Properties Homework ```Name: ______________________________ Block: _______ The Commutative and Associative Properties Homework Commutative Property of Multiplication Associative Property of Multiplication Part I: Complete the equations below. Then, identify the property you used. A. 6 + 3 = 3 + ____ _______________________________________ B. 4 • (7 • 5) = (______) • 5 _______________________________________ C. 4 • ____ = 2 • 4 _______________________________________ D. (2 + 4) + 7 = 2 + (______) _______________________________________ 3 • (6 • 4) = (______) • 4 _______________________________________ F. (3 + 4) + 8 = 3 + (______) _______________________________________ G. -7 + 3 = 3 + ____ _______________________________________ H. 3 • ____ = 5 • 3 _______________________________________ E. Part II: Determine if each statement below is true or false. If it is false, correct the statement. 1) The Commutative Property says that numbers being multiplied and divided can be reordered. 2) The Commutative Property of Multiplication states that numbers being multiplied can be regrouped without affecting the product. 3) 25 – 14 = 14 – 25. Name: ______________________________ Block: _______ Part III: Consider the expressions below.  Circle any examples of the Commutative Property.  Put a star by any examples of the Associative Property.  Put an “x” through any examples that are neither commutative nor associative. Hint: Keep in mind that it is possible for a set of expressions to model two different properties. A) 6+8 8+6 B) (2 • -3) • 4 2 •(-3 • 4) C) 5 + (3 + 2) + -1 5 + 3 + (2 + -1) D) 5-4+2 4-5+2 E) (9 • 3) + 5 (3 • 9) + 5 F) (7 ÷ 4) • 9 (4 ÷ 7) • 9 G) 3 + -5 • 4 3 + 4 • -5 H) (3 • 4) • (2 • 6) (4 • 2) • (6 • 3) I) (7 + 2) + 9 (2 + 7) + 9 J) 7-8-3 8-7–3 K) (10 + -32) • 16 10 + (-32 • 16) L) 8 + (6 + 4) + 2 8 + 2 + (4 + 6) ```<\|endoftext\|>	4.8125
269	Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Creating the Exam: I use the free features on Problem Attic to create an exam. At that site, you can peruse released exam items from dozens of state tests. There's also a paid Common Core version of the site, but I've yet to check it out. I choose approximately 60 - double the number of questions I need - problems that look like they're fair and applicable to the learning targets, then I revise the exam to balance the number of problems that will assess each SLT. I try to use a few problems that address multiple SLTs, and, because I'm not grading on % correct, I can throw a few devilishly hard problems, just to give kids exposure and see how they do. Giving the Exam: Nothing special here, and I'm sure you have your own exam procedures. Students arrive to class, I distribute the exam, and they go for it.<\|endoftext\|>	3.671875
168	Unit 2: How Trade and Travel Changed the World Lesson F: Renaissance and Reformation Activity 9: Reformation Cause and Effect You learned about the key causes and effects of the religious Reformations. Assess your knowledge by completing the assignment below. Directions: Listed below are causes of the Reformation. Read each of the actions or "causes" and then match it with the correct impact or "effect." When you have completed the activity, print a copy of the answers and keep it in your notebook for reference. The religious and political changes in Europe during the 16th and 17th century provided a preview of the global transformations that occurred in the Modern Era. The consequences of these changes were influenced by the spread of printed information and innovation that accelerated the impacts of the Renaissance and Reformations.<\|endoftext\|>	4.125
1,216	Last updated: December 24, 2016 Job - Card Sorting - Grade Level: - Upper Elementary: Third Grade through Fifth Grade - Social Studies - Lesson Duration: - 30 Minutes - Common Core Standards: - 4.L.1, 4.L.1.f, 4.L.3, 4.L.3.a, 4.L.4.c - State Standards: - Thinking Skills: - Remembering: Recalling or recognizing information ideas, and principles. Understanding: Understand the main idea of material heard, viewed, or read. Interpret or summarize the ideas in own words. Applying: Apply an abstract idea in a concrete situation to solve a problem or relate it to a prior experience. Analyzing: Break down a concept or idea into parts and show the relationships among the parts. Students sort jobs of Bent's Fort into different categories and discuss their reasoning. This activity is mainly to provoke discussion amongst students about the different jobs and roles that were at Bent's Fort. There were many different workers at Bent's Fort. All of these workers helped make Bent's Fort a successful fur trading company from 1833-1849. The main part of the trading post was to trade Buffalo Robes with the plains Native American tribes (mostly Cheyenne, Arapaho, Kiowa, Comanche, Apache) and trade those buffalo robes in the east for a profit. Bent's Fort was the center of the trade for the Bent, St. Vrain Company, though the company did have smaller posts. The owners of the Bent, St. Vrain company were William Bent, Charles Bent and Ceran St. Vrain. William Bent married Owl Woman, a Cheyenne Indian, which helped with trade negotiations with the Cheyenne tribe. Although, Bent's fort was always the scene of some trading activity with visiting Indians, by far the most profitable aspect of Bent's business was he field operations. Here he might employ as many as 4 or 5 men to visit scattered Indian camps and trade for robes. These traders, of course, were carefully picked because of their knowledge of and friendship with various tribes and a list of their names includes many of the most famous pioneers of the Southwest. Employed at one time or another by William Bent were Tom Boggs, William Boggs, Kit Carson, Lucien Maxwell, John Smith, Jim Beckwourth, John Hatcher, William Guerrier, "Uncle Dick" Wootton, Sol Silver, and "old" Bill Williams. Mexicans were hired as laborers for their knowledge of working with adobe and experience for other laborer duties, such as taking care of livestock. It was recorded that the people working at Bent's Fort was largely hispanic. Mexico was across the Arkansas River located close to Bent's Fort. Company trappers and free trappers spent the majority of their time in the mountains catching beaver and other mammals to trade. Domestic workers (cooks) usually consisted of the wives of the workers. There is only one record of a Euroamerican woman working as a cook at Bent's Fort. Charlotte was a black slave of the Bent family. She was known for making the best pumpkin pies and being the "Belle of the Ball" during Fandangos (dances). Otherwise, the women of the fort consisted of Indians and Mexicans. There could potentially be a male cook. The carpenter and blacksmith usually work closely. Their main job was to fix the wagons. The wagons carried all of the trade goods. The blacksmith would also make and fix horse, mule and oxen shoes when needed. It was rare for a trained doctor to go west for work. Usually men on the Santa Fe Trail had some medical knowledge (of the time) to help each other in different medical situations. A doctor at the fort may have been hired as a different job but with little medical training and a medical book available, he could become the fort doctor. One recorded doctor at the fort was Doctor Hempstead. The fort was mostly a seasonal fort. Workers may only live and work at the fort temporarily. Other travelers on the Santa Fe Trail would, only rarely, be able to make arrangements to sleep inside the fort. Lewis Gerrard wrote in his book Wah-To-Yah and The Taos Trail, that he arranged to stay in the fort for $1 a day. It was rare that a person learned how to read and write. Unless you are using this activity as an introduction to jobs at Bent's Fort, the students should be prepared with some knowledge of the various roles. Students learn about the various 19th century jobs at Bent's Fort either through individual research, a class trip to Bent's Old Fort NHS or class discussions and lessons. List of jobs you can print and cut out. - Students are in small discussion groups. - Each group receives cards, (or are told to write and make their own) with roles of people who were at Bents Fort. (Carpenter, Blacksmith, Laborer, Company Trader, Owner, Cook, Laborer’s wife, Indian Agent, Doctor, Guard, Clerk, Native Americans, Mexicans, Euroamericans). - Give the students different situations to sort these roles into. (Interpretation could vary for each question). Students discuss their reasoning. - Who sleeps inside the fort? - Who eats in the dining room? - There was basically a caste system, sort the cards into which workers are considered higher in society at Bents Fort. - People who had more opportunity to decide their own role. - People who needed an education for their role. - Other places these people could live and work. Information about various jobs could be found in books, YouTube clips and talking to living historians at Bent's Old Fort National Historic Site.<\|endoftext\|>	3.6875
666	# How Do You Know How Many Terms Are In An Expression? ## How to know how many terms are in a sentence? 3x 2 + 2y + 7xy + 5 In an algebraic expression, the terms are elements separated by plus or minus signs. In this example, four terms, 3x 2 , 2y, 7xy and 5. The terms can be composed of variables and coefficients or constants. ## How to find the members of an expression? A term is a single number or variable, or numbers and variables multiplied together. Now we can say that this expression has only two terms, or the second term is a constant, or are you sure that the coefficient is really 4? ## How many terms does a mathematical expression have? A term is a single number or variable, or numbers and variables multiplied together. Now we can say that this expression has only two terms, or the second term is a constant, or are you sure that the coefficient is really 4? ## What are the terms in an expression? A term is a single mathematical expression. It can be a single number (positive or negative), a single variable (a letter), or multiple variables that are multiplied but never added or subtracted. Some terms contain variables preceded by a number. The number before the term is called the coefficient. ## What is an expression and an equation? An expression is a number, a variable, or a combination of numbers, variables, and operational symbols. An equation is made up of two expressions joined by an equals sign. ## What is the constant term? In mathematics, a constant term is a term in an algebraic expression that has a constant value or cannot change because it contains no variable variables. For example, in a quadratic polynomial. 3 is a permanent member. ## How many terms does the example expression contain? In an algebraic expression, the terms are elements separated by plus or minus signs. In this example, four terms, 3x 2 , 2y, 7xy and 5. The terms can be composed of variables and coefficients or constants. In algebraic expressions, letters represent variables. ## What is a trinomial expression? Trinomial: An expression with 3 terms is called a trinomial. For example: 3x + 4yz, 5xy + 4yz + 4, a + b -c. Polynomial: An expression with more than 3 terms is called a polynomial. ## How do you classify similar terms? Like terms are terms whose variables (and their exponents, such as 2 in x 2 ) the same. In other words, terms that seem similar. Note: The coefficients (numbers multiplied by, for example, 5 times 5) can be different. ## Are the constants like terms? Like terms are terms that contain the same variable to the same extent. In 5x + y 7 the terms are 5x, y and 7, which have different variables (or no variables), so there are no like terms. Constants are terms without variables, so 7 is a constant. … The coefficients are 4, 5 and 3 and the constant is 6.<\|endoftext\|>	4.78125
818	# Chapter 6 Hypotheses texts. Central Limit Theorem Hypotheses and statistics are dependent upon this theorem. ## Presentation on theme: "Chapter 6 Hypotheses texts. Central Limit Theorem Hypotheses and statistics are dependent upon this theorem."— Presentation transcript: Chapter 6 Hypotheses texts Central Limit Theorem Hypotheses and statistics are dependent upon this theorem Central Limit Theorem To understand the Central Limit Theorem we must understand the difference between three types of distributions….. A distribution is a type of graph showing the frequency of outcomes: Of particular interest is the “normal distribution” Different populations will create differing frequency distributions, even for the same variable… There are three types of distributions: 1. Population distributions There are three types of distributions: 1. Population distributions There are three types of distributions: 1. Population distributions There are three types of distributions: 1. Population distributions 2. Sample distributions There are three types of distributions: 1. Population distributions 2. Sample distributions There are three types of distributions: 1. Population distributions 2. Sample distributions 3. Sampl ing distributions There are three types of distributions: 1. Population distributions The frequency distributions of a population. There are three types of distributions: 2. Sample distributions The frequency distributions of samples. The sample distribution should look like the population distribution….. Why? There are three types of distributions: 2. Sample distributions The frequency distributions of samples. There are three types of distributions 3. Sampl ing distributions The frequency distributions of statistics. There are three types of distributions: 2. Sample distributions The frequency distributions of samples. The sampling distribution should NOT look like the population distribution….. Why? Suppose we had population distributions that looked like these: Say the mean was equal to 40, if we took a random sample from this population of a certain size n… over and over again and calculated the mean each time…… We could make a distribution of nothing but those means. This would be a sampling distribution of means. Central Limit Theorem If samples are large, then the sampling distribution created by those samples will have a mean equal to the population mean and a standard deviation equal to the standard error. Type I and Type TT errors –Type I ： reject the correct original Hypothesis ， called Producer's Risk –Type II ： accept the wrong original Hypothesis ， called Consumer’s Risk Population condition conclu sion H o true H a true AcceptH 0 correct type TT error conclusion RejectH 0 Type I error correct conclusion We denote the probabilities of making the two errors as follows: α——the probability of making a Type I error β——the probability of making a Type TT error In practice ， the person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error ， called the level of significance for the test 。 Common choices for the level of significance are α=0.05 orα= 0.01 。 Type I and Type TT errors Sampling Error = Standard Error The sampling distribution will be a normal curve with: and This makes inferential statistics possible because all the characteristics of a normal curve are known. Errors: Type I Error: saying something is happening when nothing is: p = alpha Type II Error: saying nothing is happening when something is: p = beta Steps of Hypothesis Testing 1.Determine the null and alternative hypotheses. 2.Specify the level of significance . 3.Collect the sample data and calculate the test statistic. Using the p -Value 4.Use the value of the test statistic to compute the p - value. 5.Reject H 0 if p -value < . Steps of Hypothesis Testing Using the Critical Value 4.Use  to determine the critical value for the test statistic and the rejection rule. 5.Use the value of the test statistic and the rejection rule to determine whether to reject H 0.<\|endoftext\|>	4.5
397	How is a person’s blood glucose level measured? Blood glucose levels are measured with four major tests, including the fasting plasma blood glucose, random plasma blood glucose, oral glucose tolerance, and glycated hemoglobin (HbA1c) tests. The first three tests measure a person’s blood glucose in terms of milligrams per deciliter (seen as mg/dl or mg/dL), whereas the HbA1c (or A1c) test is measured in percentages. What are considered to be “normal” blood glucose levels? Although everyone’s blood glucose levels change during the day, when a person’s blood glucose levels are measured, there are some standards advocated by health care professionals. The following chart shows the target ranges most health care professionals use to make a diagnosis of diabetes or no diabetes—based on four types of tests (for more details about these tests, see the chapter “Taking Charge of Diabetes”). What are considered general blood glucose level targets for a person with diabetes? Although there are exceptions to every rule, the following lists the general blood glucose level targets for a person who has diabetes (note: these numbers may not apply to all people with diabetes): - Fasting or before-meal glucose level—90 to 130 mg/dl - After-meal glucose (or two hours after the start of the meal) level—greater than (>) 180 mg/dl - Bedtime glucose—100 to 140 mg/dl Have standard levels for blood glucose levels changed over the years? A main goal for a person with or without diabetes is to maintain certain blood glucose levels that sustain his or her health. And over the past two decades, the numbers that indicate “normal,” “prediabetes,” and “diabetes” have changed, mainly because of advancements in medical research and technology.<\|endoftext\|>	3.671875
5,236	# Linear Equation Problems Value of the unknown quantity for which from given equation we get true numerical equality is called root of that equation. Two equations are called equivalent when the multitudes of their roots match, the roots of the first equation are also roots of the second and vice versa. The following rules are valid: 1. If in given equation one expression is substituted with another identity one, we get equation equivalent to the given. 2. If in given equation some expression is transferred from one side to the other with contrary sign, we get equation equivalent to the given. 3. If we multiply or divide both sides of given equation with the same number, different from zero, we get equation equivalent to the given. Equation of the kind $ax + b = 0$, where $a, b$ are given numbers is called simple equation in reference to the unknown quantity $x$. Problem 1 Solve the equation: A) 16x + 10 – 32 = 35 – 10x - 5 B) $y + \frac{3}{2y} + 25 = \frac{1}{2}y + \frac{3}{4}y – \frac{5}{2}y + y + 37$ C) 7u – 9 – 3u + 5 = 11u – 6 – 4u Solution: A)We do some of the makred actions and we get 16x – 22 = 30 – 10x After using rule 2 we find 16x + 10x = 30 + 22 After doing the addition 26x = 52 We find unknown multiplier by dividing the product by the other multiplier. That is why $x = \frac{52}{26}$ Therefore x = 2 B) By analogy with A) we find: $y\left(1 + \frac{3}{2}\right) + 25 = y\left(\frac{1}{2} + \frac{3}{4} – \frac{5}{2} + 1\right) + 37 \Leftrightarrow$ $\frac{5}{2}y + 25 = -\frac{1}{4}y + 37 \Leftrightarrow \frac{5}{2}y + \frac{1}{4}y = 37 - 25 \Leftrightarrow$ $\frac{11}{4}y = 12 \Leftrightarrow y = \frac{12.4}{11} \Leftrightarrow y = \frac{48}{11}$ C) 4u – 4 = 7u – 6 <=> 6 – 4 = 7u – 4u <=> 2 = 3u <=> $u = \frac{2}{3}$ Problem 2 Solve the equation : A) 7(3x – 6) + 5(x - 3) - 2(x - 7) = 5 B) (x -3)(x + 4) - 2(3x - 2) = (x - 4)2 C) (x + 1)3 – (x - 1)3 = 6(x2 + x + 1) Solution: A) 21x - 42 + 5x - 15 - 2x + 14 = 5<=> 21x + 5x - 2x = 5 + 42 + 15 - 14<=> 24x = 48 <=> x = 2 B) x2 + 4x - 3x - 12 - 6x + 4 = x2 - 8x + 16 <=> x2 - 5x – x2 + 8x = 16 + 12 – 4 <=> 3x = 24 <=> x = 8 C) x3 + 3x2 + 3x + 1 – (x3 - 3x2 + 3x - 1) = 6x2 + 6x + 6 <=> x3 + 3x2 + 3x + 1 – x3 + 3x2 + 1 = 6x2 + 6x + 6 <=> 2 = 6x + 6 <=> 6x = -4 <=> $x = -\frac{2}{3}$ Problem 3 Solve the equation : A) $\frac{5x - 4}{2} = \frac{0.5x + 1}{3}$ B) $1 –\frac{x - 3}{5} = \frac{-3x + 3}{3}$ C) $\frac{x + 1}{3} – \frac{2x + 5}{2} = -3$ D) $\frac{3(x - 1)}{2} + \frac{2(x + 2)}{4} = \frac{3x + 4.5}{5}$ Solution: A) $\frac{5x - 4}{2} – \frac{0.5x + 1}{3} \Leftrightarrow$ 3(5x - 4) = 2(0.5x + 1) <=> 15x - 12 = x + 2 <=> 15x – x = 12 + 2<=> 14x = 14 <=> x = 1 B) $1 – \frac{x - 3}{5} = \frac{3(1 - x)}{3}\Leftrightarrow$ $1 –\frac{x - 3}{5} = 1 – x\Leftrightarrow$ -x + 3 = - 5x <=> 5x – x = - 3 <=> $x = -\frac{3}{4}$ C) $\frac{3(x - 1)}{2} + \frac{2(x + 2)}{4} = \frac{3x + 4.5}{5} \Leftrightarrow$ $\frac{2(x + 1) - 3(2x + 5)}{6} = - 3 \Leftrightarrow$ $\frac{2x + 2 - 6x -15}{6} = - 3 \Leftrightarrow$ -4x - 13 = -18 <=> -4x = -18 + 13 <=> -4x = -5 <=> $x = \frac{5}{4}$ D) We reduce to common denominator, which for 2, 4 and 5 is 20 $\frac{3(x - 1)}{2} + \frac{2(x + 2)}{4} = \frac{3x + 4.5}{5} \Leftrightarrow$ 30(x - 1) + 10(x + 2) = 4(3x + 4.5) <=> 30x - 30 + 10x + 20 = 12x + 18 <=> 40x - 12x = 18 + 10 <=> 28x = 28 <=> x = 1 Problem 4 Proof that every value of the unknown quantity is root of the equation: A) 7x - 13 = - 13 + 7x B) $(\frac{1}{2} – x)^2 – (\frac{1}{2} + x)^2 = -2x$ C) 3x - 3x = 26 - 2(7 + 6) D) $\frac{-3x + 4x^2}{5} = (0.8x - 0.6)x$ Solution: For one simple equation with unknown quantity x everyx is a solution, if it is reduced to the following equivalent equation 0.x = 0 or it transforms into identity a = a. Actually, in the left any value of x, when we multiply it with zero, will obtain zero, i.e. the right side or the value of x won’t influence the right or left side of the identity. A) 7x - 7x = -13 + 13 <=> 0.x = 0 <=> every x is a solution. B) $\frac{1}{4} - x + x^2 –(\frac{1}{4} + x + x^2) = - 2x$ <=> $\frac{1}{4} - x + x^2 -\frac{1}{4} – x – x^2 = - 2x$<=> -2x = -2x <=> -2x + 2x = 0 <=> 0.x = 0 <=> Therefore every x is a solution. C) 0.x = 26 - 2.13 <=> 0.x = 26 – 26 <=> 0.x = 0 <=> every x is a solution. D) -3x + 4x2 = 5(0.8x - 0.6)x <=> -3x + 4x2 = (4x - 3)x <=> -3x + 4x2 = 4x2 - 3x Therefore every x is a solution. Problem 5 Proof that the equation has no roots: A) 0.x = 34 B) 5 - 3x = 7 - 3x C) $\frac{x - 3}{4} = \frac{x + 5}{4}$ D) 2(3x - 1) – 3(2x + 1) = 6 Solution: A) For the left side we will get value 0 for every x and for the right side is 34, i.e. number different from 0. Therefore there is no such x to get a true numerical equality; B) 5 - 3x = 7 - 3x <=> 3x - 3x = 7 - 5 <=> 0.x = 2 <=> 0 = 2, which is impossible for any x C) $\frac{x - 3}{4} = \frac{x + 5}{4}$ <=> x - 3 = x + 5 <=> x – x = 5 + 3 <=> 0 = 8 => no solution; D) 2(3x - 1) - 3(2x + 1) = 6 <=> 6x - 2 - 6x - 3 = 6 <=> 0.x = 6 + 5 <=> 0 = 11 no solution. Problem 6 Solve the equation: A) 2x2 - 3(1 – x)(x + 2) + (x - 4)(1 - 5x) + 58 = 0 B) 3.(x + 1)2 – (3x + 5).x = x + 3 C) x2 – (x - 1).(x + 1) = 4 D) (x - 1).(x2 + x + 1) = (x - 1)3 + 3x(x - 1) E) (3x - 1)2 – x(15x + 7) = x(x + 1).(x - 1) – (x + 2)3 Solution: A) 2x2 - 3(x + 2 – x2 - 2x) + x - 5x2 - 4 + 20x + 58 = 0 <=> 2x2 - 3x - 6 + 3x2 + 6x + x - 5x2 - 4 + 20x + 58 = 0 <=> 0.x2 + 24x + 48 = 0 <=> 24x = - 48 <=> x = -2 B) 3(x2 + 2x + 1) - 3x2 - 5x = 3x2 + 6x + 3 - 3x2 -5x = x + 3 <=> (3 - 3)x2 + (6 - 5).x – x = 3 - 3 <=> 0 = 0 => every x is a solution C) x2 – (x2 -1) = 4 <=> x2 – x2 + 1 = 4 <=> 0 = 3 => no solution D) x3 + x2 + x – x2 – x - 1 = x3 - 3x2 + 3x - 1 + 3x2 - 3x <=> 0 = 0 => every x is a solution E) 9x2 - 6x + 1 - 15x2 - 7x = x3 –x2 + x2 – x – x3 - 6x2 - 12x - 8 <=> 0 = 9 => no solution Problem 7 Solve the equation: A) $\frac{6x - 1}{5} - \frac{1 - 2x}{2} = \frac{12x + 49}{10}$ B) $\frac{x - 3}{2} + \frac{2x - 2}{4} = \frac{7x - 6}{3}$ Solution: A)We reduce to common denominator and we get: 12x - 2 - 5 +10x = 12x + 49 <=> 22x - 12x = 49 + 7 <=> 10x = 56 <=> x = 5.6 B) $\frac{x - 3}{2} + \frac{2x - 2}{4} = \frac{7x - 6}{3}$ <=> $\frac{x -3 + x - 1}{2} = \frac{7x - 6}{3}$ <=> 3(2x - 4) = 2(7x - 6) <=> 6x -12 = 14x - 12 <=> 8x = 0 <=> x = 0 Problem 8 The function f(x) = x + 4 is given. Solve the equation: $\frac{3f(x - 2)}{f(0)} + 4 = f(2x + 1)$ Solution: We calculate f(0), f(x -2), f(2x +1), namely f(0) = 0 + 4 = 4; f(x - 2) = x - 2 + 4 = x + 2; f(2x + 1) = 2x + 1 + 4 = 2x + 5 The equation gets this look $\frac{3(x + 2)}{4} + 4 = 2x + 5$ <=> 3(x + 2) +16 = 4(2x + 5) <=> 3x + 6 +16 = 8x + 20 <=> 22 - 20 = 8x - 3x <=> 2 = 5x <=> x = 0.4 Problem 9 Solve the equation: (2x - 1)2 – x(10x + 1) = x(1 – x)(1 + x) – (2 – x)3 Solution: (2x - 1)2 – x(10x + 1) = x(1 – x)(1 + x) – (2 – x)3 <=> 4x2 - 4x + 1 -10x2 – x = x – x3 - 8 + 12x - 6x2 + x3 <=> 18x = 9 <=> $x = \frac{1}{2}$ Problem 10 Solve the equation: (2x + 3)2 –x(1 + 2x)(1 - 2x) = (2x - 1)2 + 4x3 - 1 Solution: (2x + 3)2 – x(1 + 2x)(1 - 2x) = (2x - 1)2 + 4x3 -1 <=> 4x2 + 12x + 9 – x(1 - 4x2) = 4x2 - 4x + 1 + 4x3 - 1 <=> 12x + 9 – x + 4x3 = - 4x + 4x3<=> 15x = -9 <=> $x = -\frac{3}{5}$ Problem 11 Solve the equation : (2x - 1)3 + 2x(2x - 3).(3 - 2x) – (3x - 1)2 = 3x2 - 2 Solution: We open the brackets by using the formulas for multiplication: 8x3 - 3(2x)2.1 + 3.2x(1)2 – 13 - 2x(2x - 3)2 – (9x2 - 6x + 1) = 3x2 - 2 <=> 8x3 - 12x2 + 6x - 1 - 2x(4x2 - 12x + 9) - 9x2 + 6x - 1 = 3x2 - 2 <=> 8x3 - 21x2 + 12x - 8x3 + 24x2 - 9x = 3x2 <=> 3x2 + 3x = 3x2 <=> 3x = 0 <=> x = 0 Problem 12 Solve the equation : $\left(2x - \frac{1}{2}\right)^2 – (2x - 3)(2x + 3) = x + \frac{1}{4}$ Solution: We use the formulas for multiplication, open the brackets and get: $4x^2 - 2x + \frac{1}{4} – (4x - 9) = x + \frac{1}{4}$ <=> $4x^2 - 2x + \frac{1}{4} - 4x^2 + 9 = x + \frac{1}{4}$ <=> 9 = x + 2x <=> 9 = 3x <=> x = 3 Problem 13 Proof that the two equations are equivalent: A) $\frac{x - 5}{2} + \frac{x - 1}{8} = \frac{1.5x - 10}{4}$ and $\frac{x + 6}{2} – \frac{5.5 - 0.5x}{3} = 1.5$ B) $x – \frac{8x + 7}{6} + \frac{x}{3} = -1\left(\frac{1}{6}\right)$ and $2x – \frac{6 – x}{3} - 2\left(\frac{1}{3}\right)x = -2$ Solution: A) For the first equation we get consecutively: 4(x - 5) + x - 1 = 2(1.5x - 10) <=> 4x - 20 + x - 1 = 3x - 20 <=> 5x – 3x = - 20 + 21 <=> 2x = 1 <=> $x = \frac{1}{2}$, for the second equation we have 3(x + 6) - 2(5.5 - 0.5y) = 6 . 1.5 <=> 3x + 18 - 11 + x = 9 <=> 4y = 9 - 7 <=> $x = \frac{2}{4}$ <=> $x = \frac{1}{2}$ Therefore the equations are equivalent. B) Analogical to A) Try by yourself Problem 14Solve the equation : A) (2x + 1)2 – x(1 - 2x)(1 + 2x) = (2x - 1)2 + 4x3 - 3 B) (2x - 1)2 + (x - 2)3 = x2(x - 2) + 8x - 7 C) (x + 2)(x2 - 2x + 4) + x(1 – x)(1 + x) = x - 4 D) $\frac{8x + 5}{4} – \frac{1}{2\left[2 – \frac{3 – x}{3}\right]} = 2x + \frac{5}{6}$ E) $\frac{x}{3} – \frac{x + 3}{4} = x – \frac{1}{3\left[1 – \frac{3 - 24x}{8}\right]}$ F) $\frac{x}{5} – \frac{(2x - 3)^2}{3} = \frac{1}{5}\left[5 – \frac{20x - 43x}{3}\right]$ Solution: A) 4x2 + 4x + 1 – x(1 - 4x2) = 4x2 - 4x + 1 + 4x3 - 3 <=> 4x – x + 4x3 = -4x + 4x3 -3 <=> 3x + 4x = -3 <=> 7x = - 3 <=> $x = -\frac{3}{7}$ B) 4x2 - 4x + 1 + x3 - 3x2.2 + 3x.22 - 8 = x3 -2x2 + 8x - 7 <=> 4x2 - 6x2 - 4x + 1 + 12x - 8 = - 2x2 + 8x -7 <=> -2x2 + 8x - 7 = - 2x2 + 8x - 7 <=> 0 = 0 => every x is a solution; C) x3 + 2x2 - 2x2 - 4x + 4x + 8 + x(1 – x2) = x - 4 <=> x3 + 8 + x – x3 = x - 4 <=> 8 = - 4, which is impossible. Therefore the equation has no solution; D) $\frac{8x + 5}{4} - 1 + \frac{3 – x}{6} = 2x + \frac{5}{6} \Leftrightarrow$ 3(8x + 5) - 12 + 2(3 – x) = 24x + 2.5 <=> 24x + 15 - 12 + 6 - 2x = 24x + 10 <=> -2x = 10 - 9 <=> $x = -\frac{1}{2}$ E) $\frac{x}{3} – \frac{x + 3}{4} = x - \frac{1}{3} + \frac{3 - 24x}{24} \Leftrightarrow$ 8x - 6(x + 3) = 24x - 8 + 3 - 24x <=> 8x - 6x - 18 = -5 <=> 2x = 18 - 5 <=> 2x = 13 <=> x = 6.5 F) $\frac{x}{5} – \left[\frac{2x - 3}{3}\right]^2 = 1 – \frac{20x^2 - 43x}{15} \Leftrightarrow$ 3x - 5(4x2 -12x + 9) = 15 - 20x2 + 43x <=> 3x - 20x2 + 60x - 45 = 15 - 20x2 + 43x <=> 63x - 43x = 15 + 45 <=> 20x = 60 <=> x = 3 #### More about equations in the maths forum Contact email: Copyright © 2005-2017.<\|endoftext\|>	4.875
575	Law and the Rule of Law It seems like everywhere you turn, someone is telling you what to do or not to do. In a civil society, we are expected to follow the rules, regulations, and laws to keep the peace and live amicably together as a group. Laws are primarily rules which protect us against abuse from companies, other people, and even the government itself. Laws protect our rights and ensure that we are all treated equally and fairly. The government has set forth a complex set of rules to protect every area of our lives such as food safety, speed limits and traffic rules as well as healthcare and other professional services. Without these laws, we would get sick from food not safely processed, and medical care would have no standards or and more people would die from negligence. Additionally, the roads and highways would be complete chaos without any rules of conduct. Thankfully, our laws are continually evolving to protect new groups or changes in how people represent themselves. For instance, there are now laws enacted to protect same-sex marriage, gender non-specific individuals and other organizations, which did not exist prior. The Bill of Rights guards our freedoms of speech, press, and religion. We also have laws that protect against discrimination or abuse because of color, race, disability, age or gender. Where Do Our Laws Come From? The Constitution provides the basic guidelines for most of our laws, however, it does not cover every contingency. Therefore the law is evolving and frequently updated to include new technologies and changes that naturally occur in our society. For example, bullying laws had to change when social media popped onto the scene, and now it includes clauses about cyber stalking and bullying. The executive branch of government, courts, and judges interpret the law as it is laid out in the Constitution. They do not make new laws. New laws are proposed, discussed and then voted on in Congress. The Rule of Law The rule of law means that as Americans we have agreed to live in a society and abide by the government-mandated laws. We further agree to the equal and fair process of judicial resolution for disputes. The rule of law epitomizes democracy promising that each American will respect, honor, stay committed to, and be faithful to the ideals of our civil society by obeying and upholding our laws. Changes in the Law Despite our agreement as American citizens to honor our democratic society, not everyone agrees that specific things should be legal or illegal. For instance, there is a considerable debate going on about whether or not the legalization of certain recreational drugs is unconstitutional. There are other instances where state or federal laws come into question about overstepping boundaries when it comes to health and people's personal lives. Time will tell how things will continue to evolve in our democratic society of laws and how well the public will receive it.<\|endoftext\|>	3.75
160	Class Summary - Today we talked about touch. We talked about the 5 receptors (mechanorecpetors, thermoreceptors, photoreceptors, nociceptors, and chemoreceptors) which is a review from last unit. We talked about how general sensory receptors are categorized into three categories; proprioreceptors - muscle - balance and equilibrium, cutaneous receptors - skin - several types, touch and pressure, and pain receptors - skin and internal organs - pain. Homework - None Objectives - Students will categorize sensory receptors according to five types of stimuli. Students will discuss the function of propriorecepters. Students will relate specific sensory receptors in the skin to particular sense of the skin. Students will discuss the phenomenon of referred pain.<\|endoftext\|>	3.9375
1,196	JEE Main 2024 Question Paper Solution Discussion Live JEE Main 2024 Question Paper Solution Discussion Live # Coefficient of Restitution ## What is the Coefficient of Restitution? The ratio of final velocity to the initial velocity between two objects after their collision is known as the coefficient of restitution. The restitution coefficient is denoted as â€˜eâ€™ and is a unit less quantity, and its values range between 0 and 1. ### A Simple Explanation of Coefficient of Restitution Sir Issac Newton derived many mathematical equations and laws that help us make sense of what happens when two objects collide. One such important law, Newtonâ€™s law of restitution, states that when two objects collide, their speeds after the collision depend on the material they are made of. The measure of the colliding materialsâ€™ nature is represented by a number known as the coefficient of restitution. The coefficient of restitution provides us with information about the elasticity of the collision. Collisions in which there is no loss of overall kinetic energy is known as a perfectly elastic collision. This type of collision has the maximum coefficient of restitution of e = 1. Collision, where maximum kinetic energy is lost, is known as a perfectly inelastic collision. They have a coefficient of restitution of e = 0. Most real-life collisions are in between. ### Coefficient of Restitution – Video Lesson #### Coefficient of Restitution Formula The mathematical formula of the Coefficient of Restitution is given as follows: From the above equation, you notice that you always divide the smaller number by a more significant number. Therefore, the coefficient of restitution is always positive. The value is almost always less than one due to initial translational kinetic energy being transformed to rotational kinetic energy, plastic deformation, and heat. However, it can be more than one if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. Range of Values for e If e = 0, then it is a perfectly inelastic collision If 0 < e < 1, then it is a real-world inelastic collision, in which some kinetic energy is dissipated. If e = 1, then it is a perfectly elastic collision in which no kinetic energy is dissipated, and the objects rebound from one another with the same relative speed with which they approached. ## Coefficient of Restitution in Sports The Coefficient of Restitution plays a vital role in the design of sports balls. A basketball, for example, bounces more than a tennis ball because less energy is lost by the basketball when it hits the ground. We can determine the percentage of speed that the ball retains after the collision by use of the coefficient of restitution. The larger the value of the Coefficient of Restitution of the ball, the more elastic the collision. Furthermore, if we drop a ball from a certain height, balls with a higher Coefficient of Restitution will rebound higher. ## Coefficient of Restitution Example Problems A ball strikes a smooth wall with a velocity (vb)1 = 20 m/s. The coefficient of restitution between the ball and the wall is e = 0.75. Find the velocity of the wall just after the impact. Solution: The collision is an oblique impact, with the line of impact perpendicular to the plane. Thus, the coefficient of restitution applies perpendicular to the wall and the momentum of the ball is conserved along the wall. The given problem is solved by defining x-y axes along the line perpendicular to the line of impact, respectively. The momentum of the ball is conserved in the y-direction: $$\begin{array}{l}m({v_{b}}_1)\sin 30=m({v_{b}}_1)\sin\Theta\end{array}$$ Solving, we get $$\begin{array}{l}({V_{b}}_2)\sin \Theta = 10\, m/s……….(1)\end{array}$$ The coefficient of restitution applies in the x-direction: $$\begin{array}{l}e=\frac{[0-{(V_{bx})}_2]}{{[(V_{bx})}_1-0]}\end{array}$$ Solving the above equation, we get: $$\begin{array}{l}(v_{b})_2\cos \Theta =12.99\, m/s……….(2)\end{array}$$ Solving equations (1) and (2), we get $$\begin{array}{l}(v_{b})_2=(12.99^{2}+10^{2})^{0.5}=16.4\,m/s\end{array}$$ $$\begin{array}{l}\Theta =37.6^{\circ}\end{array}$$ ## Frequently Asked Questions – FAQs ### What is the coefficient of restitution? The ratio of final velocity to the initial velocity between two objects after their collision is known as the coefficient of restitution. ### What does the coefficient of restitution measure? The coefficient of restitution measures the elasticity of the collision. ### How does the coefficient of restitution affect the collision? The coefficient of restitution is important in collision because it determines whether a collision is elastic or inelastic. ### When is the coefficient of restitution zero? When the collision is perfectly inelastic, the coefficient of restitution is zero. ### Can the coefficient of restitution be negative? No, the coefficient of restitution is always positive. ### How does the coefficient of restitution relate to sports? The larger the value of the coefficient of restitution of the ball, the more elastic the collision. Furthermore, if we drop a ball from a certain height, balls with a higher coefficient of restitution will rebound higher.<\|endoftext\|>	4.40625
700	ZFS (Zettabyte FileSystem) is a file system designed by Sun Microsystems for the Solaris Operating System. ZFS is a 128-bit file system, so it can address 18 billion billion times more data than the 64-bit systems ZFS is implemented as open-source filesystem, licensed under the Common Development and Distribution License (CDDL). The features of ZFS include support for high storage capacities, integration of the concepts of file system and volume management, snapshots and copy-on-write clones, continuous integrity checking and automatic repair, RAID-Z etc. Additionally, Solaris ZFS implements intelligent prefetch, performing read ahead for sequential data streaming, and can adapt its read behavior on the fly for more complex access patterns. To eliminate bottlenecks and increase the speed of both reads and writes, ZFS stripes data across all available storage devices, balancing I/O and maximizing throughput. And, as disks are added to the storage pool, Solaris ZFS immediately begins to allocate blocks from those devices, increasing effective bandwidth as each device is added. This means system administrators no longer need to monitor storage devices to see if they are causing I/O bottlenecks. The minimum amount of disk space required for a storage pool is 64 MB. The minimum disk size is 128 MB. The minimum amount of memory needed to install a Solaris system is 768 MB. scrubbing – examines all data to discover hardware faults or disk failures resilvering :The process of copying data from one device to another device is known as resilvering. For example, if a mirror device is replaced or taken offline, the data from an up-to-date mirror device is copied to the newly restored mirror device. This process is referred to as mirror resynchronization in traditional volume management products. Limits the amount of space a dataset and its descendents can consume. This property enforces a hard limit on the amount of space used, including all space consumed by descendents,including file systems and snapshots The refquota property limits the amount of space a dataset can consume. This hard limit does not include space used by descendents, such as snapshots and clones. The minimum amount of space guaranteed to a dataset and its descendents. reservation : minmum guranted on dataset, it includes its descendents The refreservation property sets the minimum amount of space that is guaranteed to a dataset, not including its descendents. # zfs set refquota=10g tank/studentA # zfs set quota=20g tank/studentA When a block is accessed, regardless of whether it is data or meta-data, its checksum is calculated and compared with the stored checksum value If the checksums match, the data are passed up the programming stack to the process that asked for it; if the values do not match, then ZFS can heal the data if the storage pool provides data redundancy (such as with internal mirroring), # zfs mount -c home/bob /export/home/bob ZFS Pool Statuses: Repairing an Unbootable System # If the system is in a panic-reboot loop: boot -m milestone=none mount -o remount / svcadm milestone all # …perform ZFS repairs.<\|endoftext\|>	3.703125
1,195	# How To Find Reciprocal by -112 views If were given a reciprocal function in the form 1 f x then we can find the vertical asymptotes by setting f x 0 and solving for x. Define vectors b i by b i a j 2πδ ij where δ ii 1 δ ij 0 if i j If we define the. Reciprocal Construction How To Say Each Other In Spanish Youtube Spanish Teaching Resources Verbs Activities Teaching Spanish ### To find the reciprocal of a fraction we invert the fraction. How to find reciprocal. The reciprocal of a fraction can be found by interchanging the numerator and the denominator values. To get a positive result when multiplying two numbers the numbers must have the same sign. To find the reciprocal of a fraction switch the numerator and the denominator the top and bottom of the fraction respectively. In other words swap over the Numerator and Denominator. To find the reciprocal of 7 first it is re-written as frac71. The reciprocal of 34 is 43 Reciprocal of a number. Learn to find reciprocal of a fraction. To find the reciprocal of any number just calculate 1 that number For a fraction the reciprocal is just a different fraction with the numbers flipped upside down inverted. A b b a Reciprocals are considered multiplicative inverses since they will always be 1 when we multiply a number by its reciprocal. To get the reciprocal of a fraction just turn it upside down. This means that we place the numerator in the denominator and the denominator in the numerator. The reciprocal of a fraction is just switching the numerator top number and the denominator bottom number. So reciprocals must have the same sign. To get the reciprocal of a number we divide 1 by the number. To find the reciprocal of a fraction we invert the fraction. Reciprocal Formula The following formula is used to calculate the reciprocal of a fraction. A function that when multiplied by the original function results in the constant 1. Then its numerator and denominator are flipped and the reciprocal frac17. The reciprocal of a number or a function is the value or expression that results from reversing the numerator and denominators places. Find the reciprocal of 2 3. For instance the reciprocal of 3 4 is 43. The reciprocal of 2 is ½ a half. A reciprocal function is the multiplicative inverse of a regular function. Reciprocal of XY YX As can be see from the equation above the reciprocal is another word for opposite. So simply speaking the reciprocal of ab is ba. Therefore the reciprocal of a fraction 23 is 32. Reciprocal Lattice The reciprocal lattice is the set of vectors G in Fourier space that satisfy the requirement G T 2πx integer for any translation Tn 1n 2 n 1 a 1 n 2 a 2 n 3 a 3 in 3D How to find the Gs. Any number times its reciprocal will give you 1. If f x is your function is the reciprocal function. The reciprocal of 23 is 32. This means that we place the numerator in the denominator and the denominator in the numerator. To find the solution we will follow the following steps. So reciprocals must have the same sign. To get a positive result when multiplying two numbers the numbers must have the same sign. A b b a 1. The reciprocal is simply swapping the placement of the numerator and the denominator. The negative reciprocal takes the negative of that number. However when given a reciprocal function in its general. The reciprocal of a number is 1 divided by the number. Or use the formula x 1x where 231213. The reciprocal of a fraction is a fraction obtained by switching the values in the numerator and the denominator of the given fraction. To find the reciprocal take 1 and divide by the entire quantity of the fraction. How To Find The Reciprocal Of A Whole Number Shorts Math Videos Math Find How To Find The Reciprocal Of A Fraction An Example With 1 17 Shorts Math Videos Fractions Find Dividing Fractions And Reciprocal Youtube Fractions Dividing Fractions Student Teaching Video With A Math Song About How To Divide Fractions Using A Reciprocal Math Methods Math Songs Dividing Fractions Fraction Math Find The Reciprocal Worksheet Education Com Math Fractions Fraction Practice Upper Elementary Math Explanation Of Why Dividing By A Fraction Is The Same As Multiplying By The Reciprocal Middle School Math Homeschool Math Fifth Grade Math How To Find The Multiplicative Inverse Of A Number By Finding The Reciprocal Of The Given Number Home Simplifying Expressions Combining Like Terms Math Videos Find The Reciprocal Of A Rational Numbers Rational Numbers Subtraction Mathematics Math Reciprocal Of A Mixed Number Youtube Mixed Numbers Mini Lessons Math Reciprocal Pythagorean Theorem Calculus Mathematics Math Geometry How To Divide Fractions Just Make Sure You Explain How Dividing And Multiplying By The Reciprocal Are The Same Thin Math Methods Learning Math Homeschool Math Find The Reciprocal Www Brainrush Com Fun Adaptive Learning Games Play To Learn Learning Games Learning Why Is Dividing By A Fraction Equivalent To Multiplying By The Reciprocal Repinned By Chesapeake College Ad Math Methods Equivalent Fractions Fractions Reciprocal Trig Functions Worksheet Trigonometry Notes Trigonometric Functions In 2020 Kids Worksheets Pr In 2021 Math Tutorials Trigonometry Worksheets Tutorial Class Creating The Equation Of Reciprocal Linear Functions Algebra Fun Linear Function Algebra 2 Quotient And Reciprocal Identities Printable Math Worksheets Identity Math Worksheets Pin On Math Analysis The Actual Math Behind Multiplying By The Reciprocal Keep Change Flip Works For A Reason Training By Ptnmath Math Train It Works READ: How To Find Maximum Revenue<\|endoftext\|>	4.5625
831	# What is the difference between Integers, whole numbers, and natural numbers? • Last Updated : 03 Sep, 2021 Number System, any of various sets of symbols and therefore the rules for using them to denote numbers, which are wont to state what percentage objects are there during a given set or in other words Number system is a mathematical presentation of numbers of a given set. Let’s look at the different types of numbers in the decimal number system, ### Types of numbers There are different types of numbers in the number system, complex numbers include all types of numbers, both real and imaginary, in real numbers, there are rational and irrational numbers. There are integers that go from negative infinity to positive infinity, the integers that go from 0 to positive infinity are whole numbers, the integers that generate from 1 and go up to infinity are natural numbers. Integers If a set of all-natural numbers are constructed, whole numbers, and negative numbers then such a set is called an Integer set. It can be positive, negative, or zero. For Example : 2 , 5 , -9 , -17 , 112 , etc. Rational Numbers A rational number can be defined as any number which can be represented in the form of p/q where q is greater than 0. For Example 2/5, 22/7, 4/3, etc. Whole Numbers The whole numbers are the part of the number system in which it includes all the positive integers from 0 to infinity. These numbers exist in the number line. Hence, they are all real numbers. The complete set of natural numbers alongside ‘0’ are called whole numbers. Whole numbers are 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , … Natural Numbers Natural numbers are a part of the number system which includes all the positive integers from 1 to infinity. Natural numbers are 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , … ### Difference between Integers, whole numbers, and natural numbers The most basic difference between whole numbers and natural numbers is that whole numbers include 0 and natural numbers generate from 1 and therefore do not contain 0. However, Integers include numbers from negative infinity to positive infinity. Below are the point showing major difference between Integers, whole numbers, and natural numbers, • For Integers, counting starts from negative infinity, in natural numbers counting starts from 1 while in whole numbers counting starts from 0. • It can be said that each one natural numbers are whole numbers but all whole numbers are not natural numbers. However, all natural and whole numbers are Integers but not vice versa. • For Example : {-∞,…-1, 0, 1, 2,….+∞} is a set of Integers, {0, 1, 2, 3, 4,…} is a set of whole number while {1, 2, 3, 4,…} is a set of natural numbers. • The integers are represented as I, natural numbers are represented as N, whole numbers are represented as W. W = {I} – {-∞, ….-1} N = {W} – {0} ### Sample Problems Question 1: Can we say that 7 is both whole number and natural number? Yes, 7 is both whole number and natural number. Question 2: State whether the given statement is true or false: “all natural numbers are whole numbers”? True, all natural numbers are whole numbers but not vice versa. Question 3: Is 5 a whole number and a natural number? Yes, 5 is both a natural number and whole number. Question 4: Is 0.7 a whole number? No, it is a decimal. Question 5: “-17” is whole number or natural number ?<\|endoftext\|>	4.5
578	# Evaluating logarithms using logarithm rules ### Evaluating logarithms using logarithm rules #### Lessons • 1. a) Which of the following correctly states the "product law"? i) $\log_2 8 + \log_2 4 = \log_2 12$ ii) $\log_2 8 + \log_2 4 = \log_2 32$ iii) $\log_2 8 \cdot \log_2 4 = \log_2 32$ b) Which of the following correctly states the "quotient law"? i) $\log_b 15 - \log_b 3 = \log_b 5$ ii) $\log_b 15 - \log_b 3 = \log_b 12$ iii) ${{\log_b \sqrt{8}} \over {\log_b \sqrt{32}}} = \log_b(\sqrt{1 \over 4})$ c) Which of the following correctly states the "power law"? i) $(\log 100)^3 = \log 100^3$ ii) $(\log 100)^3 = 3\log 100$ iii) $\log 100^3 = 3\log 100$ • 2. Evaluate and state the laws involved in each step of the calculation: ${5 \log_2{^3}\sqrt{80} \over 5 \log_2{^3}\sqrt{20}}$ • 3. Express as a single logarithm: ${\log A-3\log B-\log C}$ • 4. Evaluate logarithms: a) Determine the value of ${\log_n ab^2, }$ if ${\log_na=5}$ and ${\log_nb=3}$ b) Given: $\log_5x = y$ express$\log_5125{x^4}$ • 5. a) $\log_3 \sqrt{15}- {1\over2} \log_35$ b) $\frac{({a^{\log_a8})}({a^{\log_a3}})}{a^{\log_a6}}$ • 6. a) if ${\log_3x^2 = 2}$ and ${2\log_b\sqrt{x} = {1\over3},}$ then the value of $b$ is ____________________ . b) if ${\log_5x^2 = 4}$ and ${\log_2y^3 = 6 ,}$ and ${\log_bx+\log_by = {1\over2}}$ where x, y > 0, then the value of b is ____________________ .<\|endoftext\|>	4.4375
372	In Structures, students explore materials and designs of towers, bridges and other structures. A good overall resource for the unit is the Exploritorium's Structures Around the World. In one activity in this unit, students build towers from 3" x 5" file cards. Can students top one made by teachers? An excellent interactive site for bridges is Nova Online's Super Bridge. Check it out. You might want to visit two of New York State's most famous bridges, the Peace Bridge and the Brooklyn Bridge. The Association of Bridge Construction and Design has a lot of interesting links. A good student book on the subject is Bridges by Ken Robbins. For information on skyscrapers, visit Skyscrapers: Twentieth Century. Two other good sources are The Skyscraper Page and the Skyscraper Museum. The PBS series "Building Big" has Web resources on how bridges, skyscrapers, domes, dams and other large structures work. Architecture in America has pictures of a variety of buildings as does Architecture.com, a British site. Also of interest are the PBS pages on Frank Lloyd Wright, perhaps America's most famous architect. Building structures requires engineering skills andThe Sightseer's Guide to Engineering gives some places to go to learn about engineering and to see its results. For information about concrete, visit Concrete Basics developed by the Portland Cement Association. For information on how houses are built, see How House Construction Works. The Franklin Institute has exhibits related to structures and a page with useful links. The Lemelson Center's Invention at Play has online activities that relate well to the creative inventing and building that this unit involves. Below are some pictures of structures built by Ms. Vigdor's third grade class in Penfield, NY, using this unit.<\|endoftext\|>	3.84375
484	Scientists imaged the moonlet on Aug. 15, 2008, and then confirmed its presence by finding it in two earlier images. The moonlet is too small to be resolved by Cassini's cameras, so its size cannot be measured directly. However, Cassini scientists estimated the moonlet's size by comparing its brightness to another small Saturnian moon, Pallene. Hedman and his collaborators also have found that the moonlet's orbit is being disturbed by the larger, nearby moon Mimas, which is responsible for keeping the ring arc together. Aegaeon is the smallest known moon of Saturn. It's not quite spherical, but is instead roughly football-shaped. The tiny moonlet has a mean radius of 0.2 miles (0.12 km) and orbits Saturn at an average distance of 104,080 miles (167,500 km) in 0.808 Earth days, at an inclination of 0.001 degrees to Saturn's equator, with an eccentricity of nearly zero—meaning it's almost perfectly circular. At this distance from Saturn, Aegaeon is embedded within a partial ring, or ring arc, previously found by Cassini, the G ring. Debris knocked off the moon forms a bright arc near the inner edge, which in turn spreads to form the rest of the ring. Saturn's rings were named in the order they were discovered. Working outward they are D, C, B, A, F, G, E and a very distant disk or ring encompassing the orbit of the moon Phoebe. The G ring is one of the outer diffuse rings. Within the faint G ring there is a relatively bright and narrow, 150-miles (250-km-wide) arc of ring material, which extends 90,000 miles (150,000 km), or one-sixth of the way around the ring's circumference. Based on its interaction with the dust particles that make up the G ring arc Aegaeon is embedded in, this small moon most likely has a density around half that of water ice. But unlike ice, Aegaeon is very dark. It's the least reflective of any Saturnian moon inward of Titan. How Aegaeon Got its Name Originally designated S/2008 S1, Aegaeon is named for a fierce giant with many heads and arms who helped conquer the Titans.<\|endoftext\|>	4.0625
628	# Graphing Proportional Relationships Worksheet Have you ever struggled with graphing proportional relationships? Well, look no further because we have the perfect resource for you! Our Graphing Proportional Relationships Worksheet is designed to help you master this concept with ease. Whether you’re a student looking to improve your graphing skills or a teacher searching for engaging practice materials, this worksheet is just what you need. So, let’s dive in and explore the world of graphing proportional relationships together! In the world of mathematics, graphing proportional relationships can sometimes be a daunting task. It’s easy to get overwhelmed by the various concepts and calculations involved. However, with our handy worksheet, you can bid farewell to all those challenges. With step-by-step instructions and practice problems, we’ve got you covered. Say goodbye to confusion and hello to confidence in your graphing abilities! Now, you might be wondering, what exactly is a proportional relationship? A proportional relationship is a mathematical connection between two quantities that can be expressed as a ratio. In other words, when one quantity increases or decreases, the other quantity changes in direct proportion. Our worksheet will help you understand this concept better by providing you with clear examples and exercises to practice on your own. By the time you’re done, you’ll be a pro at identifying and graphing proportional relationships! If you’re ready to take your graphing skills to the next level, check out these interesting links that can complement your learning experience. For a fun break, you can try playing Retro Bowl Unblocked, a popular online game that will keep you entertained while still sharpening your mathematical mind. Additionally, if you’re interested in exploring more topics related to math and gaming, visit the Gaming category on our website. It’s filled with informative articles and resources that can enhance your understanding of graphing proportional relationships and beyond. • #### 1. What is a proportional relationship? A proportional relationship is a mathematical connection between two quantities that can be expressed as a ratio. When one quantity changes, the other quantity changes in direct proportion. • #### 2. How can graphing proportional relationships be helpful? Graphing proportional relationships allows us to visually represent the connection between two quantities. It helps in understanding patterns, making predictions, and solving real-world problems. • #### 3. How can the Graphing Proportional Relationships Worksheet benefit students? The Graphing Proportional Relationships Worksheet provides step-by-step instructions and practice problems to help students improve their graphing skills. It offers a structured approach to understanding and mastering proportional relationships. • #### 4. Where can I find more math-related games and resources? You can visit the Gaming category on our website, which offers a wide range of math-related games and resources to make learning math more enjoyable and interactive. ## Conclusion of Graphing Proportional Relationships Worksheet Graphing proportional relationships doesn’t have to be a challenging task anymore. With our Graphing Proportional Relationships Worksheet, you can gain a solid understanding of this concept and excel in your mathematical endeavors. So why wait? Dive into the world of graphing and unlock your full potential today!<\|endoftext\|>	4.53125
361	# Given cos (x+y)-1=0, verify if sin(2x+y)-sinx=0? giorgiana1976 \| Student We'll start by expanding the cosine of the sum: cos (x+y) = cos xcos y - sin xsin y From enunciation, we know that: cos xcos y - sin xsin y - 1 = 0 We'll add sin xsin y + 1 both sides: cos xcos y = sin xsin y + 1 (1) Now, we'll expand the function sin (2x+y): sin (2x+y) = sin 2xcos y + sin ycos 2x We'll re-write the factor sin 2x: sin 2x = sin(x+x) = 2sin xcos x We'll re-write the factor cos 2x: cos 2x = cos (x+x) = 1 - 2(sin x)^2 We'll re-write the sum: sin (2x+y) = 2sin xcos xcos y + sin y[1 - 2(sin x)^2] We'll substitute the product cos xcos y by (1): sin (2x+y) = 2sin x(1 + sin xsin y) + sin y[1 - 2(sin x)^2] We'll remove the brackets: sin (2x+y) = 2sin x + 2(sin x)^2sin y + sin y - 2(sin x)^2*sin y sin (2x+y) - 2sin x - sin y = 0<\|endoftext\|>	4.4375
688	PULLEYS - Newton's Laws - SAT Physics Subject Test ## Chapter 3 Newton”s Laws ### PULLEYS Pulleys are devices that change the direction of the tension force in the cords that slide over them. For the purposes of this text and the SAT Physics Subject Test, we”ll consider each pulley to be frictionless and massless, which means their masses are so much smaller than the objects attached to the ends of them, that they can be ignored. In the case of two single masses m1 and m2 that are attached to a pulley and cord, the downward forces are due to the weight (mass and gravity exerted on it) of the masses. The upward forces are due to the tension (T) in the cord. 18. In the diagram above, assume that the tabletop is frictionless. Determine the acceleration of the blocks once they”re released from rest. Here”s How to Crack It There are two blocks, so we need to draw two free-body diagrams. To get the acceleration of each one, we use Newton”s second law, Fnet = ma. Notice that there are two unknowns, FT and a, but we can eliminate FT by adding the two equations, and then we can solve for a. A quicker way of solving for the acceleration is to treat the entire system (blocks plus string) as one object. Since we are only concerned with forces acting on the object, we can ignore tension. The string is part of the object. Then we need only consider forces acting in the direction of motion (Mg) and forces opposite the direction of motion (none). Our mass in Newton”s second law becomes M + m, so Fnet = ma becomes Mg = (M + m) a, giving us the same answer for acceleration. 19. Using the diagram from the previous example, assume that m = 2 kg, M = 10 kg, and the coefficient of kinetic friction between the small block and the tabletop is 0.5. What is the acceleration of the blocks? Here”s How to Crack It Once again, draw a free-body diagram for each object. Notice that the only difference between these diagrams and the ones in the previous example is the inclusion of friction, Ff, that acts on the block on the table. As before, we have two equations that contain two unknowns (a and FT). FTFf = ma (1) MgFT = Ma (2) We add the equations (thereby eliminating FT) and solve for a. Notice that, by definition, FfFN and from the free-body diagram for m, we see that FN = mg, so Ff = µmg: Or, using our shorter metod: Substituting in the numerical values given for m, M, and µ, we find that a = g (or 7.5 m/s2). 20. In the previous example, calculate the magnitude of the tension in the cord. Here”s How to Crack It Since the value of a has been determined, we can use either of the two original equations to calculate FT. Using equation (2), MgFT = Ma (because it”s simpler), we find As you can see, we would have found the same answer if we”d used equation (1): <\|endoftext\|>	4.53125
658	Accurately identify the most likely etiology when patients present with abnormal bowel function, through history, diagnostic tests, and patient findings on examination to enable recommendation of effective treatment or referral to an appropriate provider. Use the knowledge of the pathophysiology, etiology, and common presentations of diarrhea and constipation as a primary symptom to review prescription orders for appropriateness and to accurately educate patients about their disease and its treatment. Use the knowledge of the pathophysiology, etiology, and common presentation of diseases with diarrhea and constipation as a primary symptom to accurately interpret the diagnostic process to advise regarding the most appropriate prescription therapy. Diarrhea and constipation are very common presentations in ambulatory and urgent care centers. In the United States, up to 27% of patients have experienced constipation. Similarly, 211 to 375 million cases of diarrhea are reported annually. While there are a variety of definitions for both conditions, normal frequency of bowel movements in most adults range from three times a day to every 3 days. Even though most causes are self-limiting, both can be a symptom of a serious or life-threatening disorder. Because both disorders are so common and generally self-limiting, patients frequently try treatment with nonprescription products prior to seeking medical care. Therefore, screening patients for those more serious forms of these disorders is critical before recommending from the multitude of products available for self-care. Diarrhea, defined as three or more bowel movements within a 24-hour period, can be classified as acute, persistent, and chronic. Acute diarrhea lasts for less than 14 days; persistent diarrhea lasts for more than 14 days; and patients experiencing diarrhea for more than 30 days have chronic diarrhea. Dysentery is usually defined as acute diarrhea in which subjects have frequent watery stools, often with blood and mucous. The most common cause of acute diarrhea is infection with viral pathogens, such as norovirus and rotavirus. Common bacterial causes include Shigella and Salmonella species along with Campylobacter jejuni, Clostridium difficile, and Escherichia coli species. Giardia lamblia, Entamoeba histolytica, and Cryptosporidium species are the most common protozoal causes of diarrhea. Noninfectious causes of acute diarrhea include foods and medication. Inflammatory bowel diseases such as celiac disease, microcolitis, ulcerative colitis, and Crohn’s disease all cause chronic diarrhea. Irritable bowel syndrome is a noninflammatory condition that in two versions (diarrhea predominant and mixed) cause chronic diarrhea. Lastly, diabetic gastroparesis, acquired immunodeficiency syndrome (AIDS), and post-gastrointestinal (GI) tract surgery are uncommon causes for chronic diarrhea. While most viral diarrheas are self-limiting (typically lasting 24 to 48 hours) complications such as dehydration are uncommon, except in infants and the elderly. Some bacterial diarrheas produce toxins that result in more serious complications such as toxic megacolon, intestinal perforation, sepsis, and even death. Some parasitic infections can cause persistent diarrhea and some may migrate to other internal organs such as ...<\|endoftext\|>	3.65625
629	# If the Sum of the Squares of Zeros of Quadratic Polynomial F(X)= x^2 – 8 x + K is 40, Find k By BYJU'S Exam Prep Updated on: October 17th, 2023 If the sum of the squares of zeros of quadratic polynomial F(X)= x² – 8 x + K is 40, find k For the given quadratic polynomial F(x) = x² – 8x + K, assume α and β be the zeros of F(x). As it is mentioned that the sum of the squares of the zeros is (α² + β²) hence (α² + β²) = 40. To find the solution, use α + β = 8 and αβ = K. Further use the identity (α² + β²) = (α + β)² – 2αβ. Substitute the values into the identity: (8)² – 2K = 40. Simplify the equation and solve for K. ## If the Sum of the Squares of Zeros of Quadratic Polynomial F(X)= x^2 – 8 x + K is 40, Find k Solution: Let’s find the sum of the squares of the zeros of the quadratic polynomial F(x) = x² – 8x + K. The sum of the squares of the zeros can be found using the following relationship: Sum of squares of zeros = (α² + β²) In the quadratic polynomial F(x) = x² – 8x + K, let α and β be the zeros. Therefore, we have: α + β = 8 (from the coefficient of the linear term) αβ = K (from the constant term) We are given that the sum of the squares of the zeros is 40. So, we have: (α² + β²) = 40 To solve for K, we can express (α² + β²) in terms of α + β and αβ using the identity: (α² + β²) = (α + β)² – 2αβ Substituting the values we know, we get: (α + β)² – 2αβ = 40 (8)² – 2K = 40 64 – 2K = 40 -2K = 40 – 64 -2K = -24 K = (-24)/(-2) K = 12 Therefore, the value of K is 12. ## When the Sum of the Squares of the Zeros of the Quadratic Polynomial F(x) = x² – 8x + K is 40, the Value of K is 12 Similar Questions: POPULAR EXAMS SSC and Bank Other Exams GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 [email protected]<\|endoftext\|>	4.65625
1,371	# What does AB with a line over it mean? A line segment is a section of a line that has two points at each end, which are referred to as endpoints. The term "endpoint" refers to the fact that a line has a beginning and an end. The notation for a line segment is represented by a bar above any letter of your choosing, as seen below. If the letter AB had a bar above it, you would interpret it as "line segment AB" or something like. People often wonder what the difference is between the letters AB and AB with a line across them. The distinction between them may be seen in their respective names. The AB vector is a vector in its entirety, while the AB bar is a line segment. A vector has both a direction and a magnitude, while a line segment (scalar) just has a magnitude and no direction. ### Second, what does the letter AB with an arrow above it indicate? Lines. A line is one of the most fundamental concepts in geometry. The name of a line that passes through two separate points A and B is written as "line AB" or as the two-headed arrow over AB, which denotes a line that passes through points A and B respectively. Example: An illustration of two lines is shown below, labelled as line AB and line HG. In a similar vein, one would wonder what a symbol with a line across it means in mathematics. A vinculum is a horizontal line that is used in mathematical notation to represent a particular object or function. It may be used as an overline (or underline) over (or beneath) a mathematical expression to indicate that the expressions are to be regarded as a single unit of analysis. ### What does a line drawn across a group of numbers mean? Normally, a line drawn above a symbol for a set indicates that the set's complement is present. ### Are parallel lines compatible with one another? When two parallel lines are intersected by a transversal, the angles formed by the two parallel lines are congruent. It is true that lines are parallel if a transversal cuts two lines and the accompanying angles are equivalent; otherwise, the lines are not parallel. Interior angles on the same side of the transversal as the transversal are as follows: The name of this angle is a description of the "location" of the angles in question. ### What is the graphical representation of a line? In geometry, the following is a table of symbols: Symbol Symbol Symbol Symbol Symbol Symbol Line containing the meaning and definition line that goes on forever AB line segment line from point A to point B ray line that begins at point A arc arc from point A to point B AB line segment line from point A to point B ### What is the definition of a perpendicular line? For the purposes of introductory geometry, the connection between two lines that intersect at a right angle (perpendicularity) is defined as the attribute of being perpendicular (90 degrees). It is stated that a line is perpendicular to another line if the two lines connect at a straight angle when the two lines overlap. ### What is the best way to create a line segment? In geometry, you will create a line segment with the letters for each of the end points and a line drawn over the top of the letters to represent the segment. Consider the following example: if your end points are A and B, you would write your line segment AB with a line crossing over the top of it. ### What is the symbol for the concept of parallel? Parallel lines are two lines that are both in the same plane but never intersect. They are also known as perpendicular lines. Parallel lines are always the same distance away from one another no matter how long they are. Parallel lines are denoted by the symbol //, which stands for parallel lines. ### What is a perpendicular bisector of a line, and how does it work? Definition: A line that divides a line segment into two equal halves at a 90-degree angle. Consider the following: Drag one of the orange dots at A or B, and observe that the line AB always splits the section PQ into two equal halves, regardless of which one you choose. It is referred to as the perpendicular bisector when it is perfectly at right angles to the perpendicular axis. ### What is the difference between a line and a line segment? The term "line segment" refers to a section of a line that is limited by two different end points and includes every point on the line between those endpoints. A closed line segment contains both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes just one of the endpoints; and a half-closed line segment excludes both endpoints and one of the endpoints. ### What exactly does SX stand for? SX Definition / What Does SX Stand For? "Sex" is the definition of SX in this context. ### What exactly is the Virnaculum rule? The virnaculum of a fraction is the horizontal line that divides the numerator from the denominator of the fraction. Occasionally, it is referred to as "Bar." 7+15. ———- 10+4 = ———- ### What does the letter A with a line drawn across it represent? What Exactly Is a Diacritic, Exactly? It is customary to use diacritics to express a specific pronunciation—such as an accent, tone, or stress—as well as meaning in a word. This is particularly true when a homograph exists that does not include the highlighted letter or letters. ### In mathematics, what does a vertical bar represent? The vertical bar is referred to as a 'pipe' in certain circles. It is often used in the fields of mathematics, logic, and statistics. It is often understood to mean 'assuming that.' In probability and statistics, it is often used to denote conditional probability, although it may also denote a conditional distribution in certain situations. You may think of it as 'conditional on' in this context. ### What exactly are Vinculum numbers? There is a notion in Vedic mathematics known as Vinculum Numbers, and it refers to numbers that have at least one digit that is negative (having bar over them). Bar numbers are often referred to as bar codes. As previously stated, Normal Number may be written as follows: 2345 is equal to 2000 plus 300 plus 40 plus 5. ### What is the significance of an Overbar? An overline, overscore, or overbar is a typographical element that consists of a horizontal line drawn directly above the text and extending to the right of the text. Long before the invention of the modern computer, an overline was used in mathematical notation to indicate that particular symbols were intended to be used in conjunction.<\|endoftext\|>	4.53125
476	To find out how you can use the content, check the site's copright terms. Look for a link at the bottom of the webpage. Get tips on how narrative flow is created. Explore problem/solution scenarios and the slow or fast pacing of events. Rearrange comic strip panels and short story paragraphs in a logical order. Identify problem/solution storylines in both a comic strip and a short story. Choose pacing strategies for two short stories by identifying major and minor events and deciding which events to present in detail and which to present briefly. Craft a storyline by selecting items based on a problem/solution scenario and by using appropriate pacing. This learning object is one in a series of five objects. English > Level 5 > Language > Text structure and organisation > VCELA309 Understand how texts vary in purpose, structure and topic as well as the degree of formality English > Level 7 > Literature > Examining literature > VCELT374 Recognise and analyse the ways that characterisation, events and settings are combined in narratives, and discuss the purposes and appeal of different approaches English > Level 7 > Literature > Responding to literature > VCELT372 Compare the ways that language and images are used to create character, and to influence emotions and opinions in different types of texts Discuss aspects of texts, including their aesthetic and social value, using relevant and appropriate metalanguage Literary Devices, Narratives, Illustration, Text purpose, Text structure Students connect the sequence of events in a narrative storyline. Students identify narrative structure. Students investigate steps in a problem/solution plot structure. Students differentiate between major and minor events in narratives and identify pacing techniques. Students create a story outline using a problem/solution scenario and a pacing technique. Models techniques for narrative plot construction and well-paced story telling. Enables students to interpret, reconstruct and deconstruct model narrative structures. Enables students to construct narrative plot outlines using problem/solution and pacing techniques. Allows students to interactively rearrange narrative events, select events for storylines, classify events as 'problems' or 'solutions', select short or long versions of events, and plan a storyline using problem/solution and pacing strategies. Provides an option to print a completed plot outline.<\|endoftext\|>	4.59375
313	The Russian Revolution and subsequent Civil War were two of the most cataclysmic events of the twentieth century. In reshaping the largest nation on earth, their aftershocks are felt a century later. Civil war emanated from the Revolution. The instigators of the Revolution, the Bolsheviks, acted in desperation, instating short-sighted policies of nationalization and confiscation which antagonized most of Russian society. A loose coalition of counterrevolutionary forces, vaguely referred to as the White Movement, began fighting back in earnest in summer 1918. Their aims and means, however, always operated from the margins and often at cross purposes. As a result, no true center of opposition to Communism ever coalesced. By 1921, after three years of cruel fighting, famine, and pestilence that resulted in up to ten million military and civilian deaths, the Bolsheviks emerged as the improbable victors, which resulted in the creation of the Union of Soviet Socialist Republics. Like all nascent states that arise out of a cauldron, it sought to memorialize its achievement. The series of ten pictorial maps published towards that end represent a traditional channel of Russian communication: revolutionary posters celebrating the victory of Bolshevism over its internal and external enemies. Both charismatic and informative, they tell the story of the Civil War from the Bolshevik point of view. Hopefully their presentation will inspire further discussion on this painful episode in Eurasian history. Cartographic Reference Specialist Geography and Map Division Library of Congress<\|endoftext\|>	3.859375
1,122	Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs! #### Online Quiz (WorksheetABCD) Questions Per Quiz = 2 4 6 8 10 ### High School Mathematics - 24.8 Types of Functions - (One To One) One-One Function A function f: A->B is called a one-one function if distinct elements of A have distinct images in B, i.e if a1, a2 €A and a1 ≠a2=>f(a1) ≠ f(a2) Equivalently, we say f:A -> B is one-one if and only if for all a1, a2 € A, f(a1) = f(a2) => a1 = a2. Note 1: A one-one function is also called an injective function or injection. Note 2: If A and B are finite sets and f: A -> B is injective. Then n(A)<= n(B). Note 3: If n(A) = P and n(B) = q, then the number of possible mappings from A to B is qp. Illustrations: If A = {4, 5, 6} and B = {a, b, c, d} and if A -> B such that f = {(4,a), (5,b), (6,c)}, then f is one-one. The mapping f: R->R such that f(x) = x2 is not a one-one function since f(-2) = 4 and f(2) = 4, that is two distinct elements -2 and 2 have the same image 4. Example: Find if the following functions are one-one or not. f: R->R, defined by f(x) = x3, x € R. Example:Find whether the following functions are one-one or not. f: R->R , defined by f(x) = x3, x € R f: Z->Z, defined by f(x) = x2 + 5 for all x € Z. Method to check the injectivity of a function Step 1: Take two arbitrary elements x and y in the domain of f. Step 2: Put f(x) = x Step 3: Solve f(x) = f(y). If it yields x = y only then f: A->B is a one-one function or injection. Remark: Let f: A ->B and let x,y € A. Then x = y=>f(x) = f(y) is always true from the definition, but f(x) = f(y)=>x = y is true only when f is one-one. Solution: 1. Let x, y be two arbitrary elements of domain f, (x,y € R) such that f(x) = f(y). Then f(x) = f(y) =>x3 = y3 =>x = y 2. Let x, y be two arbitrary elements of Z such that f(x) = f(y). Then f(x) = f(y) =>x2 +5 = y5 + 5 =>x2 = y2 =>x = + or - y. Since f(x) = f(y) does ot yield a unique answer and x = y but gives x = + or - y, so f is not a one-one function. Suppose we have f(2) and f(-2), for either cases we get 9, thus two distinct elements 2 and -2 have the same image. Hence f is one-one function. Example: A one to one function is a function in which every element in the range of the function corresponds with one and only one element in the domain. Example of a one-to-one function: { (0,1) , (5,2), (6,4) } Domain: 0, 5, 6 Range: 1,2, 4 Each element in the domain (0, 5, and 6) correspond with a unique element in the range. Therefore this function is a one-to-one function. Directions: Solve the following problems. Also write at least 5 examples of your own. Q 1: Determine if the given function is one-one. To each person on earth assign the number which corresponds to his age.NoYes Q 2: {(x, y) : y = 5x - 6 }. If it is a function, what function is it?OntoOne-oneNot a function Q 3: {(x, y) : y = 4 for all values of x}, is it a function?yesno Q 4: To each country in the world which has a prime minister assign its prime minister.YesNo Q 5: Let f be defined by f(x) = x/(2x+1), find f(2)2/52/31/2 Q 6: To each country in the world, assign the latitude and longitude of its capital. Determine if it is one-one or not.NoYes Q 7: {(x,y): y greater than or equal to x-3}, x € z is a function or not, YesNo Q 8: To each book, written by only one author, assign the author. Determine if it is one-one or not.yesNo Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!<\|endoftext\|>	4.40625
711	# Question: How Many Are In A Set? ## Does a set mean two? They come packaged two speakers in a box. So a set of two means two speakers.. ## How do you define a set? A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. ## What is an empty or null set? Empty Set: The empty set (or null set) is a set that has no members. ## What are the rules of set? A set consists of three cards satisfying all of these conditions:They all have the same number or have three different numbers.They all have the same shape or have three different shapes.They all have the same shading or have three different shadings.They all have the same color or have three different colors. ## Is {} an empty set? In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. ## What is set Give 5 examples? Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z. Each object or number in a set is called a member or element of the set. Examples include the set of all computers in the world, the set of all apples on a tree, and the set of all irrational numbers between 0 and 1. ## What are the elements of set a? Elements of a SetA = {v, w, x, y, z} Here ‘A’ is the name of the set whose elements (members) are v, w, x, y, z.If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are ‘true’ or ‘false’: … If set Z = {4, 6, 8, 10, 12, 14}. ## What is cardinality of set? In mathematics, the cardinality of a set is a measure of the “number of elements” of the set. For example, the set contains 3 elements, and therefore. has a cardinality of 3. ## What is rule method? Rule Method This method involves specifying a rule or condition which can be used to decide whether an object can belong to the set. … The rule method is often preferred when defining larger sets where it would be difficult or time consuming to list all of the elements in a set. ## How much is in a set? Rep (repetition) is one complete motion of an exercise. A set is a group of consecutive repetitions. For example, you can say, “I did two sets of ten reps on the crunches” This means that you did ten consecutive crunches, rested, and then did another ten crunches. ## What do we call the members of a set? A set has members (also called elements). A set is defined by its members, so any two sets with the same members are the same (e.g., if set and set have the same members, then ). ## How many elements are there in set a only? c) The set {1, {1}} has two elements: 1 and the set whose only element is 1.<\|endoftext\|>	4.5
332	Created by Selam H over 5 years ago In Spanish there a 4 definite articles (all meaning "the.") This is different from English which has the used for both female, male, plural, and singular words: la → feminine (singular) el → masculine (singular) las → feminine (plural) los → masculine (plural) In English, these are the definite articles: the an a some* sometimes used What is unique about Spanish is that not only is there feminine or masculine people, other nouns have genders as well. ends with -o → feminine ends with -a → masculine ends with -e → both When making a noun plural, add -s if it already ends with a vowel. If it ends with a consonant, add -es. el ajedrez*doesn't end in -a, -o, and -e To say you like something add the verb gustar (like) and a definite article before the noun. To talk about a noun as a definite article and the noun. gusta → singular gustan→ plural EXAMPLES: Me gustan las verduas. I like vegetables. No me gusta la pasta. I don't like pasta. You must have the articles in sentences even when the word "the" isn't used.EXAMPLES: El helado es delicioso. Ice cream is delicious. Los animales son interesante. Animals are interesting. Quiz on Nouns & Definate Articleswww.examtime.com/en-US/p/369853<\|endoftext\|>	3.671875
628	Nearly three of every one thousand babies are born with some form of hearing loss. In most cases, however, hearing issues aren’t discovered in children until they are at least two years old. The first two years of a child’s life are hugely important in physical development as well as in forming emotional, learning and communication skills. Because of this, babies with moderate to severe hearing loss often experience major developmental setbacks. Newborn hearing loss is very difficult to detect, which is why many cases go undiagnosed until the child reaches talking age. Often, the only way to identify and treat a hearing loss problem when it truly counts is to take your baby to an audiologist for an infant hearing screening. Despite many years of developing and refining these hearing tests for babies that are just a few months old, studying infant hearing loss still presents many challenges. If you’re a new parent preparing for a newborn hearing screening, there are several important factors and facts you need to know before your appointment. - There are two common hearing tests used for newborns. The first is called an Auditory Brainstem Response (ABR) test, which measures the response of a baby’s hearing nerve using electrodes. The second is the Otoacoustic Emissions (OAE) test, which uses a microphone and earphone to calculate an infant’s hearing abilities by measuring the reflection of a sound’s echo as it passes through the ear canal. - False positives for hearing loss are common in newborn screenings. Your baby’s first hearing screening will likely be performed within a few hours or days of birth. Statistics show that between 2–10 percent of infant hearing tests indicate hearing loss, while only 0.003 percent of infants actually suffer from a permanent hearing impairment condition. Rather than a diagnosis, these tests are administered in order to help parents identify a potential problem as early as possible, promoting the prevention of developmental disorders. - Hearing loss in infants is usually the result of a temporary, treatable condition. The tests used in newborn hearing screenings are accurate; however, they can’t reveal what is causing the irregular results. In most cases, a hearing loss–positive result from the tests indicates an easily treatable problem like a fluid buildup, earwax blockage or ear infection. In other cases, the cause is never identified. Infants with irregular test results will be directed to an audiologist for a more in-depth examination. - It’s important to continue following up on irregular newborn hearing screening results. Since one in 10–50 newborn hearing screenings indicate a potential hearing loss, many parents are left wondering about their child’s hearing. It’s critical to identify permanent hearing loss symptoms as early as possible in infants, so regularly retesting your infant’s hearing is highly advisable. Talk to an audiologist about your situation to find out how frequently you should test your child’s hearing to detect possible hearing loss problems. Contact one of our convenient locations for more information or to schedule an appointment.<\|endoftext\|>	3.75
588	1. ## STRONG induction i have this question that i couldnt solve its about STRONG INDUCTION i need ur help pls. Prove by strong induction that any natural number n greater than 7 can be written as a sum of 3's and 5's, being understood that no 3's or no 5's is also accepted, as in 12 = 3 +3 + 3 + 3. thank you 2. ??? 9 can't. Do you mean sum or difference of 3s and 5s? 3. Originally Posted by Debsta ??? 9 can't. Do you mean sum or difference of 3s and 5s? Why not? 9 = 3+3+3 is an accepted form, according to his post. Base case: n=8. 8=5+3 so we are good to go. Assume that $n \in \mathbb{N}$ can be written as a sum of 3's and 5's. We want to prove that we can also express n+1 as such a sum. Lemma: The implementation of n as such a sum has either at least three 3s or one 5 in it. Proof: If n has a 5 in its sum, we are done. Assume it doesn't, then n = 3m for some positive integer m. But we only take n > 7, therefore $m\geq 3$ and the lemma is correct. According to the inductive assumption, we know that $n = 3k + 5m$ where $k\geq 3 \ \text{or} \ m \geq 1$. Substituting that into n+1, we get: $n+1 = 3k + 5m + 1$. Now, if: (I) $k <3$: We know that there is at least one 5 in the sum. This gives us that: $n+1 = 3k + 5(m-1) + 5 + 1 = 3k + 5(m-1) + 6 = 3(k+2) + 5(m-1)$ and we are done. (II) $m < 1$: Then, $n = 3m: \ m \in \mathbb{N}, \ m\geq 3$. This gives us that: $n+1 = 3m+1 = 3(m-3)+3\cdot3 + 1 = 3(m-3)+10 = 3(m-3) + 2\cdot 5$ and then n+1 is also a sum of 3,5s. Therefore it always is, and we are done.<\|endoftext\|>	4.4375
2,219	NetLogo Models Library: This model shows how lightning is generated. The process consists of two phases: the production of the electric field, and the air ionization to create the bolt. This model represents the latter phase. Lightning is one of the most visually impressive and frequently occurring natural phenomena on Earth. However, very few people actually have a solid understanding of what causes lightning, how it works, and why it occurs. This model attempts to illustrate lightning strikes from beginning to end at the level of individual charges. The user will observe how the behavior and interaction of extremely small charges can lead to very powerful and visually impressive action of a lightning strike. In order to thoroughly understand the phenomenon of lightning one must have a strong understanding of electrical physics and thermodynamics. In order to be accessible this model avoids such complicated topics and instead provides a more general basis for charge and particle behavior. The emphasis in this model is to understand lightning as it relates to individual charges, not the underlying forces behind the charges themselves. The Earth (green squares) and sky (black squares) serve as the layout for the world. The cloud appears as a gray cloud-shape at the top of the skyline. Among these are three main attributes that influence the production of lightning: step leaders at the bottom of the clouds (blue negative circles), dust particles in the air (dark gray squares), and positive streamers on the surface of the Earth (red positive circles). The initial cause of a lightning strike is a separation of charge within a cloud. Positively charged particles accumulate at the top of the cloud while negatively charged particles concentrate themselves at the bottom of the cloud. Within the cloud the charges move in a random manner; however, they are constrained to their charge regions. The negative charges in the bottom of the cloud have such a high concentration that they force the electrons on the Earth's surface deep into the ground. This also has the effect of pulling the positive charges in the Earth's surface. As the strength of the electric field increases the air around the cloud breaks down and converts into plasma, or ionized air. The positive and negative components of the air itself are pulled apart and separated from each other. This separation allows the electrons to flow through the plasma much more easily than they could through normal air. As electrons flow through the plasma they force the air around them to become plasma in turn. In this way electrons in the cloud "burn out" paths towards the Earth. The paths are known as step leaders, and grow from cloud to ground in a tentacle-like manner. Impurities in the air may cause some patches of air to turn into plasma more easily than others. Rather than direct lines from cloud to ground, lightning takes the path of least resistance. In this model this is illustrated by distribution of dust particles throughout the sky. Streamers are the positive equivalent of the negative step leaders created by clouds. However, streamers are not self-sufficient and thus do not grow indefinitely towards the cloud. All objects on the Earth's surface will emit a streamer, though depending on the size and material of the object the streamer's length may vary. Streamers are released much more quickly than a step leader since they are much smaller and only extend a very limited distance. Step leaders progress towards the ground until they encounter either the ground or a streamer. In both cases the "circuit" is completed and charge may flow freely between cloud and ground. The large concentration of positive charges on the Earth's surface flow very quickly through the plasma stream towards the sky and neutralize the electrons in the cloud. The flash of light that is seen is the rapid movement of charge through the air, exactly the same as the light you see during a spark of static electricity. Once the massive negative charge in the cloud has been neutralized the flow of charge through the air comes to a stop. The plasma becomes de-ionized and once again becomes air as the step leaders are destroyed. The environment is now ready to begin the process anew. There are a series of breeds that live in the world that simulate lightning. Trees - These are the trees on the Earth's surface. They stand above the Earth's surface and release positive streamers. Positive Cloud Ions - These are the positive ions that bounce around the top of the cloud. As molecules of water evaporate, they collide into one another and exchange charges. Those with positive charges move to the top of the cloud. Negative Cloud Ions - These are negatively charged ions at the bottom part of the cloud. They move around the cloud and become step leaders. Positive Ground Ions - These are the positive ions on the Earth's surface. As the electric field forms around the cloud it pulls the positive charges to the top of the Earth's surface. Step Leaders - These are the negative ions that branch out from the bottom of the cloud. The ions are drawn to the positive charge on the Earth's surface, particularly the positive streamers. These particles leave a path of ionized air as they travel; they tend to move towards the Earth's surface or impurities, such as dust, in the air. When the step leaders reach the Earth's surface, a positive streamer, or a tree, the path is closed and lightning strikes. There is a chance that the step leader will die and the path will fade. Positive Streamers - These are the positive ions that branch upwards from the Earth's surface. They tend to move towards the electric field formed by the cloud or step leaders. If the positive streamer reaches the path of ionized air or connects with a step leader then lightning strikes. These breeds live together following specific rules elaborated below: At every time step, each step leader moves towards the Earth ionizing a blue path of air along the way. This path is found using the following criteria: - if one of the immediate neighbors is a dust particle, there is a chance that it will move to that patch - if there is no dust, then if one of the patches in front of it is a positiveStreamer, it moves one step towards it - otherwise, the leader will move in a random direction towards the Earth's surface with the highest probably being directly downwards After the step leaders have made one move we check their new position. If they have moved of the edge of the world, then their path fades. Some paths also fade by chance. If the path does not fade we check to see if they have closed a connection by hitting any of a tree, a positive streamer, a positive streamer's path, or the Earth's surface. If so, then lightning strikes. When lightning strikes, the patch at which the connection took place will turn yellow to indicate a connection. From there each patch will ask its neighbors to check if they've been ionized (or are a blue or violet color). If they have been ionized, they too will turn yellow and ask their neighbors about their color. If the neighboring patches have not been ionized they will turn white to highlight the glow of the lightning bolt. This cycle continues until the the surface of the Earth and the top of the cloud are reached. This indicates that the charge has been released, and the paths are removed from the world. Since the positive streamers (from the ground) grow at a slower rate than the step leaders, after 40 ticks have passed the streamers begin to grow up by one step for every tick. The direction of their growth is random, but most likely in the upward direction. The ions in the clouds shift their location to indicate the motion of the particles in the cloud. SETUP button - sets up the step leaders, dust particles and positive streamers in the world GO button - runs the model STRENGTH-OF-FIELD slider - changes the strength of the electric field produced within the cloud SIZE-OF-CLOUD slider - changes the size of the cloud NUMBER-OF-TREES slider - changes the number of trees on the Earth's surface DUST slider - changes the number of dust particles in the sky between the cloud and the Earth's surface The model runs until all paths have died or a bolt of lightning has been produced. Notice the charges on the earth's surface and how they relate to the cloud's field and position. Notice how the trees and dust particles change the path taken from the cloud. The location and size of the trees influence the pull on the ionized air. The positive streamers and the step-leaders have similar behavior as they grow; however, the speed, distance, and strategy in which they do so is quite different. Can you tell how these paths differ? Try different settings for the strength of field. What do you notice about the amount of lightning that is formed? How are the strength of field and size of the cloud related? Does the cloud size have any effect on the lightning formed? Does varying the amount of dust or impurities in the air have influence on the path taken by the step leaders? Consider other features that influence lightning paths, such as lightning rods. How do these influence the path or amount of lightning produced? The positive streamers have potential to grow towards the electric field of the cloud. How does their growth influence the paths? Currently, the shape of the cloud does not affect the direction that a step leader starts out traveling. Modify the code to allow a user to change the shape of the cloud, and cause step leaders to progress perpendicularly from its surface. "Climate Change" has some similarities. In both cases, there is a relationship to cloud behavior. "Percolation" also has similar features to the way elements move through their world. Wikipedia on Lightning: https://en.wikipedia.org/wiki/Lightning How Stuff Works on Lightning: https://science.howstuffworks.com/nature/natural-disasters/lightning.htm National Geographic on Lightning: http://environment.nationalgeographic.com/environment/natural-disasters/lightning-profile.html If you mention this model or the NetLogo software in a publication, we ask that you include the citations below. For the model itself: Please cite the NetLogo software as: Copyright 2011 Uri Wilensky. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at [email protected].<\|endoftext\|>	4.0625
3,278	# Perimeter of a Rectangle Word Problems There are different types of geometry word problems. One of the most common involves the perimeter which is the distance around the outside of a two-dimensional geometric shape. In this lesson, our focus is on the perimeter of a rectangle in which we’ll find the relationship between the sides of a rectangle, that is, the length and the width. The key to solving perimeter word problems is to know two things: 1) The formula of the perimeter of a rectangle. • Perimeter of a Rectangle Formula: ${\textbf{\textit{P = 2L + 2W}}}$ • Remember that the length (L) is the longest side of the rectangle while the width (W) is the shortest side. 2) Being able to express the length in terms of the width and vice versa, depending on the word problem. Let’s tackle this topic by working on a variety of examples involving the perimeter of a rectangle. Just a heads up, the first five examples (word problems 1-5) can be solved by multi-step equations with one or single variable while the last three (word problems 6-8), can be solved by systems of equations with two variables. Example 1: The perimeter of a rectangle is 128 centimeters and the length is 47 centimeters. Find the width of the rectangle. The first thing that we need to do is to construct or draw a diagram based on the information given in the problem. Looking back at our problem, we were given the measurement for the length as well as the perimeter of the rectangle. With this visual, we can easily spot what pieces of information are already provided to us and what else is missing that we need to solve. Clearly, the width is unknown so we’ll represent this value with the variable $W$. Let’s now solve for the width ($W$) using the formula for the perimeter of a rectangle. Since we already know the value for the length ($L$) and the value for the perimeter ($P$), we will simply replace the variables ($L$) and ($P$) with their values. Answer: The width of the rectangle is 17 centimeters. Before we proceed to the next problem, let’s find out if our answer is correct. We can do this by simply substituting the values that we have for the perimeter, length, and width into the perimeter formula, then verify if each side of the equation equals the other. Check: Great! We’re able to confirm that the measurement of the width which is 17 cm, indeed is the correct value. Example 2: A rectangle has a width of 7 feet and a length of 132 inches. Find the perimeter of the rectangle both in feet and inches. This problem is asking us to express the perimeter of the rectangle using two different units of measurement, i.e. in feet and in inches. To get both measurement units, we’ll solve the problem in two parts. PART 1: Express the perimeter of the rectangle in feet. We are given the following information below: Since the width is already in feet, we don’t have to do anything with it. However, for the length, we must convert 132 inches to feet. We know that 1 ft = 12 in. Therefore, Perfect! We were able to get the measurement of our length in feet. So now, we have: • Width = 7 ft • Length = 11 ft To find the perimeter, we simply have to plug in the values into the Perimeter Formula then simplify. Part 1 Answer: The perimeter of the rectangle is 36 feet. PART 2: Express the perimeter of the rectangle in inches. We will use the same methodology in Part 1 to find our answer for this second part. Again, the following pieces of information are given to us in the original problem: • Width = 7 ft. • Length = 132 in. This time, the length is the side that has its measurement already in inches. Since we want to find the rectangle’s perimeter in inches, we’ll focus on converting the width from feet to inches. Remember again that 12 in. = 1 ft. So now we have, The dimensions of our length and width are now both in inches. Let’s now find the perimeter of the rectangle by substituting the values into the perimeter formula then simplify. Part 2 Answer: The perimeter of the rectangle is 432 inches. Example 3: The width of a rectangle is 23 meters less than the length. The perimeter of the rectangle is 94 meters. Find the dimensions of the rectangle. For us to solve this word problem, we need to be able to substitute values or expressions to the $L$, $W$, and $P$ variables of the perimeter formula, $P=2L+2W$. In our problem, we can easily spot the value of the perimeter ($P$) which is 94 meters. On the other hand, the width of the rectangle is expressed in terms of length. The width of the rectangle is 23 meters less than the length. We need to translate the algebraic sentence above into an algebraic expression so we can substitute it for the width in the perimeter formula. So if ($W$) stands for the width and ($L$) is for the length, our algebraic expression for the width is, ${W = L – 23}$ How about the length? Well, at this time, our length is unknown. So in this case, we will keep the variable $L$ to stand for the length. Let’s construct a diagram to get a better visual of all the information that we have so far. Now that it’s clear to us what we’ll substitute the $L$, $W$, and $P$ variables with, we can proceed to solve for $L$ using the perimeter of a rectangle formula. Since the problem is asking us to find the dimensions for both the length and the width, let’s use the value of $L$ to also get the value for the width. • Length ($L$)= $35$ • Width ($W$)= $L-23={\color{red}35}-23=12$ Answer: The length of the rectangle is 35 meters and the width is 12 meters. I advise my students to always check their answers no matter how confident they feel that they got the correct one. For this problem, we simply have to substitute the values of $L$, $W$, and $P$ in the perimeter formula. If the left and the right side of the equation both equal each other, then our answers are correct. Check: Example 4: The length of the rectangle is 12.5 feet more than the width. If the perimeter of the rectangle is 101 feet, how long is the length of the rectangle? Right off the bat, we can see that the value of the perimeter is 101 ft and that the length is defined using the width as given in the following statement. The length of the rectangle is 12.5 feet more than the width. Just like in our previous example, we need to algebraically express the length in terms of the width so we can substitute this algebraic expression for the length ($L$) in the perimeter formula. Translating the statement above, we get $L = W+12.5$ Let’s put together the information that we’ve gathered so far in a diagram. As you can see, the width ($W$) is currently unknown. Therefore, the next step that we have to do is solve for ($W$) using the formula for the perimeter of a rectangle. Now that we know the value of the width ($W$), let’s use this value to find the dimension for the length. • Width ($W$)= $19$ • Length ($L$)= $W+12.5={\color{red}19}+12.5=31.5$ Going back to our original problem, how long is the length of the rectangle? Answer: The length of the rectangle is 31.5 ft. This time, I will leave it up to you to check if our answer is correct. Start by substituting the values of $L$, $W$, and $P$ in the perimeter formula, $P=2L+2W$, then simplify. If both sides of the equation equal each other, then we got the correct answer. Example 5: A certain rectangle has a length of half a yard more than twice the width. The perimeter of the unknown rectangle is 55 yards. Find its width. Let’s start by determining the values and expressions that are provided to us in the problem. • Length = half a yard more than twice the width = $2W +\Large {1 \over 2}$ • Width = ? • Perimeter = 55 Using the perimeter formula, we will solve for the unknown which is the width ($W$). Before we formally answer the question from our word problem, let’s find out the dimensions of the rectangle using the value of the width. • Width ($W$)= $9$ • Length ($L$)= $2W +{\Large {1 \over 2}}=2({\color{red}9})+{\Large {1 \over 2}}=18+{\Large {1 \over 2}}=18{\Large{1 \over 2}}$ Answer: The width of the rectangle is 9 yards. Example 6: When you subtract the width from the length of the rectangle, the difference is 8 inches. What are the dimensions of the rectangle if its perimeter is 72 inches? This example is a little bit more complex than the previous ones. As you may have noticed, neither the length nor the width is expressed in terms of the other. So to proceed, we will start by discussing the important statements that are given to us in this problem. When you subtract the width from the length of the rectangle, the difference is 8 inches. Let’s use again the variable $L$ to stand for length and $W$ for the width. Translating this algebraically, we have the following equation $L – W = 8$ The perimeter of the rectangle is 72 inches. We are already familiar with the formula for the perimeter of a rectangle so we can simply write this as $P = 2L + 2W$ $72 = 2L + 2W$ By just going through both statements, we were able to come up with two equations: • Equation 1: $L – W = 8$ • Equation 2: $72 = 2L + 2W$ It is clear at this point that we are dealing with two equations with two unknowns, namely, $L$ and $W$. In other words, we are going to solve systems of linear equations with two variables. Observe that we can use Equation 1 to express the length in terms of the width, then substitute the expression into Equation 2. The result will be a multi-step equation having the width ($W$) as the only variable in the equation. Let’s do this step-by-step. 1) Start with Equation 1 and solve for $L$. Notice that to clean up the left side of the equation, we added $W$ to both sides of the equation. Then we applied the Commutative Property of Addition, that is, $8 + W = W + 8$. 2) Next, substitute the expression of $L$ into Equation 2. We now have the value for the width. But since we are asked to find both dimensions of the rectangle, let’s plug the value of the width which is 14 into Equation 1 to find the value for the length ($L$). So, what are the dimensions of the rectangle if its perimeter is 72 inches? Answer: The length of the rectangle is 22 inches while the width is 14 inches. Calculator Check: Example 7: The difference of the length and three times the width of a rectangle is 5 centimeters. Find the length and the width of the rectangle if its perimeter is 82 centimeters. We have the same situation here as our previous example. Neither the length nor the width is expressed in terms of the other. Therefore, let’s break down the problem again to find out what important pieces of information are given to us. First, we are told that The difference between the length and three times the width of a rectangle is 5 centimeters. Using $L$ and $W$ again as our unknown variables, we can translate this algebraic sentence as $L – 3W = 5$ Next, our second statement says that The perimeter of the rectangle is 82 centimeters. This one is easy. Using the formula for the perimeter of a rectangle we have, $P = 2L + 2W\,\,\, \to \,\,\,82 = 2L + 2W$ Before we proceed, let’s look at our two equations: • Equation 1: $L – 3W = 5$ • Equation 2: $82 = 2L + 2W$ Our next step is to express $L$ in terms of the width. 1) Solve for $L$ using Equation 1. We are now ready to plug in the expression of $L$ into our second equation. 2) Substitute $L$ with $3W + 5$ in Equation 2. Let’s now also see what the value of the length is using the value we got for the width which is 9. Answer: The length of the rectangle is 32 centimeters and the width is 9 centimeters. Calculator Check: Let’s verify again real quick using a calculator if both values will give us a perimeter of 82 cm when plugged into the formula of the perimeter of a rectangle. If the sum of twice the length ($2L$) and twice the width ($2W$) is 82, then both of our answers are correct. Example 8: The sum of the length and one-half of the width is 42 yards. The rectangle’s perimeter is 100 yards. What is the width of the rectangle? You should be familiar by now with what to do first when we have perimeter word problems where neither each side of the rectangle is defined using the other. Let’s delve into the important statements right away and translate them into an algebraic equation. The sum of the length and one-half of the width is 42 yards. • Equation 1: $L + {\Large{1 \over 2}}W = 42$ The rectangle’s perimeter is 100 yards. • Equation 2: $P = 2L + 2W\,\,\, \to \,\,\,100 = 2L + 2W$ What’s next? Well, it’s time for us to define the length using the width by solving for $L$ using Equation 1. Let’s now substitute $L$ with $42 – {\Large{1 \over 2}}W$ in Equation 2. The word problem is only asking us for the measurement of the width. So we’ll say, Answer: The width of the rectangle is 16 yards. I’ll leave it up to you to do the checking if our answer is correct. Make sure to find the value of the length first by using the value of the width ($W$) which is 16. Then, move on to check if both values, when plugged into the perimeter of a rectangle formula ($P=2L+2W$), will give you a perimeter of 100 yards as stated in our original word problem. You may also be interested in these related math lessons or tutorials: Perimeter of a Rectangle Area of a Rectangle<\|endoftext\|>	4.875
1,372	# If $a+b+c=6$ and $a,b,c$ belongs to positive reals $\mathbb{R}^+$; then find the minimum value of $\frac{1}{a}+\frac{4}{b}+\frac{9}{c}$ . 11,340 ## Solution 1 We can use also AM-GM. For $a=1$, $b=2$ and $c=3$ we get a value $6$. We'll prove that it's a minimal value. Thus, it's enough to prove that $$\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\geq6.$$ Indeed, by AM-GM: $$\frac{1}{a}+\frac{4}{b}+\frac{9}{c}=a+\frac{1}{a}+b+\frac{4}{b}+c+\frac{9}{c}-6\geq$$ $$\geq2\sqrt{a\cdot\frac{1}{a}}+2\sqrt{b\cdot\frac{4}{b}}+2\sqrt{c\cdot\frac{9}{c}}-6=2+4+6-6=6.$$ Done! ## Solution 2 Cauchy-schwarz inequality state that : • For any $x_1, y_1, z_1; x_2, y_2, z_2 \ \in \mathbb{R}^+$; we have: $$\left(\sqrt{x_1 \cdot x_2} + \sqrt{y_1 \cdot y_2} + \sqrt{z_1 \cdot z_2}\right) ^ 2 \leq \left(x_1 + y_1 + z_1\right) \cdot \left(x_2 + y_2 + z_2\right) \ .$$ So we can conclude that: \begin{align} & \left( \sqrt{\frac{a}{a}} + \sqrt{\frac{4b}{b}} + \sqrt{\frac{9c}{c}} \right) ^ 2 & \leq \ \ \ \ & \left(\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\right) & \cdot \ \ \ \ \ \ \ \ & (a+b+c) & \Longrightarrow \\ & \ \ \ \ \ \ \ \ \ \ \ \left( 1 + 2 + 3 \right) ^ 2 & \leq \ \ \ \ & \left(\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\right) & \cdot \ \ \ \ \ \ \ \ & \ \ \ \ \ \ \ \ (6) & \Longrightarrow \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 6 \right) ^ 2 & \leq \ \ \ \ & \left(\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\right) & \cdot \ \ \ \ \ \ \ \ & \ \ \ \ \ \ \ \ (6) & \Longrightarrow \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6 & \leq \ \ \ \ & \left(\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\right) \end{align} Note that this in-equlality is sharp for $a=1, b=2, c=3$; for which one can see the value of $\left(\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\right)$ is equal to $6$. ## Solution 3 Using the lagrange multiplier ($k$); $$f(a,b,c,k)=\frac1a+\frac4b+\frac9c+k(a+b+c-6)$$ We will take the derivative in respect to all the variables; $$f_k'=0$$ $$f_a'=k-\frac{1}{a^2}$$ $$f_b'=k-\frac{4}{b^2}$$ $$f_c'=k-\frac{9}{c^2}$$ $$k=\frac{1}{a^2}=\frac{4}{b^2}=\frac{9}{c^2}$$ $$\sqrt{k}=\frac1a=\frac2b=\frac3c$$ $$\sqrt{k}=\frac{1+2+3}{a+b+c}=\frac66$$ $$a=1$$ and $$b=2$$ and $$c=3$$ then $$\frac1a+\frac4b+\frac9c=1+2+3=6$$ Done!! Share: 11,340 Author by ### Tabber Updated on August 01, 2022 • Tabber 10 months If $a+b+c=6$ and $a,b,c$ belongs to positive reals, then find the minimum value of $$\frac{1}{a}+\frac{4}{b}+\frac{9}{c}$$ using AM $\ge HM$ $\frac{a+b+c}{3}\ge\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$ ${\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\ge\frac{3}{2}$ or why not $AM\ge GM$ $\frac{\frac{1}{a}+\frac{4}{b}+\frac{9}{c}+a+b+c}{6}\ge (\frac{1}{a}\times\frac{4}{b}\times\frac{9}{c}\times a\times b\times c)^\frac{1}{6}$ $\Rightarrow \frac{1}{a}+\frac{4}{b}+\frac{9}{c}\ge 6(6^\frac{1}{3}-1)$ • Tabber over 5 years Thank you Famke that answers my question. • Davood over 5 years @Quintessence ; My dear Quintessence, wellcome to math-stack-exchange .<\|endoftext\|>	4.40625
260	Working with Gifted & Talented Students Working with a gifted student can be both a joy and a frustration. To understand why, we need to be clear about definitions. A gifted student is one whose intelligence - typically described as an IQ score resulting from one or more tests - is 130 or above. That is, giftedness is a measure of innate ability, not performance. The result is a paradox. A motivated student who works hard, gets straight "A"s, and behaves well in class may not be gifted. A student who doesn't perform well, is disruptive, and clowns around in class may well be gifted. This can be frustrating for classroom teachers! Just as you adapt to the needs of disabled students, working with gifted students can require classroom and curriculum modifications. But the results can be highly rewarding for both teachers and students. These resources should be helpful in identifying and working with gifted students. How to Spot a Gifted Student - A listing of traits - both positive and negative that gifted students frequently exhibit. Strategies for Teachers - Ideas, strategies, and resources you can use to ensure that gifted students use their abilities to the fullest. Web Resources - Our annotated collection of resources dealing with giftedness and gifted education.<\|endoftext\|>	4.1875
3,165	In many games, the game objects interact with each other and very often it is important to detect when game objects collide with others, for example when a missile hits a target. A first and rather easy method to detect such collisions is to "inscribe" the polygon inside a sphere, or a circle in 2D. To compute the radius of the circle, it is possible to take the distance from the center of the polygon to each vertex and to then average those values, or to simply take the smallest or the largest one, depending on what should be achieved. An often used heuristic is to take a value between the largest possible radius and the average of the radii. This actually quite simple idea thus already leads to two mathematical problems that need to be solved: First, the centroid, or the geometrical center, of a polygon must be found and then, the distance between the centroid and different vertices must be computed (fast!). The following tutorial explains how to compute the centroid of a convex polygon. ## Centroid of a Polygon Before tackling the task of computing the centroid of a polygon, it is wise to have a look at the better known problem of computing the centroid of a triangle. #### The Barycenter Most high school students learn how to compute the coordinates of the centroid of a triangle, also called the barycenter of a triangle. The barycenter is the intersection point of the three lines going through one of the vertices and the middle of the opposite edge of the triangle (as seen in the figure above - the point G is the barycenter of the triangle). In the language of linear algebra, the coordinates of the barycenter $G$ of a triangle $ABC$, where $A=(x_A,y_A)$, $B=(x_B,y_B)$ and $C=(x_C,y_C)$ are three non-aligned points in $\mathbb{R}^2$, can easily be computed as follows: $G = \left(\dfrac{x_A+x_B+x_C}{3},\dfrac{y_A+y_B+y_C}{3}\right),$as indeed $G$ is the barycenter of the triangle, if, and only if, $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\vec{0}$. In affine geometry, the above formula can be written as $G = A + \dfrac{1}{3} \cdot \left( \overrightarrow{AB} + \overrightarrow{AC} \right)$. As an example, let $A=(0,0)$, $B=(0,3)$ and $C=(3,0)$ be three points in the plane, then the barycenter of the triangle $ABC$ is $G=\begin{pmatrix}0\\0\end{pmatrix} + \dfrac{1}{3} \cdot \left( \begin{pmatrix}0\\3\end{pmatrix} + \begin{pmatrix}3\\0\end{pmatrix} \right) = \begin{pmatrix}1\\1\end{pmatrix}.$ #### Uniform Distribution of Mass Unfortunately computing the centroid of a polygon isn't just as easy as computing the barycenter of a triangle; in the case of a polygon simply averaging over the coordinates of the vertices no longer results in the correct coordinates of the centroid - the only exception being regular polygons. While the centroid of a polygon is indeed its center of mass, the mass of a polygon is uniformly distributed over its entire surface, not only at the vertices. Note that for simple shapes, such as triangles, rectangles or the above mentioned regular polygons, the mass being evenly distributed over the surface is equivalent to the mass being at the vertices only. In the case of a convex polygon, it is easy enough to see, however, how triangulating the polygon will lead to a formula for its centroid. #### The Area of a Triangle As the mass is distributed over the entire surface of the polygon, it is necessary to compute the area of the triangles resulting from the triangulation. As above, let $ABC$ be a triangle, then the well-known formula, $V=\dfrac{l \cdot h}{2}$, for the area of the triangle, where $l$ is the length of the base of the triangle and $h$ its height, can be reformulated in the language of affine geometry using the determinant function: $V = \dfrac{1}{2} \cdot \left\| \operatorname{det}\begin{pmatrix}x_A&x_B\\y_A&y_B\end{pmatrix}\right\|.$ Note that the choice of $A$ and $B$ is irrelevant, the formula holds for any two points, i.e. let $v_1$ and $v_2$ be two vectors defining two sides of the triangle, then $V = \frac{1}{2} \cdot \operatorname{det}(v_1,v_2)$. Using, once again, the example from above, the area of the triangle $ABC$ is $V=\dfrac{1}{2} \cdot \left\| \begin{pmatrix}0&3\\3&0\end{pmatrix}\right\| = \dfrac{9}{2}$. #### Convex and Closed With the knowledge we just gathered, we can now tackle the problem of computing the centroid of a convex and closed polygon. Thus let $P$ be a convex and closed polygon defined by its $n$ vertices $V_0 = (x_0, y_0)$, $V_1 = (x_1, y_1)$, ..., $V_n = (x_n, y_n)$, noted in a counter-clockwise order, simple to make sure that the determinant computed for the area of a triangle is positive, and thus being able to omit the use of the absolute value. #### Triangle Centroids As seen above, to compute the centroid and area of a triangle in vector notation, the vectors between a fixed vertex, $V_1$, for convenience, and the other vertices of the triangle are needed. Thus let $v_i = V_{i+1}-V_1$, for $i=1$, $2$, ... $n-1$ be those vectors between $V_1$ and the other vertices. After triangulation, there are $n-2$ adjacent triangles with centroids $C_i = A_1 + \frac{1}{3}\cdot\left(v_i + v_{i+1}\right)$, for $i=1$, $2$, ..., $n-2$. #### Triangle Areas Let us denote the areas of the triangles with a lower $w$, for weight. In vector notation, the weight of the triangles resulting from the triangulation are $w_i = \frac{1}{2} \cdot \operatorname{det}(v_i, v_{i+1})$, $i=1$, $2$, ..., $n-2$. The total area $W$ of the polygon is thus $W = \sum\limits_{i=1}^{n-2}w_i = \frac{1}{2}\sum\limits_{i=1}^{n-2}\operatorname{det}(v_i,v_{i+1}).$ The name "weight" is well suited as, as visualized in the figure below, the area of the triangle measures how much of the entire mass of the polygon is contained in the triangle, thus by how much the barycenter of that triangle influences the location of the centroid of the polygon. Think of an election: the more inhabitants in a region, the more delegates that region sends to the central government, the more influence it has on global politics. #### Centroid of the Polygon To now finally compute the coordinates of the centroid $C_P$ of the polygon $P$, it is thus sufficient to divide the sum of the "weighted" centroids of the triangles by the total area of the polygon:$C_P = \frac{1}{W}\sum\limits_{i=1}^{n-2}w_iC_i.$To resemble the formula for the barycenter of a triangle in affine space, the above formula can be rewritten as follows:$C_P = A_1 + \dfrac{1}{3}\dfrac{\sum\limits_{i=1}^{n-2}\operatorname{det}(v_i,v_{i+1}) \cdot (v_i+v_{i+1})}{\sum\limits_{i=1}^{n-2}\operatorname{det}(v_i,v_{i+1})},$or, using coordinates in euclidean space:$C_P = \dfrac{1}{3}\left( \dfrac{\sum\limits_{i=1}^{n}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)}{\sum\limits_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)}, \dfrac{\sum\limits_{i=1}^{n}(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)}{\sum\limits_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)} \right).$ Note that these formulas are correct, even if the vertices are not given in counter-clockwise order: The determinants might become negative, but the computed coordinates will be correct. Obviously the formula to calculate the centroid of a polygon contains the case of the barycenter of a triangle, as in the case of $n=3$ the formula reads:$C_P = \frac{1}{W} \cdot w_1 \cdot C_1 = \frac{1}{w_1} \cdot w_1 \cdot C_1 = C_1.$ #### Source Code In C++, the discussed ideas translate to the following code: void Geometry::computeCentroid(const std::vector<D2D1_POINT_2F>& vertices, D2D1_POINT_2F* centroid) { float centroidX = 0, centroidY = 0; float det = 0, tempDet = 0; unsigned int j = 0; unsigned int nVertices = (unsigned int)vertices.size(); for (unsigned int i = 0; i < nVertices; i++) { // closed polygon if (i + 1 == nVertices) j = 0; else j = i + 1; // compute the determinant tempDet = vertices[i].x * vertices[j].y - vertices[j].xvertices[i].y; det += tempDet; centroidX += (vertices[i].x + vertices[j].x)tempDet; centroidY += (vertices[i].y + vertices[j].y)tempDet; } // divide by the total mass of the polygon centroidX /= 3det; centroidY /= 3*det; centroid->x = centroidX; centroid->y = centroidY; } #### Examples As a first example, consider the three points $A=(1,0)$, $B=(2,0)$ and $C=(0,3)$ in the euclidean plane, then the barycenter of the triangle $ABC$ is the point $G=(1,1)$: Clearly, the above formula leads to the same result: $C_P = \frac{1}{W} \cdot w_1 \cdot C_1 = \frac{2}{3} \cdot \frac{3}{2} \cdot (1,1) = (1,1)$, or in coordinate form:$G=\dfrac{1}{3}\left(\dfrac{3 \cdot 0 + 2 \cdot 6 + 1 \cdot (-3)}{0+6-3},\dfrac{0 \cdot 0 + 3 \cdot 6 + 3 \cdot (-3)}{3} \right) = \frac{1}{3} \cdot (\frac{9}{3},\frac{9}{3})=(1,1).$ For a more complicated example, let $V_1=(1,0)$, $V_2=(2,1)$, $V_3=(0,3)$, $V_4=(-1,2)$ and $V_5=(-2,-1)$ be five points in the euclidean plane and $P$ the polygon defined by those five points: To compute the centroid $C_P = (x_C,y_C)$ using the coordinates of the five vertices, it is a good idea to first compute the "weights": $w_1 = 1 \cdot 1 - 0 \cdot 2 = 1$, $w_2=6$, $w_3=3$, $w_4=5$ and $w_5=1$. The $x$-coordinate of the centroid can now be computed as follows: $x_C = \frac{1}{3} \cdot \frac{3 \cdot 1 + 2 \cdot 6 + (-1) \cdot 3 + (-3) \cdot 5 + (-1) \cdot 1}{16} = \frac{1}{3} \cdot \frac{-4}{16}=-\frac{1}{12}\approx0,08.$The $y$-coordinate of the centroid can be computed equivalently: $y_C = \frac{1}{3} \cdot \frac{1+24+15+5-1}{16} = \frac{1}{3} \cdot \frac{11}{4} = \frac{11}{12} \approx 0,92.$Thus the centroid of the above polygon $P$ is $C_P = (-\frac{1}{12},\frac{11}{12})$. # Formula to remember In later tutorials we will learn how to detect collisions between games objects. To do so, we must find the center of the game objects, which are often given as convex polygons (think of aircrafts, for example). Thus the important thing to remember from this tutorial is the following formula to compute the centroid of a convex polygon: $C_P = \dfrac{1}{3}\left( \dfrac{\sum\limits_{i=1}^{n}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)}{\sum\limits_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)}, \dfrac{\sum\limits_{i=1}^{n}(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)}{\sum\limits_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)} \right).$<\|endoftext\|>	4.78125
650	# Method of Undetermined Coefficients with complex root I'm trying to solve the following ODE and am stuck at the end. The question is: $$y''+4y=2\sin(2x)+x^2+1$$ I first solve the homogeneous part. Writing the characteristic equation: \begin{align} r^2 + 4 = 0 \implies r=\pm2i \end{align} So the complementary solution is $$Y_c=c_1\cos(2x)+c_2\sin(2x)$$ Next, I guess a particular solution of the form: $$Y_p(x)=2A\sin(2x)+2B\cos(2x)+Cx^2+Dx+E.$$ Computing its first and second derivatives yields: \begin{align} Y'_p(x) & =4A\cos(2x)-4B\sin(2x)+2Cx+D \\ Y''_p(x) & =-8A\sin(2x)-8B\cos(2x)+2C. \end{align} Substituting these into the ODE gives: $$-8A\sin(2x)-8B\cos(2x)+2C+2A\sin(2x)+2B\cos(2x)+Cx^2+Dx+E$$ $$= 2C+Cx^2+Dx+E =2\sin(2x)+x^2+1$$ I know $C=1$ and $E=1$ but then I'm unsure. Any help would be really appreciated • Since the inhomogeneous term contains $\sin(2x)$ which is part of the complementary solution, you should guess $Ax\sin(2x) + Bx\cos(2x) + Cx^2 + Dx + E$ for $Y_p(x)$ instead. – Chee Han Apr 26 '17 at 12:51 $$Y_p(x)= \color{red}{2A\sin(2x)+2B\cos(2x)}+Cx^2+Dx+E$$ The red part in your $Y_p$ above can't work because that's already a part of the solution to the homogeneous part $Y_c$ (so that will simplify to $0$...!): $$Y_c=c_1\cos(2x)+c_2\sin(2x)$$ The trick is to multiply by $x$, so take: $$Y_p(x)= \color{blue}{A\,x\sin(2x)+B\,x\cos(2x)}+Cx^2+Dx+E$$ Note that you can omit the factors $2$ since you still have the undetermined coefficients $A$ and $B$. Hoover over to see what you should get: $$\displaystyle Y_p(x)= -\frac{1}{2}\,x\cos(2x)+\frac{x^2}{4}+\frac{1}{8}$$<\|endoftext\|>	4.40625
755	# Roulette Wheel Selection Note: this page has been created with the use of AI. Please take caution, and note that the content of this page does not necessarily reflect the opinion of Cratecode. Roulette wheel selection is a popular technique in genetic algorithms to randomly choose parents for reproduction based on their fitness scores. It's like spinning a roulette wheel where each candidate in a population has a slice proportional to its fitness, and the wheel stops at a random position, selecting the parent within that slice. This method favors individuals with higher fitness scores, increasing the chances of better offspring. ## How Roulette Wheel Selection Works Imagine a roulette wheel with sectors assigned to each individual in the population based on their fitness scores. The probability of being chosen is proportional to the size of the individual's sector. Here's a step-by-step guide: 1. Calculate the total fitness of the population. 2. Calculate the relative fitness for each individual by dividing their fitness by the total fitness. 3. Calculate the cumulative probability for each individual by summing up the relative fitnesses. 4. Generate a random number between 0 and 1. 5. Select the first individual whose cumulative probability is greater than or equal to the random number. Let's go through an example to understand this better. ### Example Suppose we have a population of four individuals with the following fitness scores: ``````Individual A: 12 Individual B: 8 Individual C: 6 Individual D: 4`````` 1. Calculate the total fitness: ``Total Fitness = 12 + 8 + 6 + 4 = 30`` 1. Calculate the relative fitness: ``````Individual A: 12 / 30 = 0.4 Individual B: 8 / 30 = 0.2667 Individual C: 6 / 30 = 0.2 Individual D: 4 / 30 = 0.1333`````` 1. Calculate the cumulative probability: ``````Individual A: 0.4 Individual B: 0.4 + 0.2667 = 0.6667 Individual C: 0.6667 + 0.2 = 0.8667 Individual D: 0.8667 + 0.1333 = 1`````` 1. Generate a random number between 0 and 1: ``Random number = 0.52`` 1. Select the parent: In this case, the random number falls between the cumulative probabilities of Individual A (0.4) and Individual B (0.6667). So, Individual B is selected as a parent. Repeat the process to select another parent, and then perform crossover and mutation to create offspring. ## Implementation Here's a simple implementation of roulette wheel selection in Python: ``````import random def roulette_wheel_selection(population, fitness_scores): total_fitness = sum(fitness_scores) relative_fitness = [f / total_fitness for f in fitness_scores] cumulative_probability = [sum(relative_fitness[:i+1]) for i in range(len(relative_fitness))] rand = random.random() for i, cp in enumerate(cumulative_probability): if rand <= cp: return population[i]`````` Now you know how roulette wheel selection works and how to implement it in genetic algorithms. By using this technique, your algorithm will favor individuals with higher fitness scores, increasing the probability of producing better offspring and improving your solution over time. Hey there! Want to learn more? Cratecode is an online learning platform that lets you forge your own path. Click here to check out a lesson: What Programming Means (psst, it's free!).<\|endoftext\|>	4.8125
941	# Show that ${n\choose r}={{n-1}\choose{r-1}}+{{n-1}\choose {r}}$. [duplicate] Show that ${n\choose r}={{n-1}\choose{r-1}}+{{n-1}\choose {r}}$. My try: $$\dfrac{n!}{(n-r)!\;r!}=\dfrac{(n-1)!}{(n-r)!(r-1)!}+\dfrac{(n-1)!}{(n-1-r)!\;r!}$$ Multiplying all terms by $r!(n-r)!$ $$n!=r(n-1)!+(n-1)!(n-r)$$ Dividing everything by $(n-1)!$ $$n=r+(n-r)$$ $$n=n$$ Is this correct? • It looks like you are actually multiplying by $r!(n - r)!$. Commented Mar 21, 2017 at 1:20 • The combinatorial proof is often seen as more elegant and more desirable. Any way of choosing $r$ people from $n$ total people where one of the persons is special is either choosing the special person and $r-1$ other people from the $n-1$ non-special people or not choosing the special person while choosing $r$ people from the $n-1$ non-special people. Commented Mar 21, 2017 at 1:30 Other than the typo, I don't see a problem with it. Because you are dividing, I think it would help if you stated that $a! \neq 0.$ You have a typo on the line "multiplying all terms by...", $(r−1)!$ should be $r!$. Looks good. You have a typo on the line "multiplying all terms by...", $(r-1)!$ should be $r!$. Ideally you would write this proof backwards, starting with $n=n$ and finishing with what you are trying to prove. Combinatorial Proof: Let $A= \big\{1,2, \ldots,n\big\}$. there are $\binom{n}{r}$ ways to form r-combinations $S$ of $A$. we can count the number of r-combinations $S$ of $A$ in a different way. Every r-combination $S$ of $A$ either contains the element $1$ or not. if $1 \in S$, the number of ways to form S is $\binom{n-1}{r-1}$. if $1\notin S$, the number of ways to form $S$ is $\binom{n-1}{r}$. thus we have: $$\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r}$$ source:Principles and Techniques in Combinatorics $$\dfrac{n!}{(n-r)!\;r!}=\dfrac{(n-1)!}{(n-r)!(r-1)!}+\dfrac{(n-1)!}{(n-1-r)!\;r!}$$ Multiplying all terms by $r!(n-r)!$ (rather than $(r-1)!(n-r)!$) $$n!=r(n-1)!+(n-1)!(n-r) = r(n-1)!+(n-1)!n - r(n-1)! = n!$$ But it's usually better to start with one side and ending up at the other side. So: $${{n-1}\choose{r-1}}+{{n-1}\choose {r}} =$$ $$\dfrac{(n-1)!}{(n-r)!(r-1)!}+\dfrac{(n-1)!}{(n-1-r)!\;r!} =$$ $$\dfrac{r(n-1)!}{(n-r)!r!}+\dfrac{(n-r)(n-1)!}{(n-r)!\;r!} =$$ $$\dfrac{r(n-1)!+n(n-1)!-r(n-1)!}{(n-r)!\;r!} =$$ $$\frac{n!}{(n-r)!r!} =$$ $${{n}\choose {r}}$$<\|endoftext\|>	4.53125
221	The activity requires students to reflect on their understanding of the science ideas represented in the diagram and consider how the representation supports them to understand these ideas through its use of scientific conventions and terms. By the end of this activity, students should be able to: - identify the key science ideas in a scientific diagram - describe how the diagram supports the science ideas through its use of scientific conventions and terms - use information from the scientific diagram to identify the key science ideas - write a scientific explanation of how a heat pump works using the key science ideas represented in the diagram. Download the Word file (see link below) for: - introduction/background notes - what you need - what to do - extension idea - student handout. Nature of Science Scientists use a range of conventions to represent scientific ideas, and students need to be conversant with these to make meaning of this information. As part of the development of student ability to communicate in science, teaching and learning must expose students to a range of scientific representations. Find out more about the science capabilities.<\|endoftext\|>	4.1875
2,009	Mean, Median, and Mode MEAN, MEDIAN, and MODE by Dr. Carol JVF Burns (website creator) Follow along with the highlighted text while you listen! Statistics is the discipline devoted to organizing, summarizing, and drawing conclusions from data. Given a collection of data, it is often convenient to come up with a single number that somehow describes its center; a number that in some way is representative of the entire collection. Such a number is called a measure of central tendency. The two most popular measures of central tendency are the mean and the median. Another measure sometimes used to describe a ‘typical’ data value is the mode. the MEAN of a data set The mean (or average) is already familiar to you: add up the numbers, and divide by how many there are: DEFINITION mean, average The mean (or average) of the $\,n\,$ data values $$\,\cssId{s14}{x_1, x_2, x_3, \ldots, x_n}\,$$ is denoted by $\,\bar{x}\,$ (read as ‘$\,x\,$ bar’) and is given by the formula \begin{alignat}{2} \cssId{s19}{\bar{x}}\ &\cssId{s20}{= \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n}}\\ &\cssId{s21}{= \frac{\sum_{i=1}^n\ x_i}{n}}\ \ &&\cssId{s22}{\text{(using summation notation)}}\\ &\cssId{s23}{= \frac1n\sum_{i=1}^n\ x_i}\ \ &&\cssId{s24}{\text{(an alternate version of summation notation)}}\\ \end{alignat} Thus, to find the mean of $\,n\,$ data values, you add them up and then divide by $\,n\,$. Similarly, the mean of the $\,n\,$ data values $\,y_1, y_2, y_3, \ldots, y_n\,$ would be denoted by $\,\bar{y}\,$ and read as ‘$\,y\,$ bar’. Since dividing by $\,n\,$ is the same as multiplying by $\,\frac{1}{n}\,$, the notation $\displaystyle\,\frac{\sum_{i=1}^n\ x_i}{n}\,$ is more commonly written as $\,\frac 1n\sum_{i=1}^n\ x_i\,$ or $\displaystyle\,\frac 1n\sum_{i=1}^n\ x_i\,$ . EXAMPLE: Find the mean of these data values: $2,\ -1,\ 2,\ 3,\ 0,\ 25,\ -1,\ 2$ There are $\,8\,$ data values. The mean is found by adding them up and then dividing by $\,8\,$: $$\cssId{s42}{\frac{2+(-1)+2+3+0+25+(-1)+2}{8}} \cssId{s43}{= \frac{32}{8}} \cssId{s44}{= 4}$$ As discussed in Average of Three Signed Numbers, the mean gives the balancing point for the distribution, in the following sense: if eight pebbles of equal weight are placed on a ‘number line see-saw’: two pebbles at $\,-1\,$, one pebble at $\,0\,$, three pebbles at $\,2\,$, one pebble at $\,3\,$, and one pebble at $\,25\,$; then the support would have to be placed at $\,4\,$ for the see-saw to balance perfectly! Notice in the previous example that the number $\,25\,$ seems to be unusually large, compared to the other numbers. An outlier is an unusually large or small observation in a data set. A drawback of the mean is that its value can be greatly affected by the presence of even a single outlier. If the outlier $\,25\,$ is changed to $\,250\,$, then the new mean would be $\,32.125\,$, which does not seem at all representative of a ‘typical’ number in this data set! the MEDIAN of a data set The median, on the other hand, is quite insensitive to outliers. Just as the median strip of a highway goes right down the middle, the median of a set of numbers goes right through the middle of the ordered list. Of course, only lists with an odd number of values have a true middle: the middle number in the ordered list $\,5,\ 7,\ 20\,$ is $\,7\,$. See how the definition below solves the problem when there are an even number of data values: DEFINITION median To find the median of a set of $\,n\,$ data values, first order the observations from least to greatest (or greatest to least). If $\,n\,$ is odd, then the median is the number in the exact middle of the list. That is, the median is the data value in position $\,\frac{n+1}{2}\,$ of the ordered list. If $\,n\,$ is even, then the median is the average of the two middle members of the ordered list. That is, the median is the average of the data values in positions $\,\frac{n}{2}\,$ and $\,\frac{n}{2}+1\,$ of the ordered list. EXAMPLE: Question: Find the median of these data values: $\,2,\ -1,\ 2,\ 3,\ 0,\ 25,\ -1,\ 2$ (This is the same data set as in the previous example.) Solution: Begin by ordering the eight data values from least to greatest: $$\cssId{s84}{\underset{\text{position 1}}{\underset{\uparrow}{-1,\strut}}}\ \ \ \ \cssId{s85}{\underset{\text{position 2}}{\underset{\uparrow}{-1,\strut}}}\ \ \ \ \cssId{s86}{\underset{\text{position 3}}{\underset{\uparrow}{0,\strut}}}\ \ \ \ \cssId{s87}{\overset{\text{the two ‘middle’ members}} {\ \ \overbrace{ \underset{\text{position 4}}{\underset{\uparrow}{2,\strut}}\ \ \ \ \underset{\text{position 5}}{\underset{\uparrow}{2,\strut}} }}}\ \ \ \ \cssId{s88}{\underset{\text{position 6}}{\underset{\uparrow}{2,\strut}}}\ \ \ \ \cssId{s89}{\underset{\text{position 7}}{\underset{\uparrow}{3,\strut}}}\ \ \ \ \cssId{s90}{\underset{\text{position 8}}{\underset{\uparrow}{25\strut}}}$$ There are an even number of values, so we average the values in positions four and five: the median is $\,\frac{2+2}{2} = 2\,$. Note that, for this data set, the median seems to do a better job than the mean in representing a ‘typical’ member. Note also that if the outlier $\,25\,$ is changed to $\,250\,$, it doesn't affect the median at all! the MODE of a data set Finally, a mode is a value that occurs ‘most often’ in a data set. Whereas a data set has exactly one mean and median, it can have one or more modes. For example, consider these data values: $\,2,\ -1,\ 2,\ 3,\ 0,\ 25,\ -1,\ 2\,$ Re-group them occurring to their frequency: \begin{align} \cssId{s102}{2,\ \ 2,\ \ 2,}\ \ \ \ &\cssId{s103}{\text{three occurrences of the number 2}}\\ \cssId{s104}{-1,\ \ -1,}\ \ \ \ &\cssId{s105}{\text{two occurrences of the number -1}}\\ \cssId{s106}{0,}\ \ \ \ &\cssId{s107}{\text{one occurrence of the number 0}}\\ \cssId{s108}{3,}\ \ \ \ &\cssId{s109}{\text{one occurrence of the number 3}}\\ \cssId{s110}{25}\ \ \ \ &\cssId{s111}{\text{one occurrence of the number 25}} \end{align} The mode of this data set is $\,2\,$, since this data value occurs three times, and this is the most occurrences of any data value. Every member of the data set $\,3,\ 7,\ 9\,$ is a mode, since each value occurs only once. The data set $\,3,\ 3,\ 7,\ 7,\ 9\,$ has two modes: $\,3\,$ and $\,7\,$. Each of these numbers occurs twice, and no number occurs more than two times. Master the ideas from this section<\|endoftext\|>	4.75
858	On Saturn, it may be a very long wait for the calm after a storm. As big and destructive as hurricanes on Earth can be, at least they don't last long. Not like those on Saturn, where storms may rage for months or years. Viewed from space, hurricanes on Earth and the huge atmospheric disturbances observed on Saturn look similar. But their differences are greater and offer intriguing insights into the inner workings of the ringed world being investigated by scientists on NASA's Cassini mission. Earth's hurricanes and Saturn's storms each have swirling clouds, convection, rain and strong rotating winds. "Hurricanes on Earth are low pressure centers at the ground and high pressures at the top where the storms flatten out," says Dr. Andrew Ingersoll, member of the Cassini imaging team and professor of planetary science at the California Institute of Technology in Pasadena, Calif. "Storms on Saturn could be like hurricanes if what we're seeing is the top of the clouds." The frequency of storms on Saturn seems to be about the same as on Earth, and the fraction of planet occupied by storms is also similar. Not surprisingly, since Saturn is so much larger than Earth -- nine Earths would fit across its equator -- its storms are bigger. Hurricane Katrina stretched more than 380 kilometers (240 miles) across, for example, while two storms the Cassini spacecraft spotted in February 2002 each extend more than 1,000 kilometers (620 miles) in diameter, about the size of Texas or France. On Earth, hurricane winds can exceed 240 kilometers per hour (150 miles per hour), similar to the speed of the jet stream, just about the fastest wind on the planet. Though spinning furiously, hurricanes travel along at a much slower pace -- eight to 32 kilometers per hour (five to 20 miles per hour). Saturn is different because its jet stream is much stronger. "Saturn's a very windy place," says Ingersoll. "The jet stream on Saturn blows ten times faster than on Earth, up to a thousand miles per hour." Saturn's winds are like conveyor belts between which storms appear to roll like ball bearings, he explains. "While we don't know the wind speeds within the storms, a good guess is that they are slower than the winds in the jet stream." What most distinguish storms on Saturn from those on Earth are the forces that drive them and physical differences between the two planets. The heat that drives hurricanes on Earth comes from the oceans, vast reservoirs of solar energy. The oceans are also the source of moisture for convection, which draws energy from the ocean into the atmosphere and creates storm clouds and driving rainfall. Hurricanes quickly fade once they make landfall, once the plug is pulled on their power source. The fuel for Saturn's storms is quite different. The interior of the planet acts like an ocean and stores energy, but the energy does not come from the sun. "Saturn makes it own heat, which it got when the pieces that made the planet crashed together during the violent history of the early solar system," says Ingersoll. Saturn's atmosphere has all the ingredients necessary for hurricane-like storms including heat and water vapor, he continues, so there's no need for that first step in hurricane development where the ocean evaporates. And, without a solid surface like Earth's ocean, Saturn's storms behave very differently. "You'd think that when two storms merge, for example, that you'd get a bigger storm," says Ingersoll, "but they seem to stay the same size. They can also split apart. They may go on forever, merging and splitting." Scientists will be able to study Saturn's storms more closely next year, when the Cassini spacecraft tours a region in the southern hemisphere mission scientists that call storm alley. With the exception of a few storms, like the dramatic Dragon Storm observed by the Cassini spacecraft last year, most of Saturn's storms are unnamed, unlike those on Earth. That may change, says Ingersoll, when scientists get to know them better. Written by Rosemary Sullivant Media Contact: Carolina Martinez (818) 354-9382 Jet Propulsion Laboratory, Pasadena, Calif.<\|endoftext\|>	3.890625
697	Home readers, sight words, NAPLAN….while these words are just an ordinary part of the schooling experience for some, for others, they bring huge stress and angst. Often difficulties with these experiences arise from a breakdown in reading. Despite consistent efforts with daily homework, some children continue to be slow, effortful, frustrated readers. A question we often receive from our families, is: “why is it so hard for us to grasp?”. Unfortunately, there isn’t a straightforward answer to this question, because reading is such a complex process. To be a successful reader a child (or adult) has to perform two tasks. Firstly – they must accurately decode (or sound out, or recognise) the words on the page. Secondly – they must be able to extract and hold onto the information on the page (that is, they must comprehend what they have read). Children may have difficulty with either or both of these tasks. Many different processes are involved in reading. As a speech pathologist working in literacy, it is our role to examine each of these areas and to consider how they may be contributing towards the reading difficulty a child may have. Because of this specialist role, speech pathologists are able to help with reading and literacy in ways that other health and educational professionals may not. An assessment should specifically consider: - Phonological Awareness: the ability to access and manipulate the individual sound units that make up syllables and words - Phonological Memory: a type of working memory which stores sounds as a reader reads a word. E.g., to sound out ‘cat’ as ‘c-a-t’ a reader has to be able to hold onto each sound that they have read out – phonological memory makes this possible. - Working Memory: short-term memory system which allows for information to be held onto and manipulated at the same time - Rapid Naming: the ability to extract information from the long-term memory system – critical for reading fluency. A break down in any of these areas will lead to difficulties with reading at the functional level. Research also indicates that children who have difficulties with both rapid naming and phonological awareness will have greater difficulties than a child who had a breakdown in only one domain. And the functional reading abilities of: - Reading Rate: how quickly and smoothly a child is able to read across the page - Reading Comprehension: the ability to hold onto and understand information that is encoded into words that have been read - Reading Accuracy: how accurately a child can read or decode the words on the page. When we consider that a child has to not only read the words, but understand their meanings and how all of the words on the page interact together to contribute toward the main idea, we quickly realise why oral language abilities are critical for reading success. So many children that we work with present to us in their early years for language development, and then go on to have reading difficulties because of literacy and language’s close relationship. When we consider how fast-paced the curriculum is, and just how complex reading is, it becomes clear how critical early intervention is for our kids. While any person may become frustrated when learning a new skill, if your child is experiencing ongoing frustrations, or is just not picking up on their reading as you feel they should, contact us today to see how we can help!<\|endoftext\|>	3.859375
538	New New Year 5 # Use efficient division strategies to solve problems I can use a range of strategies to solve division problems New New Year 5 # Use efficient division strategies to solve problems I can use a range of strategies to solve division problems Share activities with pupils Share function coming soon... ## Lesson details ### Key learning points 1. Look at the dividend and divisor to decide which strategy is most effective and explain why. 2. Use partitioning if the dividend can be partitioned into multiples of the divisor. 3. Use short division if the dividend cannot be partitioned into multiples of the divisor. ### Common misconception Pupils may always choose short division to solve division equations. Encourage pupils to think about other strategies that they know. Ask questions such as 'Is this the most efficient method?' ### Keywords • Short division - Is a formal method of division often used when dividing any number by a one digit number. • Partition - Partition means splitting a number into smaller parts. A quick recap of strategies for division at the beginning of the lesson will be helpful. It is important that pupils understand that there may not always be one best approach. Encourage debate as to why one method may be more efficient than another. Teacher tip ### Licence This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2). ## Starter quiz ### 6 Questions Q1. Identify the multiples of 4. 25 Q2. 4 hundreds can be regrouped for tens. 400 4 Q3. Your divisor is 4, which numbers will result in regrouping? A Q4. Choose the correct short division for this equation: 633 ÷ 8. Correct Answer: An image in a quiz Q5. If the partial quotient written above the 100s digit is a 0 then the divisor is __________ than the 100s digit in the dividend. less than Q6. Look at the short division, what is the missing number? Write your answer in numeral form. ## Exit quiz ### 6 Questions Q1. Identify the number which is not a multiple of 11. 11 22 33 132 Q2. Look at the diagram, choose the correct division equation which the image could represent. 1 ÷ 2 2 ÷ 1 4 ÷ 2 Q3. Look at the diagram, choose the correct division equation. 8 ÷ 0<\|endoftext\|>	4.65625
535	Folate , All You Need to Know Folate (folic acid) is a type of vitamin B that aids in the development and repairing of DNA. It equally helps in the production of red blood cells (RBCs) in the body. Folic acid is not stored in our fat cells unlike vitamin B12, because of its solubility in water. In essence, the body needs folic acid always since no reserve of it stored. During urinary excretion, water-soluble vitamins are released, making it an impossible mission for vitamins like folic acid to remain in the body for long. Folic acid deficiency or insufficiency can lead to anaemia ( i.e megaloblatic anaemia), showing symptoms like; pale skin, diarrhea, breathlessness, tiredness, emotional changes and heart palpitations. Folate deficiency can equally result to child defect during pregnancy. Subtle symptoms of folate insufficiency include; growth problems, grey hair, mouth sores, fatigue and tongue swelling. Folate deficiency is caused by poor intake or malabsorption of folate in the body. Folic acid insufficiency can occur for a number of reasons, including: - Excessive intake of alcohol, which reduces the ability of the body to make use of folate. - Blood conditions leading to the damage of red blood cells in the body. - Digestive disorders resulting to inflammatory diseases, hence reducing the ability of the body to absorb folate. - The use of certain medicines that can interfere with the level of folate in the body. - Extra amounts of folate are normally needed during pregnancy. Treatment involves increasing the intake of folate in our diets. To avoid folic acid deficiency, nutritious diets should be eaten, and such foods include; brussel sprouts, peas, spinach, broccoli, shellfish, poultry, fortified cereals, eggs, tomato juice, mushrooms, beans and legumes, citrus, fruits, kidney and liver meat, asparagus and wheat bran. It is recommended that women who may become pregnant should take folate supplement, because of its importance for fetal growth. People who take drugs that cause folate deficiency, should equally take folate supplements. Seeing your doctor before any action is usually the best! The information and reference materials contained here are intended solely for the general information of the reader. It is not to be used for treatment purposes, but rather for discussion with the patient’s own physician. The information presented here is not intended to diagnose health problems or to take the place of professional medical care.<\|endoftext\|>	3.6875
1,432	THEORETICAL PROBABILITY OF COMPOUND EVENTS About "Theoretical probability of compound events" Theoretical probability of compound events : Recall that a compound event consists of two or more simple events. To find the probability of a compound event, We have to write a ratio of the number of ways the compound event can happen to the total number of equally likely possible outcomes. The formula given below can be used to find the theoretical probability. Theoretical probability of compound events - Examples Example 1 : Jacob rolls two fair number cubes. Find the probability that the sum of the numbers he rolls is 8. Solution : Step 1 : List out all the possible outcomes when two cubes are rolled. There are 36 possible outcomes in the sample space. Step 2 : Create a table where each cell represents the sum on two number cubes. Then, circle the outcomes that give the sum of 8. Step 3 : Find the number of outcomes in which the sum is 8. Number of outcomes in which the sum is 8 = 5 Step 4 : Find the required probability. P (for sum 8) = 5 / 36 Hence, the probability that the sum of the numbers is 8 is 5/36. Example 2 : A deli prepares sandwiches with one type of bread (white or wheat), one type of meat (ham, turkey, or chicken), and one type of cheese (cheddar or Swiss). Each combination is equally likely. Find the probability of choosing a sandwich at random and getting turkey and Swiss on wheat bread. Solution : Step 1 : Make a tree diagram to find the sample space for the compound event. Step 2 : Find the number of possible outcomes in the sample space. From the above tree diagram, number of possible outcomes in the sample space is 12. Step 3 : Find the number of possible outcomes for turkey and Swiss on wheat bread. From the above tree diagram, number of possible outcomes for turkey and Swiss on wheat bread is 1. Step 3 : Find the probability of choosing turkey and Swiss on wheat bread at random. Probability = 1/12 Example 3 : The combination for Khiem’s locker is a 3-digit code that uses the numbers 1, 2, and 3. Any of these numbers may be repeated. Find the probability that Khiem’s randomly-assigned number is 222. Solution : Step 1 : List all the codes that start with 1 and have 1 as a second digit. (1, 1, 1) (1, 1, 2) (1, 1, 3) Step 2 : List all the codes that start with 1 and have 2 as a second digit. (1, 2, 1) (1, 2, 2) (1, 2, 3) Step 3 : List all the codes that start with 1 and have 3 as a second digit. (1, 3, 1) (1, 3, 2) (1, 3, 3) Step 4 : Codes start with 2(2, 1, 1)(2, 1, 2)(2, 1, 3)(2, 2, 1)(2, 2, 2)(2, 2, 3)(2, 3, 1)(2, 3, 2)(2, 3, 3) Codes start with 3(3, 1, 1)(3, 1, 2)(3, 1, 3)(3, 2, 1)(3, 2, 2)(3, 2, 3)(3, 3, 1)(3, 3, 2)(3, 3, 3) Step 5 : Find the number of outcomes in the sample space by counting all the possible codes. There are 27 such codes. Step 6 : Find the probability that Khiem’s locker code is 222. Number of outcomes for Khiem’s locker code 222 = 1 Probability (Code 222) = 1/27 After having gone through the stuff given above, we hope that the students would have understood "Theoretical probability of compound events". Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. WORD PROBLEMS HCF and LCM word problems Word problems on simple equations Word problems on linear equations Algebra word problems Word problems on trains Area and perimeter word problems Word problems on direct variation and inverse variation Word problems on unit price Word problems on unit rate Word problems on comparing rates Converting customary units word problems Converting metric units word problems Word problems on simple interest Word problems on compound interest Word problems on types of angles Complementary and supplementary angles word problems Double facts word problems Trigonometry word problems Percentage word problems Profit and loss word problems Markup and markdown word problems Decimal word problems Word problems on fractions Word problems on mixed fractrions One step equation word problems Linear inequalities word problems Ratio and proportion word problems Time and work word problems Word problems on sets and venn diagrams Word problems on ages Pythagorean theorem word problems Percent of a number word problems Word problems on constant speed Word problems on average speed Word problems on sum of the angles of a triangle is 180 degree OTHER TOPICS Profit and loss shortcuts Percentage shortcuts Times table shortcuts Time, speed and distance shortcuts Ratio and proportion shortcuts Domain and range of rational functions Domain and range of rational functions with holes Graphing rational functions Graphing rational functions with holes Converting repeating decimals in to fractions Decimal representation of rational numbers Finding square root using long division L.C.M method to solve time and work problems Translating the word problems in to algebraic expressions Remainder when 2 power 256 is divided by 17 Remainder when 17 power 23 is divided by 16 Sum of all three digit numbers divisible by 6 Sum of all three digit numbers divisible by 7 Sum of all three digit numbers divisible by 8 Sum of all three digit numbers formed using 1, 3, 4 Sum of all three four digit numbers formed with non zero digits Sum of all three four digit numbers formed using 0, 1, 2, 3 Sum of all three four digit numbers formed using 1, 2, 5, 6<\|endoftext\|>	4.75
1,122	Arc Length of Curves in Three-Dimensional Space Examples 1 # Arc Length of Curves in Three-Dimensional Space Examples 1 Recall from the Arc Length of Curves in Three-Dimensional Space that the arc length of a curve given by the vector equation $\vec{r}(t) = (x(t), y(t), z(t))$ that traces out a continuous and bounded curve $C$ once for $a ≤ t ≤ b$ can be calculated with the following formula: (1) \begin{align} \mathrm{arc \: length} = \int_a^b \sqrt{ \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 + \left ( \frac{dz}{dt} \right )^2} \: dt \end{align} We will now look at some examples of computing arc lengths of curves. For examples can be found on the Arc Length of Curves in Three-Dimensional Space Examples 2 page. ## Example 1 Find the length of the curve defined by the vector-valued function $\vec{r}(t) = \sqrt{2}t \vec{i} + e^t \vec{j} + e^{-t} \vec{k}$ for $0 ≤ t ≤ 1$. We can rewrite our vector-valued function as $\vec{r}(t) = (\sqrt{2}t, e^t, e^{-t})$. When we differentiate each term, we get that $\frac{dx}{dt} = \sqrt{2}$, $\frac{dy}{dt} = e^t$, and $\frac{dz}{dt} = -e^{-t}$. Applying this to our formula for arc length above, and noticing that $(e^t + e^{-t})^2 = e^2t + 2e^{-t + t} + e^{-2t} = e^{2t} + 2 + e^{-2t}$ and we have that: (2) \begin{align} \quad \mathrm{arc \: length} = \int_0^1 \sqrt{ (\sqrt{2})^2 + (e^t)^2 + (-e^{-t})^2} \: dt = \int_0^1 \sqrt{2 + e^{2t} + e^{-2t}} \: dt = \int_0^1 \sqrt{(e^t + e^{-t})^2} \: dt = \int_0^1 e^t + e^{-t} \: dt = [ e^t - e^{-t} ] \biggr \rvert _0^1 = e - \frac{1}{e} \end{align} ## Example 2 Verify that the circumference of the unit circle is $2\pi$. We can describe an arbitrary unit circle by the vector-valued function $\vec{r}(t) = (\cos t, \sin t, 0)$ for $0 ≤ t ≤ 2\pi$. If we differentiate each component of this equation, we get that $\frac{dx}{dt} = -\sin t$, $\frac{dy}{dt} = \cos t$ and $\frac{dz}{dt} = 0$. Applying our formula above and we get that (3) \begin{align} \quad \mathrm{arc \: length} = \int_0^{2\pi} \sqrt{(-\sin t)^2 + (\cos t)^2} \: dt = \int_0^{2\pi} \sqrt{\sin ^2 t + \cos ^2 t} \: dt = \int_0^{2\pi} 1 \: dt = t \biggr \rvert_0^{2\pi} = 2\pi \end{align} The reader should check that the circumference of the unit circle is still $2\pi$ if instead we have $\pi ≤ t ≤ 3\pi$, or any other interval of $t$ of length $2 \pi$. ## Example 3 Find the length of the curve defined by the vector-valued function $\vec{r}(t) = (3 \cos t, t, 3 \sin t)$ for $-4 ≤ t ≤ 4$. If we differentiate component by component we get that $\frac{dx}{dt} = -3 \sin t$, $\frac{dy}{dt} = 1$ and $\frac{dz}{dt} = 3 \cos t$. Applying the formula above for arc length and we get that: (4) \begin{align} \quad \mathrm{arc \: length} = \int_{-4}^4 \sqrt{ \left ( -3 \sin t\right )^2 + \left ( 1\right )^2 + \left ( 3 \cos t\right )^2 } \: dt = \int_{-4}^{4} \sqrt{10} \: dt = \sqrt{10}t \biggr \rvert_{-4}^{4} = 4 \sqrt{10} + 4 \sqrt{10} = 8 \sqrt{10} \end{align}<\|endoftext\|>	4.53125
185	Endocarditis is an inflammation or infection of the inner surface of the heart (the endocardium). It occurs when bacteria enter the bloodstream and attach to a damaged portion of the endocardium or abnormal heart valves. But it may also develop on devices implanted in the heart, such as artificial heart valves, pacemakers, or implantable defibrillators. When normal blood components, like fibrin and platelets, stick to the bacteria, the resulting buildup is called a vegetation. Endocarditis can limit the heart’s ability to efficiently pump blood. Damaged infected valves can cause severe leaking (regurgitation) of blood back through the valve(s). In addition, small pieces of the vegetation can break off and travel through the blood vessels to other parts of the body and cause problems like stroke or tissue damage, when they lodge and block blood flow to the tissues downstream.<\|endoftext\|>	3.75
492	## Book: RS Aggarwal - Mathematics ### Chapter: 12. Circles #### Subject: Maths - Class 10th ##### Q. No. 49 of Multiple Choice Questions (MCQ) Listen NCERT Audio Books to boost your productivity and retention power by 2X. 49 ##### In the given figure, O is the center of two concentric circles of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If PA = 12 cm then PB is equal to In given Figure, OA AP [Tangent at any point on the circle is perpendicular to the radius through point of contact] In right - angled OAP, By Pythagoras Theorem [i.e. (hypotenuse)2 = (perpendicular)2 + (base)2] (OP)2 = (OA)2 + (PA)2 Given, PA = 12 cm and OA = radius of outer circle = 5 cm (OP)2 = (5)2 + (12)2 (OP)2 = 25 + 144 = 136 OP = 13 cm …[1] Also, OB BP [Tangent at any point on the circle is perpendicular to the radius through point of contact] In right - angled OBP, By Pythagoras Theorem [i.e. (hypotenuse)2 = (perpendicular)2 + (base)2 ] (OP)2 = (OB)2 + (PB)2 Now, OB = radius of inner circle = 3 cm And, from [2] (OP) = 13 cm (13)2 = (3)2 + (PB)2 (PB)2 = 169 - 9 = 160 PB = 4√10 cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55<\|endoftext\|>	4.5
732	Precalculus (6th Edition) Blitzer The simplified form of the equation is $\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$. Let us consider the expression $\frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$ And simplify: \begin{align} & \frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \left( \frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right)=\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \frac{5x-3}{{{x}^{2}}+1}+\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\ & =\frac{\left( {{x}^{2}}+1 \right)\left( 5x-3 \right)}{{{\left( {{x}^{2}}+1 \right)}^{2}}}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\ & =\frac{{{x}^{2}}\left( 5x-3 \right)+1\left( 5x-3 \right)+2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\ & =\frac{5{{x}^{3}}-3{{x}^{2}}+5x-3+2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \end{align} Therefore, $\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \left( \frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right)=\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$ Thus, the simplified form of the expression $\frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$ is $\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$.<\|endoftext\|>	4.59375
320	COMMON TEACHING MISTAKES Mistake 2: Give No Feedback to Students Feedback is critical to helping students learn. The following are important elements of effective feedback: Feedback should be timely - while certain circumstances may not allow immediate feedback, especially when patients are involved, feedback is best provided while events are "fresh" in people's minds. Feedback should be based on first-hand information - what have YOU actually witnessed in/from students? At times, you might also collect information from others for feedback, but it is important to verify the information first. Feedback should focus on behaviors that can be changed. Feedback should be specific. "You did a good job" does not help students understand what they are doing well, nor does it identify behaviors needing correction. Feedback should be based on students' decisions and actions, not upon assumptions or presumed motives. For example, instead of indicating that students "lack motivation," indicate what behaviors they exhibited that were less than ideal, e.g. did not provide timely follow up on "x" tasks. There are various approaches to providing feedback. The following link will take you to a more extensive discussion of feedback. However, one approach that is easy to remember is the "feedback sandwich." Begin by identifying something students did well, followed by an area where they can improve, followed by another positive comment. Another approach is to first ask students for a self-assessment. Based on the their response, you can identify additional positive and/or constructive comments.<\|endoftext\|>	4.0625
981	\|Project work is experiential and aims at long-term memory. It is part of the NHE Teaching Methodology, preferred methods for long term learning: - Service (local and global) - The Arts (music, dance, fine arts, crafts) - Visits and visitors coming into class and telling about their work - Literature Extensions Characteristics of project work are fluidity versus control, emergent versus prepared curriculum and questioning versus knowing the answers. Especially during project work the teacher views a the child as strong, powerful, rich in potential, as partner and driven by the power of wanting to learn, know and grow. Three types of projects - The school theme projects that all children will do in the course of one year. - The daily life projects spontaneously emerge during circle time or daily life at school. - The self-managed projects are set up for the children to do independently, alone or with a friend. They can be collective, in small groups or individual. Benefits of project work - Child’s mind is engaged in challenging work, making decisions, coordination, resolving conflicting views of contributors etc. - Project work increases children’s confidence in their own intellectual powers, and strengthens their dispositions to go on learning. - Can emerge from children’s ideas and/or interests - Can be provoked by teachers - Can be introduced by teachers knowing what is of interest to children: shadows, puddles, tall buildings, construction sites, nature, etc. - Should be long enough to develop over time, to discuss new ideas, to negotiate over, to induce conflicts, to revisit, to see progress, to see movement of ideas - Should be concrete, personal from real experiences, important to children, should be large enough for diversity of ideas and rich in interpretive / representational expression At the beginning of a project - Discussions emerge and ideas take shape. The students talk and are listened to. The children are encouraged to make their own decisions and choices, usually in cooperation with their peers, about the work to be undertaken. (Teachers take notes on what is said or tape record the conversation to transcribe later and add to the display of the documentation of the project) Role of the teacher - To co-explore the learning experience with the children - To provoke ideas, solve problems and resolve conflict - To take ideas from the children and return them for further exploration - To organize the classroom and materials together with children to be aesthetically pleasing - To document children’s progress: visual, videotape, tape recording, portfolios - To help children see the connections in learning and experiences - To help children express their knowledge through representational work - To have a dialogue about the projects with parents and other teachers - To foster the connection between home, school and community Role of teacher and child - Teacher has high expectations of child’s ability to represent their thoughts, feelings and observations with graphic skills such as drawing, painting, sculpting etc. - When adults communicate sincere and serious interest in the child’s ideas and in their attempts to express them, rich and complex work can follow, even among very young children - Children sense what adults/teachers think - Children observe how you value their conversation and work - Children sense what you care for - Teachers and children examine topics of interest to young children in great depth and detail - Teacher-child relation is focused on the project (not on routine or child’s performance and academic tasks) - Both are children - Teacher and child meet on matters of real interest to both of them - Teacher listens to child’s suggestions, questions, interacting with each other rather than over-assisting. - Keen observation will reveal a wealth of information about students, about their development and about your work as a teacher. - For children to reflect on their own work - For children to connect to and reflect on other children’s work - For adults to reflect on children’s work - For families to relate to the project study and explorations of their children - To document children’s growth over time - To develop a complex and detailed picture of the child in all developmental areas - To provide a resource for the wider community of educators to understand children’s learning better. - To share with the community at large what is happening inside the school. \|This wall collage was created by students at Vistara Primary school as part of a collective project on classroom rules. The students and teacher of each classroom come up with the rules and consequences together based on the underlying concepts of respecting each others learning and environment, etc.<\|endoftext\|>	3.96875
599	\|Autumn 1\|\|Autumn 2\|\|Spring 1\|\|Spring 2\|\|Summer 1\|\|Summer 2\| \|Year 1\|\|Name, locate, and identify the four countries and capitals of UK & surrounding seas.\| Identify places using maps, atlases, globes, aerial images and plan perspectives, make maps, devise basic symbols, Fieldwork, geographical vocabulary. \|Use aerial photographs and plan perspectives to recognise landmarks and basic human and physical features; devise a simple map; and use and construct basic symbols in a key\| Use simple fieldwork and observational skills to study the geography of their school and its grounds and the key human and physical features of its surrounding environment. \|Year 2\|\|Local Area and visit to the local park to look for different features.\|\|Name and locate the continents and oceans.\| Locate hot and cold areas of the world in relation to the North and South poles. \|Castles – land formation\| \|Year 3\|\|·\|\|Anglo-Saxon settlement\| \|Mapwork -locate the world’s countries,\| · – use the eight points of a compass, four and six-figure grid references, \|UK – using atlas, identifying rivers, mountains and counties.\| Local Area – mapwork Debate regarding land use and growth of Harborough. \|Year 4\|\|Prehistoric settlements\|\|Mountains and Volcanoes\| \|Year 5\|\|Mapwork – latitude, longitude, Equator, Northern Hemisphere, Southern Hemisphere, the Tropics of Cancer and Capricorn, Arctic and Antarctic Circle, the Prime/Greenwich Meridian and time zones (including day and night)\|\| \| Ge2/1.3a describe and understand key aspects of physical geography, including: climate zones, biomes and vegetation belts, rivers, mountains, volcanoes and earthquakes, and the water cycle \|Where did the Aztecs live?\| – Locating Mexico on a world map – Identifying the continents of North and South America – Exploring a map of Mexico to find Mexico City and what was Tenochtitlan \|Map Leicestershire – locate MH, Bosworth and Leicester – grid references\| Locating Mecca on map \|Year 6\|\|· World continents, oceans, etc. Outcome: Explorers’ world map\| · The Arctic – physical geography · The Antarctic – physical geography Use maps, atlases, globes and digital/computer mapping to locate countries and describe features studied \|Where in the world did Darwin travel? What did he find?\| \|Use maps, atlases and globes – to focus on South America\| Environment, key physical and human characteristics, countries and major cities Use four and six figure grid references, symbols and key Describe and understand key aspects of human and physical geography relating to South America<\|endoftext\|>	3.890625
241	Learn To Draw Learn all of the materials involved in sketching and drawing and also some basic shading techniques and sketching concepts. In this course, I start off going through all of the materials you're going to need for learning to sketch and draw. Once you have a better understanding of those, you can start drawing simple shapes and lines and practicing sketching with an HB, B, or 2B pencil. These are good pencils for sketching lightly but also allow you to go a little darker when needed for shading. I plan to cover perspective, values, edges, shading techniques, proportions, landscapes and many other topics of sketching and drawing for beginners and intermediate artists. • My Site: http://www.Brandon-Schaefer.com • Facebook: http://www.facebook.com/SchaeferArt • Instagram: http://www.instagram.com/SchaeferArt • Etsy: http://www.etsy.com/shop/SchaeferArt - Lectures 11 - Quizzes 0 - Duration 2 hours - Skill level All levels - Language English - Students 0 - Assessments Yes<\|endoftext\|>	3.703125
1,281	# Precalculus : Divide Polynomials by Binomials Using Synthetic Division ## Example Questions ### Example Question #1 : Divide Polynomials By Binomials Using Synthetic Division Divide the polynomial by . Explanation: Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient. We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients. To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder. with remainder This can be rewritten as Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial. ### Example Question #2 : Divide Polynomials By Binomials Using Synthetic Division Divide the polynomial by . Explanation: Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient. We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients. To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder. with remainder This can be rewritten as: Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial. ### Example Question #3 : Divide Polynomials By Binomials Using Synthetic Division Divide the polynomial by . Explanation: Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient. We mulitply what's below the line by 1 and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients. To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder. with remainder Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial. ### Example Question #4 : Divide Polynomials By Binomials Using Synthetic Division Divide the polynomial by . Explanation: Our first step is to list the coefficiens of the polynomials in descending order and carry down the first coefficient. We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficient. To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder. with reminder This can be rewritten as: ### Example Question #5 : Divide Polynomials By Binomials Using Synthetic Division Divide the polynomial by . Explanation: Our first step is to list the coefficients of the polynomials in descending order and carry down the first coefficient. Remember to place a when there isn't a coefficient given. We multiply what's below the line by and place the product on top of the line. We find the sum of this number with the next coefficient and place the sum below the line. We keep repeating these steps until we've reached the last coefficients. To write the answer, we use the numbers below the line as our new coefficients. The last number is our remainder. with remainder This can be rewritten as: Keep in mind: the highest degree of our new polynomial will always be one less than the degree of the original polynomial. ### Example Question #6 : Divide Polynomials By Binomials Using Synthetic Division Use synthetic division to divide by . Remainder Remainder Remainder Explanation: To divide synthetically, we begin by drawing a box. On the inside separated by spaces, we write the coefficients of the terms of our polynomial being divided. On the outside, we write the root that would satisfy our binomial , namely . Leaving a space for another row of numbers, we then draw a line below our row of coefficients. We then begin dividing by simply carrying our first coefficient (1) down below the line. We then multiply this 1 by our divisor (3) and write the resulting product (3) below our next coefficient. We then add the two numbers in that column and write the sum (5) below the line. We then simply continue the process by multiplying this 5 by our divisor 3 and writing that product in the next column, adding it to the next coefficient, and continuing until we finish the columns. We then need to translate our bottom row of numbers into the coefficients of our new quotient. Since the first column originally corresponded to our cubic term, it will now correspond to the quadratic term meaning that our 1 can be translated as . Similarly, our second column transitions from quadratic to linear, making our 5 become . Finally, our third column becomes the constant term, meaning 8 simply remains the constant 8. Finally, our former constant column becomes the column for our remainder. However, since we have a 0, we have no remainder and can disregard it. Putting all of this together gives us a final answer of ### Example Question #7 : Divide Polynomials By Binomials Using Synthetic Division Divide using synthetic division: Explanation: First, set up the synthetic division problem by lining up the coefficients. There are a couple of different strategies - for this one, we will put a -7 in the top corner and add the columns. _________________________ The first step is to bring down the first 1. Then multiply what is below the line by the -7 in the box, write it below the next coefficient, and then add the columns: _________________________ We can interpret this answer as meaning<\|endoftext\|>	4.71875
1,718	To explore how scientific research of one declining species can increase knowledge about the natural world at large. Known for her research with large marine mammals, Terrie Williams is considered one of the 50 most important women in science. Her research with marine mammals, particularly seals, and the recent research she did with the Hawaiian monk seal, KP2, is the focus of her book, The Odyssey of KP2: An Orphan Seal, a Marine Biologist, and the Fight to Save a Species. It is estimated that Hawaiian monk seals will be extinct in 50 years. In the book, Williams' research focuses on what factors were the likely cause of the monk seal's impending extinction. The book is one of the winners of the 2013 SB&F Prize for Excellence in Science Books. Science Books & Films (SB&F) is a project of the American Association for the Advancement of Science. Williams' story is “hands-on” science that shows how human and animal cultures play roles in the natural cycle of life. She poses for readers the question of the value and appropriateness of government intervention in helping preserve species for scientific research. Because the book focuses on the research of a few key personalities (animal and human), it provides students with an up-close look at scientists and how they go about their work, in this case with marine mammals. The presence of a female scientist opens up the possibility that girls reading The Odyssey of KP2 will be more likely to pursue science when they see it in the context of this story, especially since the narrative conveys not only Williams’ work, but also how she developed into a scientist and her passion for her work. In this lesson, students will read the book using the The Odyssey of KP2 student sheet to guide them in a question and answer format. Scientific explanations must meet certain criteria. For this to happen, scientists must be consistent with experimental and observational evidence about nature and must make accurate predictions when appropriate about systems being studied. Their explanations should be logical and respect the rules of evidence and be open to criticism. Scientists should also report their methods and procedures and make their knowledge public. Explanations on how the natural world changes based on myths, personal beliefs, religious values, mystical inspiration, superstition, or authority may be personally useful and socially relevant, but they are not scientific. In this book, Williams shares her experiences and frustrations trying to balance the science of her research with the local Hawaiians' personal beliefs, culture, and view of nature and how it fit in their way of life as well as their resistance to scientific authority. Because these issues can be politically charged, students might challenge some of the theories presented in this book. You should be prepared to engage students in exploring and debating these challenges. Ideas in this lesson are also related to concepts in these Common Core State Standards: - CCSS.ELA-Literacy.RST.11-12.1 Cite specific textual evidence to support analysis of science and technical texts, attending to important distinctions the author makes and to any gaps or inconsistencies in the account. - CCSS.ELA-Literacy.RST.11-12.2 Determine the central ideas or conclusions of a text; summarize complex concepts, processes, or information presented in a text by paraphrasing them in simpler but still accurate terms. You should read the book The Odyssey of KP2: An Orphan Seal, a Marine Biologist, and the Fight to Save a Species by Terrie M. Williams and review the questions students will answer on the The Odyssey of KP2 student sheet. The questions guide the students as they read the book. To introduce students to Williams and her work, they should use their Odyssey of KP2 student esheet to watch this Interview with Terrie Williams, which will introduce students to her work with KP2 and what she and her team learned about monk seals and their physiology. After viewing this interview, lead a discussion with the students about Williams' conclusions from her research. Begin by asking them: - How did KP2 come into Williams' life? - (Williams was in Antarctica when she received an email asking her if she wanted a Hawaiian monk seal. Since she had been wanting to study these animals, she jumped at the chance to be able to do research with KP2.) - What did Williams learn about the physiology of monk seals and how did these conclusions shift the research team's plans for KP2? - (She learned how much food he needed to eat and the temperatures at which he could comfortably live. She learned that monk seals are Hawaiian animals and that they are not going anywhere because the temperatures they live in are warm water temperatures. They cannot dive to deep, deep depths, like some other marine animals.) - What did Williams mean when she said KP2 was a "sign from the ocean"? - (The Hawaiians saw him as a sign from the ocean. They believed that he was going to tell the local kids and adults what was happening in the ocean and how to protect the animals of Hawaii.) - How is KP2 an ambassador for the species? - (He's teaching children about the biology of monk seals and about what monk seals need to live in the wild.) - What was it like to write the book for Williams? - (It was an adventure for her because she recorded what was happening as it was going on. She often didn't know what would happen to KP2. So, the book comes across as in the moment. It could often be heart wrenching.) - What does Williams think of her job? - (She says that she has the best job in the world. She is still working with monk seals. She's currently working with an adult male monk seal, KP18, who had to be removed from the wild because he was hunting other monk seals.) In this part of the lesson, students will learn more about Terrie Williams' research and her work with Hawaiian monk seals by reading independently The Odyssey of KP2 and answering the questions on The Odyssey of KP2 student sheet. These questions are divided into three main sections that follow the structure of the book. You may want to schedule a "check-in" after each part is read to discuss the questions. Students should read the first two parts of the book, Part 1: Destiny and Part II: Passages, and answer the questions on the student sheet. Before going on to Part III: Survival, however, students should use their esheet to watch the video, Terrie Williams Reads from The Odyssey of KP2. In this video, Williams reads from pp. 202-204 of the book. Williams says the laws that protect endangered species from human disturbance also keep scientists from working with these species. She states that researchers and the government have the same conservation goals regarding the ocean’s largest animals, but they go about accomplishing those goals differently. Researchers feel that bureaucrats are out of touch with the realities facing these ocean animals, and bureaucrats feel that researchers ignore the laws governing marine mammal protection. Ask students to write an essay about these statements and what they see as possible solutions to this impass. Consider developing a detailed rubric for assessment of the essay. There are several resources on the Internet that describe the use of rubrics in the K-12 classroom, a few of which are highlighted here. To learn more about rubrics in general, see Make Room for Rubrics on the Scholastic site. For specific examples of rubrics, more information, and links to other resources, check out the following sites: - Kathy Schrock’s Guide to Everything: Assessment and Rubrics - Assessment: Creating Rubrics - Rubrics for Web Lessons Finally, you can go to Teacher Rubric Makers on the Teach-nology.com website to create your own rubrics. At this site you can fill out forms to create rubrics suitable for your particular students, and then print them instantly from your computer. Tigerland and Other Unintended Destinations is a memoir of a scientist who studies and defends endangered species. Mammoth Extinction explores various hypotheses, particularly infectious disease, that caused the extinction of the wooly mammoth. In this TEDx Talk, Terrie Williams talks about her work with marine mammals and what led her to doing research in this area.<\|endoftext\|>	3.734375
853	# Operations of Complex Numbers in Trigonometric Form 1 teachers like this lesson Print Lesson ## Objective SWBAT multiply and divide complex numbers in trigonometric form. #### Big Idea Operations on complex numbers are easier when the complex number is in trigonometric form. ## Bell Work 5 minutes Yesterday students found the trigonometric form of complex numbers. Today students see how complex numbers in trigonometric form can make multiplying and dividing easier. I start by giving students 2 complex numbers to convert to trigonometric form To make the work quicker I divide. Each students receives a card with either a #1 or #2 on the card. Students work the problem and compare with someone who has the same number. This cooperative activity helps students learn to communicate with other students in their class. After students have compared with at least one person (3-4 minutes),l I pick 2 students to share the answers. ## Develop multiplication and division of complex numbers. 15 minutes After sharing the trigonometric form of the equations I continue using the groups developed in the bell work throughout the class. Using the standard complex form of the numbers, Group 1 finds the product of the numbers while group 2 finds the quotient by rationalizing the denominator. Once the product and quotient are found students put the results in trigonometric form. These results are also shared with the class. Once all the results are on the board, I am ready to have students to determine how to use the trigonometric form to find the product and quotient of complex numbers. I put all the trigonometric forms on the board. I now ask students to compare the 2 original vectors with the product. • Is there a way we can use the trigonometric forms to find the product? • Look at the coefficients what do you notice? • What about the angles what do you notice? Students see how the coefficient of the product is the product of the complex number coefficient but have a little trouble seeing the angle relationship. I guide the class by asking "Is there another angle that is at the same position as zero degrees? Let's rewrite the product with that angle? Do you see a relationship now?" Once we change the angle to 360 degrees students see how the product angle is the sum of the original angles. Using what you just noticed could a rule be developed for finding the product if the complex numbers are in trigonometric form? As a class the students determine a rule. I use the same strategy as above to find a rule for division. Students discuss ideas for 3-4 minutes and then share out ideas. Once we verify the rule we write it out on the board. ## Finding products and quotients 10 minutes It is now time to have students practice using the two rules they have found. Students work on a problem. When I do this problem I give students the product first. After finding the trigonometric form I have student convert to standard form. This is to remind students how this is done. To get the students thinking about properties I ask, "If we switch the order in the product would it change the result?" I have students justify their reasoning. During the reasoning I remind students that the trigonometric form is just a different way to write a real number. For real numbers order of the factors does not change the result. I now give the students the division part of the question. After finding the result I ask "If we switch the numerator and denominator will the result be the same?" Some students may have already stated that division will be different. I will ask "What will change in division?" The questioning at this point is helping students see the importance of order and how different representations of a number do not change the outcome of an operation. ## Closure 5 minutes As class comes to an end I give students the following problems as homework: From Larson's "Precalculus with Limits, 2nd ed." p. 476-477 #47, 50, 54, 60 These problems give students some basic practice in find the product and quotient of complex numbers in trigonometric form.<\|endoftext\|>	4.78125
394	CBSE Class 10 Sample Paper for 2019 Boards Class 10 Solutions of Sample Papers for Class 10 Boards Question 18 The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and intersecting the larger circle at P on producing. Find the length of AP Transcript Question 18 The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and intersecting the larger circle at P on producing. Find the length of AP Let’s first draw the figure We need to find length of AP Join OD Now, OD is the radius of smaller circle and BD is the tangent to the smaller circle So, OD ⊥ BD (Tangent is perpendicular to the radius) ∴ ∠ ODB = 90° Also, in bigger circle AB is the diameter, and point P is a point in the semi-circle of the bigger circle ∴ ∠ APB = 90° (Tangent is perpendicular to the radius) In Δ ABP and Δ OBD ∠ APB = ∠ ODB Both 90°) ∠ ABP = ∠ OBD (Common) Δ ABP ~ Δ OBD (AA Similarity) Now, sides of similar triangles are proportional 𝐴𝑃/𝑂𝐷 = 𝐴𝐵/𝑂𝐵 𝐴𝑃/8 = 26/13 (Since AB is diameter of bigger circle, AB = 2 × 13 = 26 cm) 𝐴𝑃/8 = 2 AP = 2 × 8 AP = 16 cm We need to find length of AP<\|endoftext\|>	4.5
1,690	High School Physics Numerical problems on Vertical motion Last updated on April 16th, 2021 at 10:06 am In this post, we will solve a few ‘harder’ numerical problems on Vertical motion. Vertical motion under the influence of gravity is one of the most interesting chapters of mechanics. We have a detailed tutorial on this. Now its turn to explore some tricky but challenging numerical problems from this chapter. I hope you will like the compilation. This is the first cut of this post. I’ll add new numericals on a regular basis. You can also share similar problems and solution to my email id (see contacts). I will add those too here if found appropriate. These problems are good to practice for the students going for different exams like JEE including WBJEE, NEET, AP Physics, etc Numericals on Vertical Motion 1) A ball is thrown vertically in the air at 120 m/s. After 3 seconds, another ball is thrown vertically. What velocity must the second ball have to pass the first ball at 100m from the ground? Solution: First, we’ll frame the height formulas for both the balls: The formula is height h = ut – (1/2)g t^2 If we take g as 9.8 m/s^2, the equation becomes: h = ut – 4.9 t^2 . (Here is the initial velocity) For the first ball: h1=120t − 4.9t^2 ……………….(1) For the second ball: Lets say it was thrown with a velocity v. As it got 3 sec less time compared to the first ball, so the equation for the second ball would be: h2=v(t−3) −4.9(t−3)^2 ……………..(2) Now set h1 equal to 100 and solve: 100= 120t − 4.9t^2 => −4.9t^2+120t−100=0 Solving the equation we get, t=0.863801, t=23.626 We know that the time at which the balls pass each other must be at least 3 because the second ball is not thrown until t=3. Therefore, the first solution is invalid, and thus t=23.626 sec See also Numerical Problems based on combined Gas law or Gas equation & ideal gas equation Putting the value of t into the second equation (also set it equal to 100 like before). 100=+v(23.626−3) −4.9(23.626−3)^2 100=+20.626v −4.9(20.626)^2 v=105.916 m/sec [ this is the velocity with which the second ball was thrown ] Analysis of Q1- numerical problems on Vertical motion I believe after solving or while solving a numerical problem using mathematical formulas, every student should spend a while to visualize the situation. Like the above problem, can you visualize when and where the 2 balls actually meet at 100 m height? The first ball is thrown vertically upwards with 120 m/s. You know that it is under a retardation of approximately 10 m/s^2 when it is moving upwards.(Gravity or earth’s gravitational force on the ball acting in the opposite direction when it is moving upwards. This causes this retardation). That means a reduction of velocity by 10 m/s every second. So obviously the ball can continue its upward movement for 120/10 = 12 sec. You can similarly find this time for the second ball as well using the value of its velocity which we found out above. As the 2 balls meet after 23 secs so obviously they were on their way downwards. Vertical motion tutorial numerical problems on Vertical motion with solution 2) A balloon which is ascending at the rate of 12m/s is 30.4 meters above the ground when a stone is dropped. What time will the stone take to reach the ground? g = 10 m/s^2 Solution: When the stone is dropped from the balloon, the balloon and the stone were having an upward motion with velocity 12 m/s. Due to Inertia of Motion, the stone will continue its upward motion until its velocity becomes zero under the influence of Gravity, which is pulling the stone downwards. So we will first find out 2 data, related to the upward motion of the stone. We know: During this upward motion, initial velocity = U = 12 m/s Origin or starting point of upward motion=the point where it is thrown which is 30.4 m above ground Findings: The time taken by the stone to reach the maximum height from the point of throw= T1 = U/g = 12/10 sec = 1.2 sec………….(1) and The height reached with respect to the point of throw =h= U2 / (2g) = 12×12/(2×10) = 7.2 m Now the stone will fall from the height = H=7.2 + 30.4 m = 37.6 m. During the fall, its initial velocity = 0 Say, Time to fall through this height = T2 Using the formula H = (1/2) g T22 T2 = √ [2H/g] = √ [(2 x 37.6)/10] = 2.74 sec…………(2) So total time taken by the stone to reach the ground= T1 + T2 = 1.2 + 2.74 sec = 3.94 sec (answer) Vertical Motion – numericals (for AP physics, JEE, NEET, WBJEE) 3) A parachutist descending at a constant rate of 2.0 m/s drops a smoke canister at a height of 300 m. Find the time for the smoke canister to reach the ground and its velocity when it strikes the ground. Then find the time for the parachutist to reach the ground, the position of the parachutist when the smoke canister strikes the ground, and an expression for the distance between the smoke canister and the parachutist. Solution: Say, the time for the smoke canister to reach the ground = t We will use the following equation to find out this t S = ut + (1/2) g t^2……………………(1) As the canister had a velocity 2 m/s when its thrown, so u = 2 m/s S = 300 m g = 9.8 m/s^2 So from eqn 1 we get, So, 300 = 2t + (1/2)9.8 t^2 or, 4.9 t^2 + 2t – 300 =0…………..(2) Solving the quadratic equation we get a positive value of t = 7.6 sec The time required for the smoke canister to reach the ground = 7.6 sec Say,Velocity of the canister when it reaches the ground V. so, V = u + g t = 2 + 9.8×7.6 = 76.48 m/s As the parachutist is coming down with a constant velocity, the time for the parachutist(say T) to reach the ground is found using this equation, S = vT 300 = 2.T T= 150 sec When the canister strikes the ground, the parachutist has dropped 2.0 x 7.6 m = 15.2 m and is 284.8 m above the ground. The expression for the distance between the canister and the parachutist is X= distance traveled by the canister – distance traveled by the parachutist or, X = [2t + 4.9t^2] – 2t X = 4.9 t^2 Freefall in physics Scroll to top error: physicsTeacher.in<\|endoftext\|>	4.59375
510	Home > Grade 1 > Subtracting Two-Digit Numbers (Elementary) # Subtracting Two-Digit Numbers (Elementary) Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to make the smallest (or largest) difference. ### Hint What does the number on the left represent? What does the number on the right represent? 98 – 12 is the largest difference. There are several answers for the smallest difference including 81-79. Source: Robert Kaplinsky ## Sum to 1,000 – Two Addends Directions: Arrange the digits 1-6 into two 3-digit whole numbers. Make the sum as close … 1. Wouldn’t 20-19 be the smalls difference? or am I missing a given requirement? Thanks! • “smallest” I meant. 🙂 Typo! • Sorry…I see what I did wrong! My mistake! • Hi Kim. I think it comes down to the reality that “small” and “large” are not mathematically precise words. I have experimented with using phrasing like “least difference” but that seemed to do more harm than good. So, I have stuck with smallest difference and then wind up having a conversation about what is meant by that. 2. I’m doing this now with my 2nd graders and their first question was, “Can we go into the negatives?!” After thinking about this for a moment, I figured that you absolutely could go into the negatives! Is my thinking correct? Here is what they came up with: 12 – 98 = -86 and 97 – 86 = 11. Thanks for stretching our brains! • Your explorations hit at the heart of the challenge between language that feels comfortable to say out loud and language that is mathematically precise. Specifically, what does “smallest difference” mean? Is -90 a smaller difference than 1? Perhaps “least difference” would be more precise? Another way to express what I meant by the problem would be to say that if you put the two whole numbers on a number line, then I want the space between them to be the least (or smallest) possible. Where it gets icky is how -86 < 2 but in this context, is a larger difference. Thoughts? • Would it be okay to preface the problem with the statement, Two numbers are on a number line, make the largest or smallest difference between them…<\|endoftext\|>	4.40625
801	DETERMINE IF THE TABLE REPRESENTS LINEAR OR EXPONENTIAL FUNCTION Linear function : If the growth or decay involves increasing or decreasing by a fixed number(constant difference), then it should be a linear function. y = mx + b b is y-intercept and m is slope. Exponential function : If the growth or decay involves using multiplication, then it should be a exponential function. y = a(b)x a is starting value b is multiplication factor, either growth or decay. If b > 0 for growth and 0 < b < 1 when it is decay. Decide whether the table represents a linear or exponential function. Then, write the function formula. Problem 1 : Solution : To get values of y, we have to add 3. Then, it is linear function. Linear function will be in the form of y = mx + b Slope(m) = 3 and y-intercept (b) = 2 So, the required function is y = 3x + 2 Problem 2 : Solution : Since the multiplication factor is the same and it is greater than 1, it is exponential growth function. Exponential function will be in the form of y = a(b)x Initial value (a) = 3 Multiplication factor (b) = 2 So, the required exponential function is y = 3(2)x Problem 3 : Solution : Observing the values of y, 10/2 ==> 5 5/2 ==> 2.5 Since the multiplication factor is 1/2, that is lesser than 1. It should be exponential decay function Exponential function will be in the form of y = a(b)x Initial value (a) = 10 Multiplication factor (b) = 1/2 So, the required exponential function is y = 10(1/2)x Identify the function as linear or exponential and determine the slope or growth factor. Write the rule for each function and sketch a graph labeling the y-intercept. Problem 4 : Solution : Since the multiplication factor is same and it is greater than 2. It is exponential growth function. Multiplication factor = 2 Exponential function will be in the form of y = a(b)x Initial value (a) = 1 So, the required exponential function is y = 1(2)x Problem 5 : Solution : By observing the values of y, it is added by 2. -4 + 2 ==> -2 -2 + 2 ==> 0 0 + 2 ==> 2 y = -4x + 2 It is a linear function . Problem 6 : Solution : By observing the values of y, it is multiplied by 2 (1/16) x 2 ==> 1/8 (1/8) x 2 ==> 1/4 (1/4) x 2 ==> 1/2 (1/2) x 2 ==> 1 Multiplication factor = 2 a = 1/4 and b = 2 y = (1/4)(2)x It is a exponential function. Problem 7 : Solution : By observing the values of y, it is added by 2 -2 + 2 ==> 0 0 + 2 ==> 2 2 + 2 ==> 4 y-intercept = 2 y = 2 + 2x It is a linear function. Recent Articles 1. Finding Range of Values Inequality Problems May 21, 24 08:51 PM Finding Range of Values Inequality Problems 2. Solving Two Step Inequality Word Problems May 21, 24 08:51 AM Solving Two Step Inequality Word Problems<\|endoftext\|>	4.65625
619	# Welcome to Planet Infinity KHMS ## 23 May, 2008 ### Making e Snowflakes Dear students, Some of you have choosen Paper folding and cutting as the strategy for making mathematics project. I was exploring some ideas of making paper snowflakes and I noticed the following link : Making e snowflakes Use this for making snowflakes and save images and upload them on the KHMS e Mathematics.It is very enjoyable and creative project. Here is one example ## 19 May, 2008 ### Make this jigsaw Puzzle for you Join the pieces and make the above figure ### Arithmetic Progression Consider the following arrangement of numbers 3,6,9,12,..... These numbers are arranged according to a specific rule. If you observe the rule, you will find that every next number to the previous one is obtained by adding 3(except the first one). What is a sequence? A sequence is an arrangement of numbers in specific order like the above one. Now, what is an arithmetic progression? A special type of sequence in which every term except the first is obtained by adding a fixed number which may be positive/negative to the preceding term. A general A.P. (Arithmetic Progression) is given by a , a+d , a+2d , .....a+(n-1)d where a is the first term , d is the common difference, n is the total number of terms. General /nth term/last term of an A.P is given by an = a +(n-1) d Example :Find the common difference and nth term of the given A.P. -5 , -1, 3 ,7 ...... Here a = -5 d = -1-(-5) = -1+5 = 4 So, an = a+ (n-1)d = -5+(n-1)4 = -5+4n-4 = 4n-9 Visual representation of sequences.... Let us consider the following sequences (1) 4,6,8,10,......... Geometrically its representation could be like a ladder in which the height of each step from the first step is same. (2) 3,6,8,10,.......... Geometrically, its representation is like steps as shown below, but the difference of heights between two consecutive steps ia not always same. ## 01 May, 2008 ### Graphs of cubic polynomials This is the graph of a cubic polynomial x^3 + 2. Observe carefully, that it cuts the x-axis at 1 point only. So, it has only one zero which is the value of x-coordinate of the point where the curve cuts the x-axis. In the following graph, the curve intersects the x-axis at three points, so the given cubic polynomial has 3 zeroes. A cubic polynomial has degree 3.It has therefore atmost 3 zeroes.<\|endoftext\|>	4.40625
573	# How do you factor 6x^2 + 8x + 2? May 19, 2015 We can Split the Middle Term of this expression to factorise it In this technique, if we have to factorise an expression like $a {x}^{2} + b x + c$, we need to think of 2 numbers such that: ${N}_{1} \cdot {N}_{2} = a \cdot c = 6 \cdot 2 = 12$ AND ${N}_{1} + {N}_{2} = b = 8$ After trying out a few numbers we get ${N}_{1} = 6$ and ${N}_{2} = 2$ $6 \cdot 2 = 12$, and $6 + 2 = 8$ $6 {x}^{2} + 8 x + 2 = 6 {x}^{2} + 6 x + 2 x + 2$ $= 6 x \left(x + 1\right) + 2 \left(x + 1\right)$ $\left(x + 1\right)$ is a common factor to each of the terms $= \left(x + 1\right) \left(6 x + 2\right)$ =color(green)(2(x+1)(3x+1) May 19, 2015 There is another shortcut that avoids the lengthy factoring by grouping. $y = 2 \left(3 {x}^{2} + 4 x + 1\right) .$ Since a + b + c = 0, the trinomial in parentheses has one factor (x + 1) and another (x + c/a) = (x + 1/3) Factored form: $f \left(x\right) = 2 \left(x + 1\right) \left(x + \frac{1}{3}\right) = 2 \left(x + 1\right) \left(3 x + 1\right)$ Check by developing: $f \left(x\right) = 2 \left(3 {x}^{2} + x + 3 x + 1\right)$ .OK Reminder of the shortcut for f(x) = ax^2 + bx + c = 0. 1. When a + b + c = 0, one real roots is (1) and the other is (c/a) 2. When a - b + c = 0, one real root is (-1) and the other is (-c/a) Remember this TIP. It will save you a lot of time and effort.<\|endoftext\|>	4.8125
873	# Mathematics app for iphone In this blog post, we will be discussing about Mathematics app for iphone. Our website will give you answers to homework. ## The Best Mathematics app for iphone Solving the square is a mathematical technique used to find the value of a variable in a quadratic equation. The name comes from the fact that the technique can be used to draw a square on a graph, which can then be used to solve for the value of the variable. The most common way to solve the square is by using the Quadratic Formula, which states that the value of the variable is equal to the negative of the coefficient of the squared term, divided by twice the coefficient of the linear term. Solving the square can be a difficult process, but with practice it can become easier. In addition, there are many software programs and online calculators that can help to solve the square. With some patience and effort, anyone can learn how to solve the square. Algebra is the branch of mathematics that deals with the solution of equations. In an equation, the unknown quantity is represented by a letter, usually x. The object of algebra is to find the value of x that will make the equation true. For example, in the equation 2x + 3 = 7, the value of x that makes the equation true is 2. To solve an equation, one must first understand what each term in the equation represents. In the equation 2x + 3 = 7, the term 2x represents twice the value of x; in other words, it represents two times whatever number is assigned to x. The term 3 represents three units, nothing more and nothing less. The equal sign (=) means that what follows on the left-hand side of the sign is equal to what follows on the right-hand side. Therefore, in this equation, 2x + 3 is equal to 7. To solve for x, one must determine what value of x will make 2x + 3 equal to 7. In this case, the answer is 2; therefore, x = 2. A binomial solver is a math tool that helps solve equations with two terms. This type of equation is also known as a quadratic equation. The solver will usually ask for the coefficients of the equation, which are the numbers in front of the x terms. It will also ask for the constants, which are the numbers not attached to an x. With this information, the solver can find the roots, or solutions, to the equation. The roots tell where the line intersects the x-axis on a graph. There are two roots because there are two values of x that make the equation true. To find these roots, the solver will use one of several methods, such as factoring or completing the square. Each method has its own set of steps, but all require some algebraic manipulation. The binomial solver can help take care of these steps so that you can focus on understanding the concept behind solving quadratic equations. This gives us x=4. We can then check our work by plugging 4 in for x in the original equation. Doing so should give us a true statement: 4+3=7. Equations can be used to solve for a wide variety of values, from simple addition and subtraction problems to more complex operations like quadratic equations. No matter what type of equation you are solving, the process is always the same: find the value of the variable that will make the two sides of the equation equal. ## We solve all types of math problems This is truly the best math app. It's so easy, instead of typing all the numbers and all, just scan and that's it! And, the solutions are 100% correct and very well explained. The steps are clear and it helped me to be better at math!! Install it everybody it will make your day a lot better. Quinanna Ross It’s so good but it needs more explanation if it had every explanation, it will be used by every single person in the earth and if you make it online you can get paid even more so it’s my advice and I love this app Emely Brooks Finite math help Solving systems by elimination solver Math experts Solve pemdas problems online Calculator app picture<\|endoftext\|>	4.59375
1,838	Pink Eye (Conjunctivitis) Pink eye, or conjunctivitis is very common. The conjunctiva is the membrane forming the moist lining of the eyelids. It is more exposed to the exterior environment than any other part of the body and therefore open to more microorganisms than any other membrane in the body. There is also a thin layer of conjunctiva covering the surface of the cornea. If it becomes infected, conjunctivitis is the result. Pink eye infection generally starts in one eye and easily spreads to the other eye. Red, irritated eyes result that go away in three days to a week. If the symptoms last longer than that you should see your eye doctor. This is important because if the infection enters the cornea (the transparent surface of the eyeball) little cloudy areas can develop that may harm your vision. About the Conjunctiva The conjunctiva is the membrane forming the moist lining of the eyelids and covering the surface of the cornea. If the conjunctiva becomes infected, conjunctivitis is the result. This protective layer is more exposed to the exterior environment than any other part of the body, and it is therefore open to more microorganisms than any other membrane. The conjunctiva is made up of flattened and columnar epithelium cells (skin and lining) and goblet cells (mucin-secreting). It also secretes tears, but less than the tear glands. It contains both nerve cells and microcapillaries that deliver nutrients and remove waste. It is connected to the sclera (the white of the eye) by a thin vascular membrane called the episclera. The kinds of conjunctiva are: - Tarsal (palpebral) conjunctiva lines the eyelids. - Fornix conjunctiva forms the junction between tarsal and ocular conjunctiva. This type of conjunctiva is very flexible and permits eyelids and eyeballs to move freely. - Ocular (bulbar) conjunctiva covers the white part of the eye and is tightly bound to the underlying layer of sclera. This type of conjunctiva moves as the eyeball moves. Types of Conjunctivitis Eye doctors refer to conjunctivitis types as follows: Allergic conjunctivitis, inflammation due to allergies, is caused by the immune system's release of histamine and other biochemicals. Viral conjunctivitis can occur along with colds or flu and is contagious. Most often, rubbing the eyes with contaminated hands causes it to spread. Bacterial conjunctivitis generally requires antibiotics for treatment. - Various organisms can cause conjunctivitis including: contagious herpes keratitis, gonococcal, and chlamydia (not common). The incidence has increased in recent years and may be due, in part, to increased antibiotic-resistant strains. - Neonatal conjunctivitis (ophthalmia neonatorum) is found in newborn children, passed from the mother's birth canal. Chemical origins may be due to an acidic or alkaline material getting in the eye. Related disorders include episcleritis and scleritis. Some symptoms of conjunctivitis include: - red, irritated eyes - morning "glued" eyelids, from night discharges - sensitivity to light - feeling of itchiness - feeling of griminess Repeated conjunctivitis can be due to chronic dry eyes. Our tears contain natural antibiotics that help neutralize normal bacteria around our eyes. With less tears, come fewer natural antibiotics. Infections, including infected insect bites and scratches, are a common cause of conjunctivitis. - Bacteria. Bacteria, such as pneumococcus, staphylococcus, or streptococcus, are the cause of some cases. Bacterial conjunctivitis is more common in children. - Virus. Pink eye can also be caused by an adenovirus that spreads by many means, including swimming pools, wet towels, or touching your eyes with dirty hands. Viral conjunctivitis is more common in adults. Insect bites, debris, and rubbing the eyes can contribute to pink eye. Chronic dry eyes may contribute to the cause. The tear film provides protective, lubricating, nutritional, and antimicrobial functions, and plays an important role in visual acuity. A lack of natural tears, therefore, reduces our ability to fight bacterial infection. Drugs that can cause dry, irritated eyes include antihistamines, acne medicine, antidepressants, Parkinson's medications, sleeping pills, birth control pills, blood pressure medications, pain relievers such as NSAIDS (for example, ibuprofen), and nasal decongestants. Allergies and sensitivities that manifest in the eyes often cause pink eye. This includes environmental pollution, pollen, and other allergens. - Vernal keratoconjunctivitis. This condition typically occurs only seasonally, and it is thought to be an allergic disorder. Patients experiencing conjunctivitis only during allergy seasons, often have other family members with allergy-related conditions, such as asthma or hay fever. It is more common in boys than girls, and more common under 20 years old. It occurs most often in hot-climate summers. - Allergic conjunctivitis. Some people are susceptible to allergic conjunctivitis brought on by the use of drugs. Topical eye antibiotics are sometimes have the side effect of causing an allergic reaction of pink or red eye. Oral antibiotics or those given intravenously for bacterial infections can also cause symptoms in some people. - Synthetic penicillins (amoxicillin and ampicillin)- one may experience some mild pink eye or red eye, itching and dryness. In rare cases they may cause blood vessel hemorrhages in the conjunctiva and in the retina - Tetracycline - in addition to pinkness or redness, one might experience light sensitivity and blurred vision - Sulfonamides - many people are allergic to "sulfa drugs". This can cause blurred vision, light sensitivity and hemorrhages in the eye. - Note: Whenever taking antibiotics make sure you take probiotics such as acidophilus or bifidus and vitamin C to help ward off some of the side effects of the antibiotics. After other potential problems have been ruled out, conventional medicine usually prescribes sulfa-based eyedrops. These usually work within three days. If not, broad-spectrum antibiotic eyedrops or ointment is prescribed; these only work with bacterial infections. Use preservative-free allergy or antihistamine eyedrops to help keep the eyes lubricated and reduce itching. Antihistamines or steroids may be prescribed by an eye doctor and should only be used under the doctor’s direction. Whenever taking antibiotics, make sure you take probiotics as well. Patients who wear contacts and those who have a sexually-transmitted disease must be treated. Diet & Nutrition Certain nutrient-providing herbs such as burdock, forsythia, goldenseal and echinacea may help reduce the symptoms of conjunctivitis. - Yogurt. The acidophilus in yogurt combats the bacterial infection. For that reason, we also suggest acidophilus supplements. Eat 1/2 cup of yogurt with the live cultures three times a day or take an acidophilus supplement (with about six billion live or probiotic organisms) three times a day. Yogurt can also be used in a soothing compress to the eyes. You don't need to put the yogurt directly in your eyes, but instead, use it in a compress. - Vitamin A. Vitamin A is especially important in promoting health in all skin and membrane tissues, including the conjunctiva of the eyelids. Do not take vitamin A if you are pregnant or have Stargardt's. - Vitamin B Complex. Conjunctivitis can be triggered by a vitamin B2 (riboflavin) deficiency, but supplementing with the entire B complex can increase the availability of vitamin B2 without inducing deficiencies of the other B vitamins. Riboflavin plays an essential role in maintaining the structure and function of the ocular surface. Riboflavin deficiency induces ocular surface damage. - Preservative-free eyedrops. To keep the eyes moist, homeopathic eyedrops are available. They can help with natural tear production and for those with dry eyes. - Use warm compresses daily. Herbs can be included in an infusion, such as burdock, chamomile, chrysanthemum flower, eyebright, echinacea, goldenseal, marigold and red raspberry leaf. Takes a moment to load ...<\|endoftext\|>	4.5
244	- MLA (Modern Language Association) style is one of the most commonly used to write papers in most academic areas especially within the liberal arts and humanities. It is currently updated to 8th edittion. Here are a few steps in MLA format. Every page should have the authors surname header on the left of every page. - MLA style is based on a general method that that can be applied to every possible source and different type of writing. But since texts are increasingly mobile, and similar documents are available in various sources there are no longer fixed rule but we have basic guidelines to assist you. - The current system uses a few basic principles, that provides a flexible method that is universally applicable. Once you are familiar with the method, you will be able to apply MLA to document and any type of source, for any paper, in any field. - Here are basic principle to follow:When citing your source, start by consulting the list of core elements. When citing, the elements should be listed in the following order: - Title of source. - Title of container, - Other contributors, - Publication date, Each element should be followed by the punctuation mark shown here.<\|endoftext\|>	3.796875
583	1. ## Completing the square Can anyone solve this quadratic by completing the square for me? $\displaystyle 16y^2+16y=1$ Steps you took? 2. ## Solution It's been a few years since I've had a problem like this, so hopefully I am going about this correctly. Start by dividing the equation through by 16 to get: y^2 + y = 1/16 Now, we figure that in order to factor the left hand side of the equation we would need something of the form (y + c)^2 with c denoting some constant value. We would like to choose something along the lines of: (y+1/2)^2 because when you multiply it out you get: y^2 + y + 1/4. But since we have this additional 1/4 on the left side of the equation, we must add it to the right side to get the full equation: y^2 + y + 1/4 = 1/16 + 1/4 This becomes: (y + 1/2)^2 = (1/4) + (1/16) which becomes: (y + 1/2)^2 = 5/16. Now take the square root of both sides to yield: y + 1/2 = +/- sqrt(5)/4 Final step: subtract 1/2 from both sides to solve for y: y = (+/- sqrt(5)/4) - 1/2 I hope this explanation is useful. 3. Most. Thanx a bunch. 4. Originally Posted by EyesForEars Can anyone solve this quadratic by completeing the square for me? $\displaystyle 16y^2+16y=1$ Steps you took? Edit: This is late, but here it is anyway. First thing, we have to make the coefficient of the quadratic term 1. So divide all terms by 16. $\displaystyle 16y^2+16y=1$ $\displaystyle y^2+y=\frac{1}{16}$ Now, take half of the coefficient of the linear term, square it, and add it to both sides. $\displaystyle y^2+y+\frac{1}{4}=\frac{1}{16}+\frac{1}{4}$ $\displaystyle (y+\frac{1}{4})^2=\frac{5}{16}$ Now, take the square root of both sides. $\displaystyle y+\frac{1}{2}=\frac{\pm \sqrt{5}}{4}$ $\displaystyle y=\frac{\pm \sqrt{5}}{4}-\frac{1}{2}=\frac{\pm \sqrt{5}-2}{4}$<\|endoftext\|>	4.65625
1,545	Persistence is an integral habit of mind for the completion of a new task, unfamiliar or challenging to students. During exploratory learning, in coding, students are constantly interacting with new, unfamiliar and challenging problems. Each student's ability to persist through their frustration with unfamiliar algorithms, key concepts and building blocks of coding is paramount for his or her success in gaining foundational coding skills. In coding course E, students learn persistence is important because when interacting with computer science he or she will be solving unfamiliar problems and perhaps problems, which have not been solved before. As explorers in a new domain, students learn in the lesson titled Lesson 3: Building a Foundation that persistence is an essential habit of mind as computer scientists. In the lesson, students are presented with the code.org definition of persistence - trying again and again, even when something is very hard. During the lesson, students are encouraged to persist while attempting to build a structure with the supplies provided (toothpicks and Dots candy) that can hold their fourth grade grammar text book for 10 seconds. Throughout the lesson students persisted, determined to complete the task presented. The lesson plan titled Lesson 3: Building a Foundation for explicit teaching of persistence is displayed below. Task - Building a Foundation In this challenge, we'll work to construct towers that are strong enough to hold your fourth grade grammar textbook for at least 10 seconds, using toothpicks and candy dots. Use only the supplies provided to build a tower. The tower can be any shape, but it has to be at least as tall as the toothpick. The tower must support the weight of your fourth grade grammar textbook for a full 10 seconds. The Code.org coding curriculum, provides ample opportunities for students to relearn and revisit the habit of mindset of persistence in the classroom. All plugged in lessons provide students with a sequence of puzzles to solve, targeting a specific skill as a programmer. Students are constantly encouraged to interact with unfamiliar problems and persist through lessons for an end product of a coding sequence or program. Therefore, the curriculum provides opportunity for students to constantly interact with relevant objectives and vocabulary to the habit and mindset of persistence. Additionally the curriculum provides activities, which allow students to demonstrate they have internalized persistence that has led to their personal growth. Coding Curriculum Example A Coding Curriculum Example A is the first lesson and an uplugged activity, which gets students used to the habit of persistence as it relates to writing and rewriting code, adjusting for any found mistakes in the coding sequence. Students in the lesson, need to persist as coders, be comfortable with making mistakes. and apply different methods to solve the problem. In the lesson, students are tasked with writing a sequential code for his or her "robotic friend." The friend needs to be able to follow the code without conferring with his or her coder. Therefore, students need to work towards precision in code only listing instructions, which are relevant to his or her desired outcome. The lesson plan titled Lesson 1: Programming: My Robotic Friends is displayed below. InLesson 1: Programming: My Robotic Friends, an unplugged lesson, students are practicing persistence with their introduction to coding sequence. Students in this lesson have to try again and again to create a sequence (debugging if he or she detects any errors), which works for his or her "robotic friend" to build their desired stack of cups product. The image demonstrates the "robotic friends" applying the and re-applying code written from his or her persisting programming friend. Coding Curriculum Example B Coding Curriculum Example B is also illustrated in the access portion of my portfolio on the sub-page: Access Example #4: Exploratory Learning (Code In The Schools). Lesson 2: Sequences in Maze is the second lesson in the coding curriculum, but first plugged in lesson for student learning. Therefore, in this lesson students begin to grasp the fundamentals of programming. Students receive introductory objectives and vocabulary, in addition to receiving encouragement to work with a partner and persist through the online puzzles. In this lesson, students are interacting for the first time with blockly code, in addition to working with a partner to share ideas and different attempts at moving the pig to the end point on the puzzle. The lesson template below for Lesson 2: Sequences in Maze highlights the points of persistence teaching and student internalization and action throughout the lesson. In the video, students are working in pairs to solve the puzzles provided for Lesson 2: Sequences in Maze. The video demonstrates each student's persistence as coder, as he or she works with their partner to solve each puzzle provided in the online sequence. Example #2: Persistence as a Writer In addition to the coding curriculum, students apply persistence as writers. Students engage in the writing process submitting several drafts to meet content standards and writing mechanics expectations. The drafts are edited by the teacher and by peers before the final draft is written. One example of persistence in writing are students "Why NACA?" statements. Students were asked to share "Why?" students chose to attend NACA I. Students received their “Why NACA?” statements back with teacher edits, at the end of the week and were instructed to complete the edits and make necessary revisions. By making revisions students are learning persistence as a writers and becoming familiar with the writing process. The students writing samples are now displayed outside of my fourth grade English Language Arts and Social Studies classroom. The multiple drafts on display, show students persistence as writers, participating in a process of editing, which requires repeated attention to ones writing mechanics and expression. Example #3: Hokusai's great wave: A Lesson of Persistence In the Hokusai's great wave: A Lesson of Persistence students learn about persistence from the artwork of Hokusai Katsushika. Specifically, in the lesson students learn the importance of persistence as an artist. Students learn that the Japanese artist Hokusai Katsushika spent three years capturing thirty-six illustrations of Mt. Fuji, the tallest mountain in Japan and a mountain deemed sacred by the artist and others. Following students close reading of the article in the lesson, students apply their habit and mindset of persistence to attempt to recreate Hokusai Katsushika's cherished "The Great Wave of Kanagawa" masterpiece. The lesson plan for Hokusai's great wave: A Lesson of Persistence is displayed below. During the lesson, students read a Khan Academy article, titled Hokusai, Under the Wave of Kanagawa (The Great Wave)students are instructed to use the questions on the page above to read closely drawing key concepts and details from the text. In addition, to practicing persistence as writers students also exercise persistence as students read and reread for details in a text. Parent/guardian permission is granted for each student displayed in the image. Once students learned from the article that it had taken Hokusai Katsushika three years and thirty-six views of Mt. Fuji to capture his masterpiece or desired outcome, students applied great care and persistence to their artwork. Students in the lesson, were tasked with recreating Hokusai Katsushika's masterpiece, applying persistence as artists. Students used the image of the painting on the computer to trace carefully and capture all details. Costa, A. L. & Kallick, B. (2009). Habits of mind. Alexandria, VA: Association for Supervision and Curriculum Development.<\|endoftext\|>	4.09375
1,184	Associated Topics \|\| Dr. Math Home \|\| Search Dr. Math ### Intersection of Lines ``` Date: 06/28/98 at 05:46:40 From: Andrew C. Subject: Intersection of lines Hi. This question has been puzzling me for a couple of days now and I was wondering if you could help me out. Thanks. n co-planar lines are such that the number of intersection points is a maximum. (i) How many intersection points are there? (ii) If n such lines divide the plane into Un regions; show that Un = U(n-1) + n. Hence deduce that Un = 1 + 1/2n(n+1) How many of these regions have finite area? Un is "U subscript n" and "U(n-1) is U subscript n minus 1" Andrew ``` ``` Date: 06/28/98 at 11:40:04 From: Doctor Anthony Subject: Re: Intersection of lines >(i) How many intersection points are there? Two lines cut in 1 point. A third line will cut the other two lines in 2 more points, giving 1 + 2 = 3 points. A fourth line will cut the other 3 lines in 3 more points, giving 3 + 3 = 6 points. So the series goes n = 1, 2, 3, 4, 5, 6, ....... Number of points 0, 1, 3, 6, 10, 15, ....... The gap increases by 1 each time. This is the sequence of triangular numbers which has the nth term given by n(n-1)/2. If you made up a difference table, the second differences would be constant, so the nth term is a quadratic in n. If you assume f(n) = an^2 + bn + c, where f(n) is the nth term n = 1 a + b + c = 0 n = 2 4a + 2b + c = 1 n = 3 9a + 3b + c = 3 and solving these 3 equations for a,b,c a = 1/2, b = -1/2, c = 0 so f(n) = n^2/2 - n/2 = n(n-1)/2 as given above. So with n lines there are n(n-1)/2 intersection points. >(ii) If n such lines divide the plane into Un regions; show that > Un = U(n-1) + n. Hence deduce that Un = 1 + 1/2n(n+1) > How many of these regions have finite area? With n = 1 we divide the plane into 2 regions. With n = 2 we have 4 regions, with n = 3 we get 7 regions. A fourth line will meet the other 3 lines in 3 points and so traverses 4 regions, dividing them into two parts and adding 4 new regions. In general the nth line will u(1) = 2 u(2) = 4 u(3) = 7 u(4) = 11 and so on, where u(n) = number of regions with n lines. We get the recurrence relationship u(n+1) = u(n) + (n+1) We get the following chain of equations u(n) - u(n-1) = n u(n-1) - u(n-2) = n-1 u(n-2) - u(n-3) = n-2 ........................ ........................ u(4) - u(3) = 4 u(3) - u(2) = 3 u(2) - u(1) = 2 -------------------------- Add u(n) - u(1) = 2 + 3 + 4 + ..... + (n-1) + n all other terms on the left cancel between rows. u(n) = u(1) + 2 + 3 + 4 + ....+ n and u(1) = 2 u(n) = 1 + [1+2+3+4+...+n] = 1 + n(n+1)/2 = (2 + n^2 + n)/2 So u(n) = (n^2+n+2)/2 The number of 'open' regions with 2 lines is 4, with three lines it is 6, with 4 lines it is 8 and so on. With n lines the number of open regions is 2n. So the number of finite regions is given by: 1 + n(n+1)/2 - 2n - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ ``` Associated Topics: High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home \|\| Math Library \|\| Quick Reference \|\| Math Forum Search<\|endoftext\|>	4.4375
399	The awareness of changes in the internal & external environment What is perception? The conscious interpretation of sensation stimuli Somatosensory System organization * Input comes from: exteroceptors, proprioceptors, & interoceptors * Three main levels of neural integration: receptor level, circuit level, & perceptual level Receptor level processing * The sensory receptors * Must have specificity for the stimulus energy * The receptor's receptive field must be stimulated * Stimulus energy must be converted to a grated potential * Generator potential in sensory neuron must reach threshold Sensory receptor adaptation * Occurs when sensory receptors are subjected to unchanging stimulus * Receptor membrane becomes less responsive * Receptor potentials decline in frequency or stop Which sensory receptors adapt quickly? The receptors that respond to pressure, touch, & smell Which sensory receptors adapt slowly? The receptors resonding to Merkel's discs, Ruffini's corpuscles, & interoceptors that respond to chemical blood changes Which sensory receptors do NOT adapt? The receptors responding to pain & proprioceptors * Soma reside in dorsal root or cranial ganglia * Conduct impulses from the skin to the spinal cord or brain stem * Somas reside in dorsal horn of spinal cord or medullary nuclei * Transmit impulses to thalamus or cerebellum * Located in thalamus * Conduct impulses to the somatosensory cortex of the cerebrum How occurs at the perceptual level Thalamus projects fibers to the somatosensory cortex & sensory association areas Detecting a stimulus has occurred & requires summation How much of a stimulus is acting Identifying the site or pattern of the stimulus Used to identify a substance that has specific texture or shape Ability to identify submodalities of a sensation (like sweet or sour) Ability to recognize patterns in stimuli (like a melody or a familiar face)<\|endoftext\|>	3.765625
485	## Michigan Papyrus The Michigan papyrus 620 is a Greek papyrus written in the second century. It contains three arithmetic problems. One of these problems follows: Four numbers: their sum is 9900; let the second exceed the first by one-seventh of the first; let the third exceed the sum of the first two by 300; and let the fourth exceed the sum of the first three by 300. Find the four numbers. Source: mathcontest.olemiss.edu 4/30/2007 SOLUTION Let $a,b,c,d$ be the four numbers. Let’s represent the first number $a$ as an apple pie divided into 7 equal pieces as follows: We don’t know the numerical value of each piece. If we do, we will know the value of $a$ . All we know is that $a=7\;pieces$ Because the second number $b$ exceeds the first number by one-seventh of the first, $b=8\;pieces$ The third number $c$ exceeds the sum of the first two by 300, $c=\left (7\;pieces+8\;pieces\right )+300$ $=15\;pieces+300$ The fourth number $d$ exceeds the sum of the first three by 300, $d=\left (7\;pieces+8\;pieces+15\;pieces+300\right )+300$ $=30\;pieces+600$ The sum of the four numbers is 9900, $9900=7\;pieces+8\;pieces+\left (15\;pieces+300\right )+\left (30\;pieces+600\right )$ $=60\;pieces+900$ Subtracting 900 from both sides, $60\;pieces=9000$ $1\;piece=9000\div 60=150$ Now we can compute the value of the numbers: $a=7\left (150\right )=1050$ $b=8\left (150\right )=1200$ $c=1050+1200+300=2550$ $d=1050+1200+2550+300=5100$ Check $1050+1200+2550+5100=9900$<\|endoftext\|>	4.8125

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