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# If $\mathop {\lim }\limits_{x \to - \infty } \left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} - \sqrt {{x^6} - 2{x^5} + {x^3} + 1} } \right) = 2$ then A) b=$- 2$ B) a=$- 3$ C) a=$2$ D) b=$- 3$
Last updated date: 13th Jun 2024
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Hint: Rationalize the given function and put $\dfrac{1}{x} = 0$ in the obtained result because $x \to - \infty$ .Then sove to find the value of a and b.
Given function is $\mathop {\lim }\limits_{x \to - \infty } \left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} - \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right) = 2$
It is in $\infty - \infty$ indeterminate form. So we can solve it by rationalizing.
On rationalizing the given function we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} - \sqrt {{x^6} - 2{x^5} + {x^3} + 1} } \right) \times \left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}}{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}} = 2$ We know that ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ , On applying this in the above equation we get,
$\Rightarrow$ $\mathop {\lim }\limits_{x \to - \infty } \dfrac{{\left( {{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} } \right)}^2} - {{\left( {\sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}^2}} \right)}}{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}} = 2$
On solving we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{{x^6} + a{x^5} + b{x^3} - cx + d - \left( {{x^6} - 2{x^5} + {x^3} + x + 1} \right)}}{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}} = 2$
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{{x^6} + a{x^5} + b{x^3} - cx + d - {x^6} + 2{x^5} - {x^3} - x - 1}}{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}} = 2$
On taking the coefficients of the same terms common we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{\left( {a + 2} \right){x^5} + \left( {b - 1} \right){x^3} - \left( {c + 1} \right)x + d - 1}}{{\left( {\sqrt {{x^6} + a{x^5} + b{x^3} - cx + d} + \sqrt {{x^6} - 2{x^5} + {x^3} + x + 1} } \right)}} = 2$
On taking ${x^6}$ common in the denominator we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{\left( {a + 2} \right){x^5} + \left( {b - 1} \right){x^3} - \left( {c + 1} \right)x + d - 1}}{{{x^3}\left( {\sqrt {1 + \dfrac{a}{x} + \dfrac{b}{{{x^3}}} - \dfrac{c}{{{x^5}}} + \dfrac{d}{{{x^6}}}} + \sqrt {1 - \dfrac{2}{x} + \dfrac{1}{{{x^3}}} + \dfrac{1}{{{x^5}}} + \dfrac{1}{{{x^6}}}} } \right)}} = 2$
On taking ${x^3}$ common from numerator we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{{x^3}\left[ {\left( {a + 2} \right){x^2} + \left( {b - 1} \right) - \dfrac{{\left( {c + 1} \right)}}{{{x^2}}} + \dfrac{{d - 1}}{{{x^3}}}} \right]}}{{{x^3}\left( {\sqrt {1 + \dfrac{a}{x} + \dfrac{b}{{{x^3}}} - \dfrac{c}{{{x^5}}} + \dfrac{d}{{{x^6}}}} + \sqrt {1 - \dfrac{2}{x} + \dfrac{1}{{{x^3}}} + \dfrac{1}{{{x^5}}} + \dfrac{1}{{{x^6}}}} } \right)}} = 2$
On simplifying we get,
$\Rightarrow \mathop {\lim }\limits_{x \to - \infty } \dfrac{{\left[ {\left( {a + 2} \right){x^2} + \left( {b - 1} \right) - \dfrac{{\left( {c + 1} \right)}}{{{x^2}}} + \dfrac{{d - 1}}{{{x^3}}}} \right]}}{{\left( {\sqrt {1 + \dfrac{a}{x} + \dfrac{b}{{{x^3}}} - \dfrac{c}{{{x^5}}} + \dfrac{d}{{{x^6}}}} + \sqrt {1 - \dfrac{2}{x} + \dfrac{1}{{{x^3}}} + \dfrac{1}{{{x^5}}} + \dfrac{1}{{{x^6}}}} } \right)}} = 2$
Now since it is given that $x \to - \infty \Rightarrow \dfrac{1}{x} = 0$
So all the values multiplied with $\dfrac{1}{x}$ in the denominator and numerator will be zero and for the limit to be finite as x tends to infinite the quantity $a - 2 = 0$ $\Rightarrow a = 2$ .So the limit will only exist for $b - 1$
Then on putting the values we get,
$\Rightarrow \dfrac{{\left[ {\left( {b - 1} \right)} \right]}}{{\left( {\sqrt 1 + \sqrt 1 } \right)}} = 2$
On simplifying we get,
$\Rightarrow \dfrac{{b - 1}}{2} = 2 \\ \Rightarrow b - 1 = 4 \\ \Rightarrow b = 5 \\$
So, only option C is correct.
Note: Here we rationalize the function to make it easier to solve the limit as the function gives indeterminate form on putting the limit. Indeterminate forms are such forms which cannot be determined so we try to find another method to change the indeterminate form. Here we have used rationalization. In some questions, we use L’ Hospital rule which states that the limit of derivative of the given function is equal to the limit of indeterminate form.<|endoftext|>
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After several relatively quiet years in the equatorial Pacific Ocean, El Niño may be on its way back.
A new research study published in the Proceedings of the National Academy of Sciences (PNAS, February 2014) utilized a novel, long-range statistical approach to El Niño forecasting and found a 75% likelihood that El Niño conditions will begin to present by the end of 2014.
El Niño is the warm phase of a larger ocean-atmosphere cycle called the El Niño Southern Oscillation (ENSO). During an El Niño event, the waters of the eastern equatorial Pacific off the coast of Central and South America become anomalously warm. During the opposite phase of the cycle, La Niña, those same waters become anomalously cold (see figure at right, credit: NASA).
This fluctuation in water temperature may seem like a relatively localized phenomenon, but because the ocean and atmosphere are coupled (interconnected) and circulate the entire globe, an increase in water temperatures off the coast of Peru is not only devastating to the local fishing industry, but is also the most important driver of natural interannual climate variability across the entire planet.
Globally, El Niño conditions result in a major shift in atmospheric circulations and, consequently, weather patterns (see figure below right, credit: NOAA), as well as an increase in globally averaged temperatures.
In the northern hemisphere, El Niño conditions typically result in
Conventional El Niño forecasting techniques rely on dynamical and statistical climate models that analyze observations of sea surface temperatures (SSTs) and wind patterns. Although this forecast method can be quite accurate when making predictions a few months out, its skill is rather limited at longer-range forecasting. Accurate long-range forecasting is critical, however, to preparing for and mitigating the economic effects of El Niño events. For example, in the agricultural sector, farmers need to be able to plan which crops to plant based on expected weather conditions (e.g. hotter than normal, wetter than normal, drier than normal, etc) to reduce the likelihood of crop failure.
The study published by Ludescher, et al. this month claims to have developed a forecasting technique that can accurately predict ENSO fluctuations up to a year in advance by relying solely on statistical correlations between air temperatures across the Pacific region and upcoming changes to equatorial Pacific SSTs (i.e. upcoming El Niño or La Niña events). Although the study’s authors tout its long-range predictive ability, it cannot currently predict the magnitude (severity) of those upcoming events.
The study’s authors say that their technique accurately predicted the absence of El Niño during 2012 and 2013, but because the forecasting methodology is so new, it has yet to be tested in a prediction of non-neutral ENSO conditions. Many atmospheric scientists not involved with the study remain skeptical of the skill of the new technique for that reason as well as for the fact that the study does not propose an explanation as to why the statistical correlation should work. In other words, the study does not advance scientists’ understanding of the physical mechanisms that drive the ENSO cycle.
If the new statistical forecasting technique proves successful, though, it may alleviate a problem that has been plaguing conventional ENSO forecasters for the last couple of years and that may now be negatively affecting the skill of seasonal climate (e.g. ENSO) forecasts.
The National Oceanic and Atmospheric Administration (NOAA) maintains a network of moored buoys in the tropical Pacific Ocean to monitor real-time ocean temperatures for input into the climate models used to forecast El Niño and La Niña. Since budget cuts in 2012 forced NOAA to reduce its maintenance schedule of the buoys, however, over half of them have failed. With less detail about ocean temperatures in this critical location being provided by the thinning buoy network, forecast models may suffer a loss of accuracy.
Only time will tell. The dynamical and statistical climate models used to provide conventional ENSO forecasts are also beginning to predict an increased likelihood of El Niño conditions beginning in late 2014. It’s still too early for significant confidence, but those readers who are involved in weather-sensitive industries should monitor the situation closely and consider planning early for possible El Niño conditions beginning in late fall 2014.
We at Blue Skies Meteorological Services were fortunate to escape (by a margin of about 50 miles) the late-January winter storm that crippled much of the southeastern US last week. That storm, which brought one to three inches of snow to areas that rarely receive even a trace of frozen precipitation, clearly demonstrated the danger and damage that can occur when weather does not agree with climate norms.
A common, if somewhat over-simplified, explanation of the difference between weather and climate is:
Climate is what you expect;
Weather is what you get.
The day-long traffic gridlock in Atlanta, GA, on Jan 28th epitomizes the hazards of getting climatologically unexpected weather (and of failing to incorporate updated forecast information into emergency management decisions, but that’s outside the scope of this article). The magnitude of the weather event itself is relatively unimportant (as several colleagues from the Upper Midwest have noted, “Two inches of snow is just a normal Monday commute back home”). What matters in terms of societal impacts is the deviation of the event from normal, expected values. And the reason for this is simple – we prepare for what we expect.
As a student at Purdue University, receiving a couple inches of snow was an almost weekly winter occurrence. Snow in northern Indiana is a climate norm, so everyone is generally well prepared for it. Life proceeds without interruption thanks to stockpiles of salt/sand, fleets of snowplows, and battalions of snow plow drivers.
It’s a different story in the South, though, where 50 degrees is deemed parka weather, and where ice is typically found in tidy cubes in your sweet tea, not in impenetrable sheets coating your car windshield. Frozen precip is an anomaly, and as such, residents and municipalities don’t maintain the infrastructure nor have the experience to deal with it as “business as usual”.
However, what is lacking in infrastructure can be addressed through effective planning. By understanding the range of extreme weather events and their climatological recurrence intervals, businesses and municipalities can develop and implement emergency plans that mitigate the damage and disruption that inevitably accompanies extreme weather events.
Two inches of snowfall will always be a big deal in Atlanta (and normal January weather in the Midwest), just like temperatures above 100 degrees will always be a big deal in Maine (and a standard late July afternoon in Oklahoma). When planning for and understanding the impacts of extreme weather in any location, we must remember that it’s not the absolute magnitude of the event that matters – it’s the deviation from the norm. It’s the difference between what you expect and what you actually get.
Note: Blue Skies Meteorological Services provides climate analyses, including of extreme weather recurrence intervals, in support of business and municipal emergency planning and hazard mitigation.
Welcome to Blue Skies or Gray, the official blog of Blue Skies Meteorological Services!
After our website redesign at the beginning of 2014, Blue Skies Meteorological Services is excited about the opportunity to communicate with current and potential clients via our new blog. We’ll be focusing on weather and climate-related news, with a special emphasis on legal, environmental, and business implications and concerns. Please check back often for the latest updates, and navigate through our archives using the toolbars to your right.<|endoftext|>
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Definition: Monetary policy is how central banks manage the money supply to guide healthy economic growth. The money supply is credit, cash, checks, and money market mutual funds. The most important of these is credit, which includes loans, bonds, mortgages, and other agreements to repay. Monetary policy is the macroeconomic policy laid down by the central bank. It involves
The NRB implements the monetary policy through open market operations, bank rate policy, reserve system, credit control policy, and moral persuasion and through many other instruments. Using any of these instruments will lead to changes in the interest
For instance, liquidity is important for an economy to spur growth. To maintain liquidity, the NRB is dependent on the monetary policy. By purchasing bonds through open market operations, the RBI introduces money in the system and reduces the interest rate.
Central banks have three main tools of monetary policy: open market operations, the discount rate, and a bank’s reserve requirement. However, most banks have many more at their disposal. Here’s what they are, and how they all work together to sustain healthy economic growth.
- Open Market Operations
Open market operations are when central banks buy or sell securities from the country’s banks. When the central bank buys securities, adds cash to the banks’ reserves. This gives them more money to lend more. When the central bank sells the securities, it simply places them on the banks’ balance sheets and reduces its cash holdings. The bank now has less to lend. A central bank buys securities when it wants expansionary monetary policy, and sells them when it executes contractionary monetary policy.
- Reserve Requirement
The reserve requirement refers to the deposit a bank must keep on hand overnight, either in its vaults or at the central bank. A low reserve requirement allows the banks to lend more of their deposits. That is expansionary because it creates more credit. A high reserve requirement is contractionary since it gives banks less money to lend. It’s especially hard on small banks since they don’t have as much to lend in the first place. Central banks rarely change the reserve requirement because it’s expensive and disruptive for member banks to change their procedures.
Instead, central banks are more likely to adjust their targeted lending rate because it achieves the same result. The Fed funds rate is perhaps the most well-known of these tools. Here’s how it works. If a bank can’t meet the reserve requirement, it borrows from another bank that has excess cash. The interest rate it pays is the Fed funds rate. The amount it borrows is called the Fed funds. The Federal Open Market Committee (FOMC) sets a target for the Fed funds rate at its meetings.
Central banks have several tools to make sure the Fed funds rate meets that target. The Federal Reserve, the Bank of England, and the European Central Bank pay interest on the required reserves and any excess reserves. Bank won’t lend Fed funds for less than the rate they’re receiving from the Fed for these reserves. Banks also use open market operations.
- Discount Rate
The discount rate is the third tool. It’s the rate that central banks charge its members to borrow at its discount window. Since the rate is higher, banks only use this if they can’t borrow funds from other banks.
How It Works
Central bank tools work by increasing or decreasing total liquidity. This includes both the total amount of capital available to invest or lend, as well as money to spend. In other words, it’s more than the money supply, which consists of M1, (currency and check deposits) and M2 (money market funds, CDs and savings accounts plus M1). Therefore, when people say that central bank tools affect only the money supply, they are understating the impact.
Compiled and collected by
Basudev Sharma Poudel(PHD)
Have a question? Ask us in our discussion forum.<|endoftext|>
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At the Construction Site, small children and their parents can find out what it takes to get everything at a construction to turn and move. It’s all about one of the key branches of physics: mechanics.
At the Experimentarium’s Construction Site, children and the young at heart can embark on a journey of discovery by getting the crane to swing into action, pedalling the Hoisting cycle, using the ball race run and trying out many other activities.
At the Construction Site, you can experience the mechanics with your own body as you turn, pull and lift the balls right up into the ball runs – and feel the force of gravity in your tummy as you follow the descent of the balls before starting a new round of the mechanics game.
The Construction Site offers the following activities:
What causes what to move? In the Crane, you apply the principles of mechanics to overcome gravity and lift the ball. The force applied by a child starts a movement. Turning the handle, it is transmitted via the drum and the pulley and lifts and finally lowers the ball.
What causes what to move? On the Hoisting cycle, you apply the principles of mechanics to overcome gravity and lift the ball. The force applied by a child starts a movement. By means of the chain, it is then transmitted via the cogs and the drum and finally lifts the ball.
The balls all roll downwards along the four different runs because, like everything else on Earth, they are subject to the force of gravity.
However, they don’t all cross the finishing line at the same time because the runs slope and twist in different ways. The flatter and twistier the run, the greater the resistance for the balls. This resistance is called friction.
The Construction Site is intended for children aged 3-6 years and their inquisitive parents.<|endoftext|>
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On the structure of the human body is a treatise on anatomy from the 16th century and one of the most influential scientific books of all time.
Written by Andrés Vesalio in 1543, when he was just 28 years old, its diagrams are some of the most perfect xylographs ever made. Various renowned artists, including Johannes Stephanus of Calcar, a disciple of Tiziano, help to make them, and they are far superior to the diagrams from the anatomic atlases from the period, commonly drawn by anatomy teachers themselves.
The painstaking care that went into the making, editing and printing of this work make it one of the best examples in terms of book production from the Renaissance, with 17 full-page designs, and some hundreds of smaller illustrations that accompany the text.
The book is based on the lessons that Vesalius dictated throughout his teachings at the University of Padua, during which he strayed from common practice and made numerous dissections of bodies to illustrate his ideas. Accordingly, the work points to the importance of dissection and of what would be come to known from then on as “anatomical” vision of the human body. His anatomical model breaks with the established norms and is one of the first and greatest steps towards the development of a modern and scientific medicine.
The work is divided into six parts, or “books’, and they comprise a complete overview of the human body. The term, “fabric”, has architectural connotations. In his description he moves beyond the bones, ligaments and muscles, that make up the bodily structure, to study the connecting systems (veins and nerves), and the life-giving systems. As well as making the first valid description of the sphenoid bone, Vesalius showed that the sternum consists of three parts and the sacrum of five or six; and he meticulously described the channel on the inside of the temporal bone. He verified Etienne’s observations around the valves in the hepatic veins, discovered the azygos vein, and discovered the channel in the foetus that communicates with the umbilical cord and the vena cava, subsequently named the ductus venosus. He also discovered the omentum, and its connections to the stomach, the spleen and the colon; he proffered the first correct notion of the structure of the pylorus; and he noted the small size of the appendix in men; he gave the first valid descriptions of the mediastinum and the pleural cavity, as well as the most accurate explication of anatomy of the brain for the time.
De Humani was reprinted in 1555 and to date there hasn’t been another print.<|endoftext|>
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Precalculus (6th Edition) Blitzer
The product of $AB={{I}_{2}}$, product of $BA={{I}_{2}}$ and $B$ is the multiplicative inverse of $A$ $B={{A}^{-1}}$.
The given expression is $A=\left[ \begin{matrix} 4 & 5 \\ 2 & 3 \\ \end{matrix} \right],B=\left[ \begin{matrix} \frac{3}{2} & -\frac{5}{2} \\ -1 & 2 \\ \end{matrix} \right]$ Now, we will compute the matrix as $\left[ AB \right]$ \begin{align} & AB=\left[ \begin{matrix} 4 & 5 \\ 2 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} \frac{3}{2} & -\frac{5}{2} \\ -1 & 2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4\times \frac{3}{2}+5\times \left( -1 \right) & 4\times \left( -\frac{5}{2} \right)+5\times 2 \\ 2\times \frac{3}{2}+3\times \left( -1 \right) & 2\times \left( -\frac{5}{2} \right)+3\times 2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{2}} \end{align} Now, we will compute the matrix as $\left[ BA \right]$ \begin{align} & BA=\left[ \begin{matrix} \frac{3}{2} & -\frac{5}{2} \\ -1 & 2 \\ \end{matrix} \right]\left[ \begin{matrix} 4 & 5 \\ 2 & 3 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{3}{2}\times 4+\left( -\frac{5}{2} \right)\times 2 & \frac{3}{2}\times 5+\left( -\frac{5}{2} \right)\times 3 \\ \left( -1 \right)\times 4+2\times 2 & \left( -1 \right)\times 5+2\times 3 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] \\ & ={{I}_{2}} \end{align} Both matrix $AB$ and matrix $BA$ are equal to the identity matrix Now, we will compute the matrix $B={{A}^{-1}}$. $B=\left[ \begin{matrix} \frac{3}{2} & -\frac{5}{2} \\ -1 & 2 \\ \end{matrix} \right]$ And, $A=\left[ \begin{matrix} 4 & 5 \\ 2 & 3 \\ \end{matrix} \right]$ Determine the inverse of matrix A. Now, using inverse formula ${{A}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right]$ Substitute the values to get, \begin{align} & a=4 \\ & b=5 \\ & c=2 \\ & d=3 \\ \end{align} So, \begin{align} & {{A}^{-1}}=\frac{1}{\left| 12-10 \right|}\left[ \begin{matrix} 3 & -5 \\ -2 & 4 \\ \end{matrix} \right] \\ & =\frac{1}{2}\left[ \begin{matrix} 3 & -5 \\ -2 & 4 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{3}{2} & \frac{-5}{2} \\ -1 & 2 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} \frac{3}{2} & \frac{-5}{2} \\ -1 & 2 \\ \end{matrix} \right] \end{align} Therefore, $B={{A}^{-1}}$.<|endoftext|>
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Mutual respect is at the heart of our values. Children learn that their behaviours have an effect on their own rights and those of others.
Our provision of SMSC can be seen in all aspects of school life and it is promoted and taught in a variety of ways:
Assemblies – Each week children are provided with an opportunity to collectively reflect and understand school and British values
School council – Pupil voice is important at Queensbridge. Each week councillors have an opportunity to raise whole school issues with the class. This allows for the expression of opinions, debate and develops listening skills. The children also learn about and have experience of democracy.
Restorative Justice – As a restorative school, Queensbridge recognises the importance of building and maintaining positive relationships so that effective learning can take place. Staff and Peer mediators are trained in resolving conflicts respectfully and with focus on finding a resolution.
Black History education – Children learn about the importance of civil and human rights. They also engage with issues such as racism and equality. Children also explore and celebrate different cultures from around the world.
Anti-bullying week and Disability History month are explored and celebrated each year as well as other events such as International Women’s Day and Martin Luther King Day
Statement of Equality
Everyone has rights and responsibilities
Everyone has equal value
Everyone in school is valued and treated fairly
We respect each others’ cultures, languages, ethnicity, appearance, age, family, intellectual ability and aptitude, talents and religious beliefs
We tackle discrimination through the positive promotion of equality by challenging bullying and stereotypes and we aim to develop a culture of inclusion<|endoftext|>
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In our world, each country follows a certain time-zone. These time-zones are crucial for expressing time conveniently and effectively. However, time-zones can sometimes be inexplicit due to variables such as daylight saving time, coming into the picture.
Moreover, while representing these time-zones in our code, things can get confusing. Java has provided multiple classes such as Date, Time and DateTime in the past to also take care of time-zones.
However, new Java versions have come up with more useful and expressive classes such as ZoneId and ZoneOffset, for managing time-zones.
In this article, we’ll discuss ZoneId and ZoneOffset as well as related DateTime classes.
We can also read about the new set of DateTime classes introduced in Java 8, in our previous post.
2. ZoneId and ZoneOffset
With the advent of JSR-310, some useful APIs were added for managing date, time and time-zones. ZoneId and ZoneOffset classes were also added as a part of this update.
As stated above, ZoneId is a representation of the time-zone such as ‘Europe/Paris‘.
There are 2 implementations of ZoneId. First, with a fixed offset as compared to GMT/UTC. And second, as a geographical region, which has a set of rules to calculate the offset with GMT/UTC.
Let’s create a ZoneId for Berlin, Germany:
ZoneId zone = ZoneId.of("Europe/Berlin");
ZoneOffset extends ZoneId and defines the fixed offset of the current time-zone with GMT/UTC, such as +02:00.
This means that this number represents fixed hours and minutes, representing the difference between the time in current time-zone and GMT/UTC:
LocalDateTime now = LocalDateTime.now(); ZoneId zone = ZoneId.of("Europe/Berlin"); ZoneOffset zoneOffSet = zone.getRules().getOffset(now);
In case a country has 2 different offsets – in summer and winter, there will be 2 different ZoneOffset implementations for the same region, hence the need to specify a LocalDateTime.
3. DateTime Classes
Next let’s discuss some DateTime classes, that actually take advantage of ZoneId and ZoneOffset.
ZonedDateTime is an immutable representation of a date-time with a time-zone in the ISO-8601 calendar system, such as 2007-12-03T10:15:30+01:00 Europe/Paris. A ZonedDateTime holds state equivalent to three separate objects, a LocalDateTime, a ZoneId and the resolved ZoneOffset.
This class stores all date and time fields, to a precision of nanoseconds, and a time-zone, with a ZoneOffset, to handle ambiguous local date-times. For example, ZonedDateTime can store the value “2nd October 2007 at 13:45.30.123456789 +02:00 in the Europe/Paris time-zone”.
Let’s get the current ZonedDateTime for the previous region:
ZoneId zone = ZoneId.of("Europe/Berlin"); ZonedDateTime date = ZonedDateTime.now(zone);
ZonedDateTime also provides inbuilt functions, to convert a given date from one time-zone to another:
ZonedDateTime destDate = sourceDate.withZoneSameInstant(destZoneId);
OffsetDateTime is an immutable representation of a date-time with an offset in the ISO-8601 calendar system, such as 2007-12-03T10:15:30+01:00.
This class stores all date and time fields, to a precision of nanoseconds, as well as the offset from GMT/UTC. For example,OffsetDateTime can store the value “2nd October 2007 at 13:45.30.123456789 +02:00”.
Let’s get the current OffsetDateTime with 2 hours of offset from GMT/UTC:
ZoneOffset zoneOffSet= ZoneOffset.of("+02:00"); OffsetDateTime date = OffsetDateTime.now(zoneOffSet);
OffsetTime is an immutable date-time object that represents a time, often viewed as hour-minute-second-offset, in the ISO-8601 calendar system, such as 10:15:30+01:00.
This class stores all time fields, to a precision of nanoseconds, as well as a zone offset. For example, OffsetTime can store the value “13:45.30.123456789+02:00”.
Let’s get the currentOffsetTime with 2 hours of offset:
ZoneOffset zoneOffSet = ZoneOffset.of("+02:00"); OffsetTime time = OffsetTime.now(zoneOffSet);
Getting back to the focal point, ZoneOffset is a representation of time-zone in terms of the difference between GMT/UTC and the given time. This is a handy way of representing time-zone, although there are other representations also available.
Moreover, ZoneId and ZoneOffset are not only used independently but also by certain DateTime Java classes such as ZonedDateTime, OffsetDateTime, and OffsetTime.
As usual, the code is available in our GitHub repository.<|endoftext|>
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Since The 1960’s, Plastic In The North Atlantic Ocean Has Tripled·
In 1957 it was a piece of trawling twine commonly used for fishing. In 1965 it was a plastic bag.
For more than 60 years, scientists in the U.K. have been collecting data on marine plastic, assembling one of the most comprehensive datasets on how much plastic has filtered into the North Atlantic ocean since it became a ubiquitous household item.
Publishing their findings in the journal Nature Communications Tuesday, the research team is the first to quantifiably confirm the drastic increase in ocean plastic since the 1990s.
How did they do it?
Calculating the amount of plastic in the ocean is difficult. Many researchers and publications refer to a 2015 study in the journal Science that estimated anywhere from 4.8 to 12.7 trillion pieces of plastic enter the ocean each year. The wide range is a result of mixed methods for calculating waste, looking at signs like waste habits, consumption, and recycling capacities.
Photo: Brian Yurasits<|endoftext|>
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The Antarctic krill is an essential part of the food chain in the Antarctic, serving as prey for many marine mammals and birds. How will rising ocean acidification affect this species?
New research has found that Antarctic krill would be largely unaffected by the higher levels of ocean acidification predicted for the next 100 to 300 years.
"Our study found that adult krill are able to survive, grow and mature when exposed for up to one year to ocean acidification levels that can be expected this century," said the study’s lead author PhD student Jess Ericson. Hailing from the Institute of Marine and Antarctic Studies at the University of Tasmania, she added that the long-term lab study was the first of its kind.
The findings of the study was published in the Communications Biology journal.
The study lasted 46 weeks, during which researchers reared adult krill in seawater in the lab at different pH levels, including present-day levels, levels predicted within 100 to 300 years and up to an extreme level.
Then, they measured various physiological and biochemical variables to find out how ocean acidification in the future might affect the krill’s survival, size, lipid stores, reproduction, metabolism and extracellular fluid.
It was discovered that adult krill could actively maintain the acid base balance of their body fluids as seawater pH levels decreased, thereby enhancing their resilience to ocean acidification.
As krill is an essential part in the Antarctic food chain, the findings on their resilience are significant.
"However, the persistence of krill in a changing ocean will also depend on how they respond to ocean acidification in synergy with other stressors, such as ocean warming and decreases in sea ice extent," Ericson cautioned.<|endoftext|>
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Aquaculture is the breeding, rearing, and harvesting of animals and plants in all types of water environments, including ponds, rivers, lakes, and the ocean. Aquaculture is used for producing seafood for human consumption; enhancing wild fish, shellfish, and plant stocks for harvest; restoring threatened and endangered aquatic species; rebuilding ecologically important habitats; producing nutritional and industrial compounds; and providing fish for aquariums. After harvests from wild fisheries reached a plateau in the mid-1980s, many parts of the world turned to aquaculture as a resource-efficient way to produce protein and meet demand for seafood. Globally, aquaculture already supplies half of all seafood produced for human consumption, and this percentage continues to rise.
Just as fisheries and agriculture are vulnerable to the impacts of climate change, so is aquaculture. However, the unique locations and practices of aquaculture can provide some level of climate resilience for the world’s food production systems. For instance, ocean-based aquaculture operations free up the land and fresh water that would be required to produce an equivalent amount of food through terrestrial agriculture. Additionally, aquaculture operations are safe from tornados, droughts, floods, and other land-based extreme events that may increase with climate change. Aquaculture also serves as an economic resilience-building strategy: it has emerged as an alternative ocean-based livelihood for fishermen whose work has changed due to negative impacts on wild fish stocks.
Projections for how aquaculture will respond to climate change vary. In some regions, warming waters may result in increased occurrences of harmful algal blooms and pathogens, and these events can be particularly detrimental for cultivation of shellfish such as oysters, mussels, and clams. However, in tropical and subtropical regions, projections indicate that ocean water temperature will remain within the optimal range for most cultured species. In these regions, warming could result in faster growth and increased regional production of cultured stocks.
Ocean acidification poses a significant risk to shellfish aquaculture, as young shellfish are less able to grow shells as the pH of their environment decreases. The impact of ocean acidification is already being felt in shellfish hatcheries in the Pacific Northwest. Hatcheries are responding to reduce their vulnerability through smart site selection, improved animal health programs, species selection, selective breeding, advanced animal nutrition, and other husbandry approaches.
Aquaculture also has the potential to play a role in reducing global climate change and its impacts. In a process referred to as bioextraction, seaweeds and filter-feeding shellfish take up carbon dioxide and nutrients from their environment, improving water quality as they grow by removing dissolved acid, nitrogen, and phosphorus. In Puget Sound, a collaborative group has undertaken an effort to farm seaweed to help mitigate ocean acidification. Seaweeds also give off oxygen, which can improve water quality in low-oxygen dead zones.
Another aquaculture strategy—the use of conservation hatcheries—can be used to help save or rebuild wild aquatic species or populations that have been depleted. For example, efforts to raise captive broodstock to rescue Snake River sockeye salmon from near-extinction have produced millions of eggs and fish for reintroduction.
As ocean conditions continue to change, aquaculture has an increasingly important role in maintaining the production and availability of seafood for the world’s growing population.
Excerpted and abridged from the following sources:
- De Silva, S.S. and D. Soto, 2009: Climate change and aquaculture: potential impacts, adaptation and mitigation. Climate Change Implications for Fisheries and Aquaculture: Overview of Current Scientific Knowledge, K. Cochrane, C. De Young, D. Soto, and T. Bahri, Eds., FAO Fisheries and Aquaculture Technical Paper, No. 530, Food and Agriculture Organization of the United States, 151–212.
- National Oceanic and Atmospheric Administration, National Marine Fisheries Service, Office of Aquaculture, accessed August 2016: Welcome to the Office of Aquaculture.<|endoftext|>
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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
Grade 5 - Mathematics5.37 Fractions Review
A fraction is another way of expressing division. The expression x/y is also written as x ÷ y. x is known as the numerator and y is known as denominator. A fraction written as a combination of a whole number and a proper fraction is called a mixed fraction or mixed number. To generate equivalent fractions of any given fraction, we proceed as follows: Multiply the numerator and denominator by the same number (other than 0) or Divide the numerator and denominator by their common factor (other than 1), if any. The two fractions are equivalent if the product of numerator of the first and denominator of the second is equal to the product of the denominator of the first and numerator of the second. Reducing fractions: To reduce a fraction, divide both the numerator and denominator by the largest factor of both. A fraction is in its lowest term, if the numerator and denominator have no common factor other than 1. Comparing fractions: Of the two fractions having the same denominator, the one with greater numerator is greater. Of the two fractions having the same numerator, the one with greater denominator is smaller. Operations with Fractions: To add or subtract unlike fractions, first we convert them into like fractions and then perform the required operation on like fractions so obtained. The product of two or more fractions is a fraction whose numerator is the product of their numerators and whose denominator is the product of their denominators. When the product of two fractions or a fraction and a whole number is 1, then either of them is called the reciprocal of the other. The number zero (0) has no reciprocal. Dividing a fraction or a whole number by a fraction or a whole number (other than zero) is the same as multiplying the first by the reciprocal of the second. Directions: Answer the following questions.
Q 1: In the fraction, 4/3, __ is the numerator.Answer: Q 2: Which of these is an equivalent fraction of 5/9 - 25/35, 15/45, 25/45, 5/25Answer: Q 3: Find the difference 1/2 - 1/8Answer: Q 4: Which is the greatest and the least among these fractions - 1, 12/5, 2/3Answer: Q 5: Find the value of y in y/3 + 4/3 = 2Answer: Q 6: What is one-third of 24?Answer: Q 7: Convert 1.3 to fraction.Answer: Q 8: 1/4 + 13/4 = ____Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!<|endoftext|>
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# Proportions on the GRE
Proportions are extremely common on the GRE. If you don’t have a strong grasp of them, and you are busy trying to figure out combinations/permutations or probability, stop. Focus your attention on mastering proportions before moving on to more tertiary concepts.
So let’s start basic. Proportions can be broken up into two groups: direct proportions and indirect proportions. In this post I am going to focus on direct proportions. They are more intuitive than indirect proportions and are also the more common on the GRE.
## Direct Proportion
Below is an example of a direct proportion:
To solve for x we cross multiply, giving us: ; .
This is a direct proportion because as the ‘5’ becomes larger (namely it quadruples to become 20) the x also gets larger (it quadruples to become 8). That is both sides are getting larger.
I can almost guarantee that you will not see such a straightforward equation on the GRE Quant section. Instead, you will be given either a word problem or a graph and you will have to translate the information into an equation like the one above.
Let’s take a look at two problems:
In 2004, 2,400 condos sold, 15% of the total housing units sold that year. If 25% of the homes sold in 2004 were four-bedroom houses, then how many four-bedroom homes sold in 2004?
(A) 3,600
(B) 4,000
(C) 4,200
(D) 4,800
(E) 6,000
Explanation
Here we want to set up an equation. ; ; . (B).
Some things to note: you can take off the last two zeroes in 2,400 to get 24, a step which will make the math easier. Remember to bring the two zeroes back, which makes sense: x =40, is clearly too low and not amongst the answer choices.
Speaking of answer choices, notice that 25% is less than double of 15%. Therefore, the number of four-bedroom houses sold has to be less than twice the number of condos sold. (D) and (E) cannot be answer. (A) 3600 is only 50% greater than 2,400, so it is probably too low as well. Elimination, esp. if you are short on time, or getting tangled up in the calculation can be very effective.
Now let’s put a spin to this question. Nothing too tricky; indeed you should be able to solve this using the method above.
In 2004, 2,400 condos sold, 15% of the total housing units sold that year. How many units sold in 2004 were not condos?
(A) 16,000
(B) 13,600
(C) 12,400
(D) 11, 200
(E) 8,600
Explanation:
To approach this question, a good idea is to find the number of units in the total market. Then to find the number that are not condos subtract the condos from the total.
My reason for this approach is it is easier to do the math when we are working with 100 vs. 85 (which would be the percent of home that are not condos).
The solution is as follows:
; . Remember to add the two zeroes: 16,000. Now we have to subtract the total condos (2,400) to find the number of units that are not condos: 16,000 – 2,400 = 13,600.
## Takeaway:
Setting up a proportion is essential to solving a range of GRE math questions. Make sure you can confidently and quickly solve this question type before more on to more challenging – but less common – concepts.<|endoftext|>
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The origins of the Thracians, and thus the Swedes, can be traced back to secular and biblical history. Chapters 9 and 10 of Genesis describe how the nations developed from Noah's three sons, Shem, Ham, and Japheth. Recorded history continually verifies the biblical account of the spread of nations.
The Genesis account, as a historical document, is fully corroborated by an overwhelming richness of documentary and other historical evidence so vast that it is unique in recorded history. No other manuscript enjoys such a wealth of detailed corroboration from such a wide-ranging variety of sources. The Indo-European peoples were all too aware of their historic and ethnic descent from the line of Japheth, Noah's 3rd son. These peoples, through carefully preserved records, could trace their lineage and race back to the time of Babel and the dispersal of the nations from the plain of Shinar. Noah's flood is generally agreed to have occurred about 2350 BC, and from here we find the beginnings of nations and empires (Egypt, Persia, Greek, etc). The evidence is striking from those early post-Flood nations of the Mesopotamian valley, who had direct contact with one another, and who preserved in written records those names that are explicitly mentioned in the Genesis record.Japheth is considered the father of Indo-European people groups (several royal European genealogies confirm this). Japheth's 7th son, Tiras, was the progenitor of the Tiracians. Historians note they probably first settled in the area of Asia Minor (present day Turkey) about 1900 BC. The transfer of words through nations and languages is prevalent in every people group. Merenptah of Egypt, who reigned during the 13th century BC, provides us with what is so far our earliest reference to the people of Tiras, recording their name as the Tursha (or Tarusha), and referring to them as invaders from the north. Herodotus (425 BC Greek historian) wrote: "The Thracian people are the most numerous of the world; the Thracians have several names, according to their specific regions, but their habits are more or less the same...and only their chronic disunity prevented them from being the most powerful of all nations."
The records that have come down to us lend their weight to the already vast body of documentary evidence that can only convince us that the Genesis record is a true and faithful historical account of the early history of mankind. What is remarkable about these records is that they mostly come from ancient historians and writers of various nationalities who had not the least intention, either consciously or otherwise, of lending support to the Genesis record. Most of them were nurtured within pagan systems that were openly antagonistic to the knowledge of God, and who had labored over many centuries to darken, if not totally erase that knowledge altogether. Their verification is therefore all the more valuable.
History attests that they were indeed a most savage race, given over to a perpetual state of "tipsy excess", more likely to be in battle than laying in their beds. They are also described as a "ruddy and blue-eyed, people", fighting with their own tribal factions. In the 3rd century BC, the Thracians were noted as having numerous tribes that rarely united, most having their own kings. Thracian dress was well known. Several descriptions were given, including illustrations on Greek vase paintings. Basic dress was tunic, cloak, cap and boots. Thracian warriors carried a shield and spears, plus a small sword (dagger) as a secondary weapon. Their mode of dress and armaments continued with their descendents, the Vikings, though modified. Thracians are mentioned by many rulers in the region they lived. After the Greek victory over the Persians (449 BC), the Persian king Xerxes (486-465 BC) established for himself a large army among whose soldiers Herodotus mentions Thracians from northwest Asia Minor, who are described as follows:
"The Thracians joined the expedition wearing fox caps, wearing long coats under their vivid colored capes. Their calf-high footwear was made of deerskin. They were equipped with spears, light shields and small daggers."Josephus (1st century AD Jewish & Roman historian) identifies them as the tribes who were known to the Romans as Thirasians, and to the Greeks as Thracians, whom they feared as marauding pirates. Dio Cassius, Roman historian in the 2nd century AD, wrote "let us not forget that a Trajan was a true-born Thracian."
Tiras himself was worshipped by his descendants as Thuras (Thor), the god of war. The river Athyras was also named after him, and the ancient city of Troas (Troi, Troy - the Trajans or Trojans) perpetuates his name, as also does the Taunrus mountain range. Thracian lands stretched from southwestern Europe to Asia Minor, a vast area historically known as Thracia. The historical Thracian genealogical tree counts over 200 tribes which had several names, according to their specific regions. Some of their tribal names were Trajans, Etruscans, Dacians, Luwians, Ramantes, Pelasgians, Besins, Odrisi, Serdoi, Maidoi and Dentheletoi. The Trajans (Trojans) founded the city of Troy which existed approximately 2400 years (about 1900 BC to 500 AD), which was destroyed and rebuilt several times. Thousands of Trojan warriors left the city of Troy during the 11th century BC. They came north and captured land along the banks of the river Don (southwestern Russia), a major trade route. The locals named the Trojan conquerors the "Aes," meaning "Iron People," for their superior weaponry. The tribes of Trojan Aes would eventually move north, settling in present-day Scandinavia. The Aes or Aesar (plural), subsequently became known as the Svear, and then Swedes. Historians refer to the Aes people as "Thraco-Cimmerians" due to their Trojan ancestry. Other tribes of Thracians remained a culture in Asia Minor and southern Europe until the 5th century AD. Many present-day Bulgarians claim to be direct descendants of ancient Thracians (different from the Slavs who arrived that region in the 6th century AD).
The name Tiras perpetuated through different languages, as in this list of names from Noah to the present-day Swedes.
Japheth's first born son was Gomer. Gomer is perpetuated through the names of Gamir, Gimmer, Gomeria, Gotarna and Goth. The tribes of Gomer are mentioned by the Jews in the 7th century BC as the tribes that dwelt in the "uppermost parts of the north". The Assyrians in the 7th century referred to them as the Gimirraya. Other names used throughout history include Gimmerai, Crimea, Chomari, Cimmer, Cimmerian. Cimmerians populated areas of the north of the Caucasus & Black Sea in southern Russia. Linguistically they are usually regarded as Thracian, which suggests a close relationship. "Thraco-Cimmerian" remains of the 8th-7th century BC found in the southwestern Ukraine and in central Europe are associated with the Aes people.
Back to History<|endoftext|>
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Some issues in implementing CLIL
Carmel Mary Coonan
In this short article the intention is to highlight certain issues that are arising from the application of CLIL in “new” situations.
The concept of CLIL
The meaning of the concept CLIL (content and language integrated learning) refers to the integrated learning of language and content in situations of bilingual or multilingual education, in other words, in situations in which there are two (the historically traditional meaning) or even more vehicular languages of instruction (1) - one being the normal language of the school, the other(s) an L2 or LS (2). Other terms (in English) used are, to name but a few, “content based language learning”, “content based language instruction”, and “language enhanced content teaching” (see, for example, Nikula 1997; Wolff 1997). As Nikula states, the term CLIL “is broad enough to cover both immersion education where all instruction is conducted through a foreign language and other types of foreign language enhanced education where students only receive certain parts of their education through the medium of a foreign language” (1997:6).
Importantly, the term CLIL captures a feature of bilingual education that necessitates attention: namely that language development (in the L2) cannot be left to chance. It needs to be nurtured. Without such attention the risks are that i) the learning of the non-language subject matter will suffer due to L2 difficulties; ii) L2 language competence will not grow. As the choice of bilingual education is generally made in order to promote the L2 language (and culture), it would be paradoxical if this were not to occur (Snow et al. 1989; Swain 1985).(3) Thus CLIL can be seen as a concept that highlights the need to find the methodological means to reach two types of objectives, those of the content and of the L2, the one through the other. Only then will the language learning potential inherent (in terms of greater exposure to the language and better conditions (4) for language learning) in bilingual education programmes be released.
Change in focus
In the situation delineated very briefly above, the L2 is the vehicle whereby the content objectives are reached. In other words, the L2 carries out the same role as that of the normal language of the school as one or more school subjects are taught through it. However, such a situation legally requires a teacher fully qualified in the L2-mediated school subject, which is not always possible. Most higher education systems in Europe do not allow for a double teaching qualification in subjects that are not related. How then are countries, convinced of the value of CLIL learning, sidestepping the issue? We can take Italy as our point of reference. Two features will be mentioned and conclusions will be drawn about possible risks inherent in the choices.
a. Team teaching
One solution is to work in partnership: two teachers (the content teacher and the language teacher (5)) are physically present during the lesson. Ideally, the role of the L1 subject teacher is to teach the content in L2 and that of the L2 teacher to intervene to smooth out language problems and help in conducting group work, etc. However, in practice what are the risks? On the basis of a survey conducted in some high schools (6) (Pavesi and Zecca 2001), it would appear that the co-presence of the L1 content teacher and the L2 native speaker (language expert) can have potentially deleterious effects on the amount of time assigned to the use (by teacher and pupils alike) of the L2.
The survey found an overall tendency to present subject matter first in the L1 and for the students to use the L1 when dealing with content they found difficult. Basically speaking, the L1 tends to be used with (and by) the subject teacher and the L2 to be used with (and by) the L2 teacher. The reason might lie in the well-defined language role each teacher has institutionally. The need for the L1 subject teacher to “change” his/her language is felt less as there is another teacher there that can carry out that role. An overall danger in such a situation is that content is not mediated through the L2 to a sufficient degree for there to be the quantitative and qualitative benefits inherent in L2 vehicular learning. Furthermore, there is the risk that the L2 interventions be limited to a mere focusing on the language itself (in fact the job of the language expert co-present). Strictly speaking therefore the L2 is reduced in its vehicular function but amplified in its teaching function.
b. The L2 teacher
In our view, the root of the above problem lies in an inadequate overall preparation for CLIL teaching. Professional development courses to prepare the teachers for the specifics of the CLIL methodology are few and far between – both in Italy and other European countries. However, even when such courses are offered, they do not manage to reach the subject teacher.(7) Those who rally are the L2 teachers eager to find out about this new language-learning environment. Some route must be found (subject specific journals, initial teacher education authorities, etc.) to capture the attention of the subject teachers, inform them of CLIL and prepare those interested.(8) As things stand at the moment, in certain areas of the country there are groups of L2 teachers keenly involved in developing CLIL experiences – perhaps with subject specialist colleagues at school.
However, in parallel to this, there is a new development: the L2 teacher is practising CLIL. Profiting from the new law reform on school autonomy which assigns a greater degree of freedom to schools in managing their curriculum, the teachers, working closely with their subject colleagues, develop thematic modules of a disciplinary or interdisciplinary nature. Legally speaking the L2 teacher cannot shoulder responsibility for the discipline but can take on board contents which represent a further exploration or extension of what is done in L1 with the subject teacher. There can be potential problems with such a choice:
However, in order to maintain the qualitative dimension inherent in CLIL (learning content and learning to learn through a L2) there is a need to guarantee that the student be involved, in L2, in the cognitive processes and the learning and study activities normally associated with the content. Thus, the authenticity and the meaning-based features of the content-based experience do not give way to the “pseudo-ness” so typical of the language classroom (see, for example, Wolff 1997:61-62).
Marsh, D. et al. (eds) 1997. Aspects of Implementing Plurilingual Education: Seminar and Field Notes. Jyväskylä: University of Jyväskylä Continuing Education Centre.
Nikula, T. 1997. Terminological considerations in teaching content through a foreign language. In: D. Marsh et al. (eds), Aspects of Implementing Plurilingual Education, 5-8.
Pavesi, M. and M. Zecca. 2001. La lingua straniera come lingua veicolare: un’indagine sulle prime esperienze in Italia , CILTA, 1, 31-57.
Snow, M. et al. 1989. A conceptual framework for the integration of language and content in second/foreign language instruction, TESOL Quarterly, 23, 2, 201-217.
Swain, M. and S. Lapkin. 1982. Evaluating Bilingual Education: A Canadian Case Study. Clevedon: Multilingual Matters.
Swain, M. 1985. Communicative competence: Some roles of comprehensible input and comprehensible output in its development. In: S. Gass and C. Madden (eds), Input in Second Language Acquisition. Rowley, MA: Newbury House, 235-253.
Wolff, D. 1997. Content-based bilingual education or using foreign languages as working languages in the classroom. In: D. Marsh et al. (eds), Aspects of Implementing Plurilingual Education, 51-64.
(1) For economy, “bilingual education” – the more traditional term - will be used to refer to situations where either two or more languages of instruction are used.
(2) Although the distinction is important for discussions of bilingual education, here we will use only the expression L2 to refer to both realities.
(3) The 1982 report by Swain and Lapkin and further work by Swain and associates highlight the need for greater attention to form in immersion situations.
(4) See, for example, Wolff 1997 for his arguments concerning the qualitative merits of immersion/content-based bilingual education /CLIL.
(5) In the Italian situation concerning the vehicular use of an L2, generally speaking such a teacher is the “language expert”, a native speaker used to working in team – but in a subordinate role - with the “real” L2 teacher.
(6) The schools involved in the survey were those that are part of two Ministerial projects: the Liceo (Classico) Europeo and the Liceo (ad indirizzo) internazionale. All in all there are about 20 such high schools.
(7) Over the past four years the Department of Linguistic Sciences at the Università di Ca’ Foscari has offered a year-long post-graduate course for teachers (future teachers) on the topic of CLIL but has had little success in attracting non-language subject teachers.
(8) IRRE Piemonte has set up a project for the teaching of science subjects through an L2. In order to participate the L2 teacher must be twinned with a science subject colleague.
(9) (English) for Specific Purposes.
(10) Indeed the acronym CLIL could refer to any learning situation where there is an integration of content and language. In the foreign language teaching field today there is an overall tendency to integrate L2 with content in order to expose the students to a more sophisticated register of language as well as the professional features of the language associated with the specialisation of the school they attend. Thus in a professional high school that prepares for jobs in the tourist industry, students will be exposed to themes and language associated with diverse aspects of the profession.
ELC Information Bulletin 9 - April 2003<|endoftext|>
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# How do you differentiate f(x)=5(3x^2+1)^(1/2) (3x+1) using the product rule?
Nov 9, 2015
Product rule:
if $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$, then $f ' \left(x\right) = g ' \left(x\right) \cdot h \left(x\right) + h ' \left(x\right) \cdot g \left(x\right)$.
Here, $g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$ and $h \left(x\right) = 3 x + 1$.
Now you need to differentiate $g \left(x\right)$ and $h \left(x\right)$.
$h \left(x\right) = 3 x + 1$
$h ' \left(x\right) = 3$
$g \left(x\right)$ is the complicated one because there, you will need the chain rule.
$g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}} = 5 u {\left(x\right)}^{\frac{1}{2}}$ where $u \left(x\right) = 3 {x}^{2} + 1$.
With the chain rule,
$g ' \left(x\right) = 5 \cdot \left(\frac{1}{2}\right) \cdot u {\left(x\right)}^{- \frac{1}{2}} \cdot u ' \left(x\right)$
$= \frac{5}{2} \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}} \cdot 6 x$
$= 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}}$
$= \frac{15 x}{3 {x}^{2} + 1} ^ \left(\frac{1}{2}\right)$
So, we have
$g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$g ' \left(x\right) = 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}}$
Last thing left to do is to apply the product rule:
$f ' \left(x\right) = g ' \left(x\right) \cdot h \left(x\right) + h ' \left(x\right) \cdot g \left(x\right)$
$= 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}} \cdot \left(3 x + 1\right) + 3 \cdot 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$= \frac{15 x \left(3 x + 1\right)}{\sqrt{3 {x}^{2} + 1}} + 15 \sqrt{3 {x}^{2} + 1}$
$= \frac{15 x \left(3 x + 1\right)}{\sqrt{3 {x}^{2} + 1}} + \frac{15 \sqrt{3 {x}^{2} + 1} \cdot \sqrt{3 {x}^{2} + 1}}{\sqrt{3 {x}^{2} + 1}}$
$= \frac{45 {x}^{2} + 15 x + 15 \left(3 {x}^{2} + 1\right)}{\sqrt{3 {x}^{2} + 1}}$
$= \left(90 {x}^{2} + 15 x + 15\right) \cdot {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$= 15 \left(6 {x}^{2} + x + 1\right) \cdot {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
I hope that this helped!<|endoftext|>
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Pictograph and Tables in Data Handling | Grade 1 | ORCHIDS
Data handling
# Data in Table for Class 1 Math
This discussion is to inform the students about the data handling and how to arrange data tables. Arranging data in table is the simplest way to represent data easily. From this concept the students will get to know about data and data table.
Also, the student will learn to
• Summarise and compare data in a planned way
• Express data using appropriate picture to make a pictograph
Each concept is explained to class 1 maths students with examples and illustrations, and a concept map is given at the end to summarise the idea. At the end of the page, two printable Pictograph worksheets with solutions are attached for students to practice. Download the worksheets and assess your knowledge.
What Is Data Handling?
We have learned about Picture graphs. Now let’s learn about handling data using the table.
## What is the difference between pictograph and table?
• Pictograph is the representation of the data in the form of pictures or symbols.
• We can arrange data using a table.
Data Table
• To create a table, we have to first collect the data, organize it properly, and then represent it in the form of a table.
• A table consists of rows and columns.
### Examples:
Table format looks like this:
Rows run horizontally and columns run vertically down. When rows and columns are put together, they make a table.
• Each box of a table is called a cell.
• ### Example:
• In the first-row, first column cell, label the objects that you want to represent in the table. And the first cell of the second column is used to indicate the number of objects or numeric values in the data.
## Why table is a good way to represent the data?
Let’s see an example.
### Examples:
Rahul’s mom asked him to bring some fruits from the market.
To remember the number of fruits, Rahul makes a list using table.
He made a table like this:
• In the first column, he mentioned the names of the fruits and in the second column, he mentioned the number of that fruit.
• From the table, Rahul can easily remember the number of the fruits and bring the correct number of fruits from the market.
A table helps us to interpret the data easily.
## How to make a table from a pictograph?
### Examples:
Here in the pictograph, the information about the number of students absent in a class is given.
Now, from this pictograph make a table.
• Step 1: Decide the labels of the columns.In the first column of the pictograph, days are mentioned. So, in the table first column will be labelled as ‘Days’. In the second column of the pictograph, symbols for the number of children are given. So, the second column of the table will labelled as ‘Number of students absent’.
• The table will look like this:
• Step 2: By using the given key, find out the numeric data.The given key indicates that one symbol represents 2 children. Monday = 4 symbols (Add the number 2 four times) = 2 + 2 + 2 + 2 = 8 children Tuesday = 1 symbol = 2 children Wednesday = 3 symbols (Add the number 2 three times) = 2 + 2 + 2 = 6 children Thursday = 2 symbols (Add the number 2 two times) = 2 + 2 = 4 children Friday = 2 symbols (Add the number 2 two times) = 2 + 2 = 4 children
• Step 3: By using this numeric data complete the table.By putting these numbers in the table, it will look like:
So, from the pictograph, we can make a table.
• -<|endoftext|>
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# Solve Multi-Step Word Problems (Inequalities)
Related Topics:
Common Core for Mathematics
Lesson Plans and Worksheets for all Grades
Examples, solutions, worksheets, videos, and lessons to help Grade 7 students learn how to use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
B. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where pq, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid \$50 per week plus \$3 per sale. This week you want your pay to be at least \$100. Write an inequality for the number of sales you need to make, and describe the solutions.
Common Core: 7.EE.4b
### Suggested Learning Targets
• I can solve simple inequalities.
• I can solve two step linear inequalities of the form px+ q > r and px+q < r, where p, q, and r are specific rational numbers.
• I can solve word problems (two-step linear inequality problems) with rational coefficients (ex, px +q = r).
• I can graph the solution sets of inequalities.
• I can interpret the solution sets of the inequality.
Solving Inequalities by Adding or Subtracting When you add or subtract the same number from each side of an inequality, the inequality remains true.
Examples:
1. Solve -21 ≥ d - 8. Check your solution.
2. Solve y + 5 > 11. Check your solution.
3. Kayla took \$12 to the bowling alley. Shoe rental costs \$3.75. What is the most he could spend on games and snacks?
Solving Inequalities Using Addition and Subtraction
Examples:
Solve and graph the solutions.
1. t + 3 < 10
2. x - 2.2 ≥ 6.8
3. 5 > 7 + V
### Solve Inequalities with Multiplication and Division
When you multiply or divide by a negative, the inequality symbol must flip
Solving Inequalities by Multiplying and Dividing
### Solve 2-Step Inequalities
Solving 2-step inequalities
How to solve and graph 2-step inequalities? Solving Two - Step Linear Inequalities
Here we solve a few inequalities that require addition / subtraction and multiplication / division. Solving Two-Step Inequalities
### Inequality Word Problems
How to Translate Word Problems Into Multistep Inequalities?
In this lesson, we'll go over some word problems, which, in order to find their solution, require us to translate them into (multi-step) inequalities -- inequalities which take more than one step to solve. Writing An Inequality From A Word Problem
The elevator of a local school has a maximum weight capacity of 2,000 pounds. If there are already 3 people on the elevator that have a combined weight of 460 pounds, write an inequality expressing the possible weights that may be added to the elevator. Use "w" to represent the weight. Example:
When John bought his new computer, he purchased an online computer help service. The help service has a yearly fee of \$25.50 and a \$10.50 charge for each help session a person uses. If John can only spend \$170 for the computer help this year, what is the maximum number of help sessions he can use this year?
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.<|endoftext|>
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Generally speaking, paint has 3 components: pigment, binder and solvent. In some instances, additives are used to improve the paint’s specific properties depending upon its intended use.
Pigments are simply the color chemical (typically a metal compound) in paint and is typically a solid. The color is derived from the chemical’s ability to absorb and reflect light waves. The binder is a chemical causing the paint particles to stick to each other and the surface upon which someone is painting. Traditionally, binders were made from natural oils but now it is common to see binders made from synthetic plastics. The solvent component of paint is the substance that makes the pigment-binder a thinner and less viscous liquid. Water is a common binder but so too are various petroleum based chemicals. The solvent will evaporate after the paint dries.
In layman’s terms, you can classify household paints as either exterior paints or interior paints. Exterior paints have pigments able to withstand extreme weather conditions. Furthermore, interior paints can be grouped into either water-based or oil-based paints. There are a variety of ways you can analyze paints to see if there is a match.
The first, most obvious and probably least precise, is the “naked eye.” A careful description of features and properties should be noted. In some instances, a visual inspection can be used to determine that there is not a color match. That’s fine but unreliable when colors are very similar as this test is dependent upon the color sensitivity of the observer. In one study reviewed for this essay, it was suggested that a microspectophotometer could be used to discriminate household paints closely similar in appearance. In this analysis, the microspectophotometer allows for the comparison of the visible reflectance spectra of the paint. While this tool can help determine whether paints are the same color, other tests will help determine the chemical composition of the paints: micro-Raman Spectroscopy; Infrared Spectroscopy; Pyrolysis Gas Chromatography; Scanning Electron Microscopy; and Solubility Tests.
The micro-Raman spectroscopy is used to determine pigment/dye content in paint samples. This is a non-destructive test that uses lasers which will interact with the vibrational frequencies of the paint molecules.
Infrared Spectroscopy is an effective tool used to characterize paints in terms of their organic and inorganic component. Here, we can measure how the paint molecules react with infrared light. This can help determine the binder used with a particular paint. Similarly, with a solubility test, the chemicals used in a paint can be determined by what substances they will or will not completely dissolve into to.
Pyrolytic Gas Chromatography is a destructive technique that can be used for binder classification and comparison. In this procedure, the sample is heated until it breaks down and smaller molecules are separated. The data from this process can be used to identify the materials found in the paint sample.
Scanning Electron Microscopy allows for the characterization of the paint sample’s morphology and its elemental composition. In this test, emitted x-rays provide information on compositional elements. Also, the electrons emitted can provide additional info on the paints topography.
PartnerAndré Bélanger, a graduate of Loyola University, is a highly-respected criminal defense attorney serving the people of Baton Rouge, Louisiana, and surrounding areas of Ascension Parish and New Orleans, for one of Baton Rouge’s top-25 law firms. In his 15 years of practicing law, Mr. Bélanger has handled thousands of criminal cases at both the pre-trial and trial stage, including approximately 200 trials. This trial experience includes homicide defense and prosecution, large drug conspiracies and fraud cases making Bélanger one of the few attorneys capable of handling even the largest, most complex federal cases. https://manassehandgill.com/andre-belanger/<|endoftext|>
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Monte Carlo Simulation
From SKYbrary Wiki
Monte Carlo Method
Monte Carlo Simulation, sometimes referred to as the Monte Carlo method, is a computerized mathematical technique that allows risk to be accounted for in quantitative analysis and decision making.
Risk analysis is an important part of almost every decision. However, many of those decisions are made in the face of uncertainty, ambiguity, and variability. Even though data and information upon which to make the decision might easily be available from multiple sources, the future cannot accurately be predicted and the ultimate outcome of the decision is still an unknown quantity. Monte Carlo simulation allows generation of all the possible outcomes of the decision before it is made, thus allowing the assessment of the impact of risk and allowing for better decision making in the face of uncertainty.
Monte Carlo simulation is a computerized mathematical technique that enables risk to be accounted for in quantitative analysis and decision making. A Monte Carlo simulation will provide the user with a range of possible outcomes and the probability of occurrence for each choice of action. In other words, it will show the potential consequence of both the most aggressive and the most conservative decision as well as providing the corresponding data for any "middle of the road" decision between the two extremes.
Early use of Monte Carlo simulation was made by scientists of the Manhattan Project - the development of the first atomic weapons during WWII - to help predict neutron penetration when they were investigating radiation shielding. Since then, it has been used in many applications in widely diverse fields such as finance, project management, energy, manufacturing, engineering, research and development, insurance, oil & gas, transportation, and the environment.
How Monte Carlo Simulation Works
A Monte Carlo simulation performs a risk analysis of any chosen decision factor that has inherent uncertainty. It does this by building models of possible results using a probability distribution; that is, by substituting a range of values for any specific factor chosen by the decision maker. It then calculates the results numerous times, each time using a different set of random values from the probability functions. Depending upon the number of factors which are considered "uncertain" and the range of possible values specified for each of them, a Monte Carlo simulation could involve thousands, or even tens of thousands, of recalculations. The final products of a Monte Carlo simulation are distributions of possible outcome values.
By using probability distributions, variables can have different probabilities of different outcomes occurring. The probability distribution chosen for the simulation will depend upon the type of problem under investigation. Common probability distributions include:
- Normal (bell curve) - The user defines the expected value (mean) and a standard deviation to describe the variation about that value. The distribution is symmetrical and values in the middle (near the mean) are most likely to occur
- Lognormal – Values are not symmetrical as is the case in a Normal distribution. Rather, they are positively skewed. A Lognormal distribution is used to represent values that will never fall below zero but have unlimited positive potential
- Uniform – All values have an equal chance of occurring. The user defines the minimum and maximum values
- Discrete – The user defines a set of specific values that may occur as well as the likelihood of each
During a Monte Carlo simulation, values are sampled at random from the chosen probability distributions. Each discrete sample set is referred to as an iteration and the resulting outcome from the calculations for that sample is recorded. A Monte Carlo simulation will repeat this process hundreds, thousands or even tens of thousands of times depending upon the complexity of the problem. The compiled results from all of the iterations is a probability distribution of the possible outcomes that lie within the parameters chosen by the user. A Monte Carlo simulation thus provides a comprehensive view of not only what could happen, but how likely it is to happen.<|endoftext|>
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Parasitic worms that go from the soil into humans and pet hosts may soon receive their eviction notices thanks to progress by UO researchers.
Testing potential new drugs that target the worms, which cause serious health problems in as many as 3 billion people mostly in economically poor regions lacking adequate sanitation, is now possible through the adaption of a device invented at the UO for other research.
UO biologist Janis Weeks led the transformation under a grant awarded to her in 2013 by the Bill & Melinda Gates Foundation's Grand Challenges Explorations program. The original device was designed to study the central nervous system in small organisms, particularly the roundworm Caenorhabditis elegans.
Many people carry one or all three of the common soil-transmitted worms Ascaris, whipworm and hookworm. They also infect livestock and pets.
In a new paper, Weeks and colleagues confirmed that the tiny microfluidic chip allows for the screening of potential compounds or natural products for new treatments in up to eight live parasitic worms at a time. During testing, the worms are maintained in fluids similar to the environment experienced by larvae in their infective stage.
"I've been working in Africa for 20 years teaching neuroscience," Weeks said. "I've seen the impacts of parasitic infections first-hand. I was working on C. elegans, and we had shown the connection of electrical signals to contractions in the worm's pharynx. I thought our device had possibilities for looking at parasitic worms so I developed a proposal for a Gates Foundation grant."
Her group's findings were published in December in a special issue of the International Journal for Parasitology: Drugs and Drug Resistance. Preliminary findings were presented previously at a conference in San Diego that devoted to the fight against the worms.
After contact with soil, the worms' larvae enter the body via the skin or ingestion. They thrive in the intestines, daily producing thousands of eggs. Their presence leads to such problems as anemia, malnutrition, vitamin-A deficiency, abdominal swelling, weight loss, diarrhea and intestinal inflammation. The effects are particularly damaging in children, whose physical and cognitive development can be stunted.
When the worms are in the device, electrical monitoring — similar to a human electrocardiogram — captures the signaling between neurons and muscles as the throat rhythmically pumps while feeding. When potentially disruptive compounds are injected, researchers can determine which ones disrupt the pumping to starve and eventually kill the worms.
Weeks and UO colleague Shawn Lockery, a biologist and co-author on the paper, initially developed the device for noninvasive studies of C. elegans, a widely used model organism. Four of the paper's eight co-authors own equity in NemaMetrix Inc., which now holds the commercial license for that device. The UO owns the intellectual property and has a patent pending.
"We need new drugs against these soil-transmitted parasitic infections," said Weeks, a member of the UO's Institute of Neuroscience and African Studies Program. "For most of the successful drugs that act on neurons or muscles we have had no way to assay for this property. Our device provides a really specific test for potential new drugs, and it is rapid."
There are shortcomings with therapeutics now used against infection, Weeks said. Among them is a growing threat of resistance. Each worm also requires separate drugs, which also pose health complications for pregnant women.
"All of the drugs used today to treat these infections in humans were initially discovered for use in animals," Weeks said. "We hope our new tool will help to speed the design and discovery of new drug treatments."
—By Jim Barlow, University Communications<|endoftext|>
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# The vertex angle of an isosceles triangle measures 42°. A base angle in the triangle has a measure given by (2x + 3)°. What is the value of x? What is the measure of each base angle?
Students were asked to answer a question at education and to express what is most important for them to succeed. One that response stood out from the rest was practice. People who are usually successful do not become successful by being born. They work hard and commitment their lives to succeeding. This is how you can accomplish your goals. shown below some question and answer examples that you may possibly make use of to boost your knowledge and gain insight that will help you to maintain your school studies.
## Question:
The vertex angle of an isosceles triangle measures 42°. A base angle in the triangle has a measure given by (2x + 3)°. What is the value of x? What is the measure of each base angle?
## Answer:
The base angles of an isosceles triangle are equal.
The value of x is 33, and the measure of each base angle is 69.
The given parameter are:
So, we have:
Collect like terms
Divide both sides by 4
Recall that:
Hence, the value of x is 33, and the measure of each base angle is 69.
Read more about isosceles triangle at:
They would probably hopefully guide the student handle the question by applying the questions and answer examples. Then may make some sharing in a group discussion and also study with the classmate in relation to the topic, so another student also experience some enlightenment and still keeps up the school learning.
READ MORE Pete's pest control charges a one-time evaluation fee and a monthly service fee. The total cost is modeled by the equation y = 150 + 55x. Which statement represents the meaning of each part of the function?<|endoftext|>
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Mesoamerica is a term used to describe the region roughly corresponding to modern day Mexico and Central America which was inhabited by various societies before Spanish colonization following the famous discovery by Christopher Columbus. The Aztec Empire was a triple alliance between the city states of Tenochtitlan, Texcoco and Tlacopan which dominated most of central Mexico as well as other territories in Mesoamerica from early 14th century till the Spanish conquest in 1521. The term Aztec is at times used to refer to the Nahuatl-speaking people of central Mexico. More specifically it describes the Mexica people, who build Tenochtitlan, the capital city of the Aztec Empire while some also include the inhabitants of its allies: the Acolhuas of Texcoco and the Tepanecs of Tlacopan. Know about the Aztec origin, rulers, conquests, religion and human sacrifice; as well as the rise and fall of their empire, through these 10 interesting facts.
#1 They were banished to an island after they sacrificed the daughter of a king
Aztlan is a legendary place, the location of which has not yet been identified. The Aztec people refer to Aztlan as their ancestral home. In Nahuatl, the language spoken by the Aztecs, the word Aztecah means “people from Aztlan” and that is from where the Aztecs get their popular name. The Mexica people were the last of Aztlan migrants to arrive in the Valley of Mexico around the year 1250 AD. The king of the city-state Colhuacan allowed them to settle in a place called Chapultepec. He later wanted to marry one of his daughters to a Mexica and appoint her to rule over their tribe. However, the Mexica instead sacrificed her to their patron god Huitzilopochtli and flayed her skin. Enraged, the king drove them out of his region. They were forced into an island in Lake Texcoco, a swampy region which no one had yet inhabited due to its difficult geography. There the Mexica people, or the Aztecs, saw an eagle nested on a nopal cactus which they interpreted as a sign from their God to build their city on that location. They called it Tenochtitlan.
#2 They build the marvelous city Tenochtitlan on one of the most difficult terrains
When the Aztecs founded the city of Tenochtitlan, it was a small swampy island in Lake Texcoco in the Valley of Mexico. They faced many challenges like creating a strong foundation for their buildings in the swampy land, bringing clean water to the city, connecting their city to the mainland, protecting it against floods and producing food for its rising population. However, despite these challenges, the Aztecs build on that location one of the greatest cities of the time. At its peak, Tenochtitlan was the largest city in the Pre-Columbian Americas. It covered an estimated area of 8 to 13.5 km2 and had an estimated population between 200,000 and 300,000. It held twice the population of London or Rome and was one of the largest cities in the world. When the Spanish arrived at Tenochtitlan, they were so mesmerized by its streets, causeways, canals, aqueducts, marketplaces, palaces and temples that they doubted whether it was real or were they in a dream.
#3 The Aztecs built one of the largest and most powerful empires in Mesoamerica
The Mexica founded the city of Tenochtitlan in 1325 AD. Initially it allied with and paid tribute to Azcapotzalco, the capital city of the Tepanec empire. In 1426, the Azcapotzalco king arranged the assassination of the Mexica ruler. The following year Tenochtitlan allied with the city-states of Texcoco, Tlacopan and Huexotzinco to wage war against Azcapotzalco. They emerged victorious in 1428. After the war, Huexotzinco withdrew. The other three city-states formed the Triple Alliance, with Tenochtitlan soon becoming the dominant power. The alliance waged wars of conquest and expanded rapidly after its formation. At its height, the Triple Alliance or the Aztec Empire controlled most of central Mexico as well as other territories. It covered 80,000 square miles and contained 25 million people in almost 500 towns and cities. It was one of the largest and most powerful empires in the history of Mesoamerica.
#4 The Aztec Empire was an informal empire which ruled by indirect means
Unlike European empires, the Aztec Empire ruled by indirect means and it didn’t claim supreme authority over the tributary provinces. The Aztecs left the local rulers of the conquered city states in power as long as they agreed to pay semi-annual tribute and supplied military forces to wage war when needed by the empire. There were two types of provinces in the empire: Strategic provinces, which aided the Aztec state with mutual consent and Tributary provinces, in which the obligations were mandatory rather than consensual. The provincial tribute system was overseen and coordinated not by the king but a separate official called the petlacalcatl. At the provincial level, the collection of tribute was overseen by an official called huecalpixque, whose authority extended over the lower-ranking calpixque.
#5 The Aztec emperor was known as the Huey Tlatoani or the Great Speaker
A ruler in the Aztec empire was called Tlatoani (“the one who speaks”). The city-states of the Aztec empire each had their own Tlatoani. The Mexica ruler of the Aztec Empire who governed from Tenochtitlan was called Huey Tlatoani (“great speaker”). The management of tribute, war, diplomacy and expansion were all under the purview of the Huey Tlatoani. While Huey Tlatoani was the ultimate authority and the external leader, there was also an internal ruler called Cihuacoatl (“female twin”), whose primary role was to to govern the city of Tenochtitlan. The Cihuacoatl was always a close relative of the Huey Tlatoani. Both these positions were not priestly but they did perform important ritual tasks. The Aztec Empire also had a four-member military and advisory Council which assisted the emperor in decision-making. The four members of the Council also served as military generals. At the time of succession, the emperor could only be selected from this Council. This was done to contain ambition among the nobility.
#6 Perhaps the greatest Aztec Emperor was Ahuitzotl
There were a total of 11 Huey Tlatoani during the 145 year reign of the Aztec Empire. Acamapichtli founded the Aztec imperial dynasty in 1375. Their fourth emperor Itzcoatl (r. 1428 – 1440) defeated the Tepanec empire and laid the foundation of the Triple Alliance. He was succeeded by Moctezuma I (r. 1440 – 1469) who consolidated the empire and organized the construction of the famous double aqueduct which supplied Tenochtitlan with fresh water. The Aztec Empire expanded rapidly under the leadership of their eighth ruler Ahuitzotl (r. 1486 – 1502), who is considered by many as the greatest military leader of Mesoamerica. His conquests opened up routes to the coastal areas leading to expansion of trade, and prosperity. He also oversaw many construction projects. The reign of Ahuitzotl was a golden era for the empire. The Aztec Empire reached its greatest extent under the reign of his successor Moctezuma II (r. 1502 – 1520), who was the ruler when the Spanish arrived in 1519. The last Aztec emperor was Cuauhtémoc, who was captured by the Spanish in 1521.
#7 The Spanish were initially welcomed by the Aztec emperor Moctezuma II
Hernán Cortés, a Spanish conquistador, arrived at Yucatán in present day Mexico in early 1519. He led a contingent of 11 ships carrying around 630 men. He used the strategy to pit native people against each other to conquer the region. After defeating the Tlaxacan and Cholula warriors by allying with other natives, he reached Tenochtitlan on November 8, 1519. The Aztec king of Tenochtitlan, Moctezuma II, welcomed Cortes and allowed him to stay in his city. After around six weeks, Cortes used an incident in which two Spaniards were killed as a pretext to take Moctezuma hostage. Cortes then ruled the city indirectly for several months. On May 20, 1520, the Spaniards under Pedro de Alvarado attacked unarmed Aztec nobles congregated at the Festival of Toxcatl and slaughtered thousands, including much of the leadership of Tenochtitlan. This led to a revolt in Tenochtitlan in which Moctezuma II was killed; and Cortes and his men were forced to leave the city.
#8 Most of the Aztec population was wiped out by disease following the Spanish conquest
After the revolt in Tenochtitlan, a smallpox outbreak hit the city killing more than 50% of the region’s population. After allying with or defeating the cities under Aztec control, Cortes began a siege of Tenochtitlan a year later. In May 1591, he attacked the city with 600 Spaniards and more than 50,000 warriors of native tribes. After a hard fought battle, Tenochtitlan fell on August 13, 1521 with its emperor Cuauhtémoc being captured. Subsequently, the Valley of Mexico was hit with two more epidemics, smallpox (1545–1548) and typhus (1576–1581), which wiped out more than 80% of the indigenous population of the region. Mexico City was built on the ruins of Tenochtitlan. A majority of modern day Mexicans are mestizos, a term used to describe a person of combined European and indigenous descent. The central emblem in the Mexican flag is based on the Aztec symbol for Tenochtitlan, the sight which inspired them to build their great city there. The Nahuatl language is still spoken by around 1.5 million people, mostly in the mountainous regions of Mexico.
#9 Cortes was aided in defeating the Aztecs by a native woman named La Malinche
In April 1519, the Spanish defeated the natives of the Chontal Maya of Potonchán, who in turn gave them 20 slave women among whom was a woman named La Malinche. She spoke two native languages, Mayan and Nahuatl. She soon learned Spanish and served as a translator for Cortes. Later, she became a mistress to Cortes and gave birth to his first son, Martin. La Malinche played a critical role in the Spanish conquest of the Aztec civilization. Apart from serving as an adviser and interpreter of Cortes, she played an important role in diplomacy helping the Spanish ally with the native tribes. Among other things, she warned Cortes of a native plan to destroy his small army leading to slaughter of the plotting tribe. One source even says that Cortes himself said that after God, Malinche was the main reason for his success. Due to her, the term “malinchist” applies to all those who feel an attraction to foreign cultures and disregard for their own national culture. Today, La Malinche is seen by some as a woman caught in between two cultures and a mother of a new race; while others regard her as a traitor who brought the end of her own civilization.
#10 Aztec religion was polytheistic and they adopted gods from other cultures
The Aztec religion was polytheistic and involved a large and ever increasing pantheon of gods and goddesses. The Aztecs often adopted gods from different cultures and allowed them to be worshiped as part of their pantheon. The most important celestial entities in Aztec religion were the Sun, the Moon, and the planet Venus. The patron god of the Mexica tribe of Tenochtitlan was the Sun god Huitzilopochtli who represented war and sacrifice. Other important gods included Tlaloc, supreme god of the rain and by extension a god of earthly fertility and of water; Tezcatlipoca, associated with a wide range of concepts; and Quetzalcoatl (“feathered serpent”), the god of wind, sky and star who was associated with learning. Religion was part of all levels of Aztec society with each level having their own rituals and deities. The word for priest in Nahuatl was tlamacazqui which meant “giver of things”. His main duty was to make sure that the gods were given their due in the form of offerings, ceremonies and sacrifices.
Aztec Human Sacrifice
The Aztecs believed that continuing sacrifice of the gods sustained the Universe. Human sacrifice was considered the highest offering to the Gods to repay their debt and it was a major part of their religion. In their ritual, the person to be sacrificed was taken to the top of the temple. He was then laid on a stone slab by four priests, and his abdomen was sliced open by a fifth priest with a ceremonial knife. The cut was made in the abdomen and went through the diaphragm. The priest would then grab the heart and tear it out, still beating. It would be placed in a bowl held by a statue of the honored god, and the body thrown down the temple’s stairs. During the ritual, the priests and the audience would stab, pierce and bleed themselves as auto-sacrifice. Their is debate about the extent and reasons of human sacrifice among the Aztecs. It is generally believed that it was practiced on a large scale. Apart from religious reasons, it is theorized that public spectacle of sacrificing warriors from conquered states was a major display of political power and a way to deter future rebellions.<|endoftext|>
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Amino Acids: Organic compounds that generally contain an amino (-NH2) and a carboxyl (-COOH) group. Twenty alpha-amino acids are the subunits which are polymerized to form proteins.Amino Acids, Aromatic: Amino acids containing an aromatic side chain.Hydrocarbons, Aromatic: Organic compounds containing carbon and hydrogen in the form of an unsaturated, usually hexagonal ring structure. The compounds can be single ring, or double, triple, or multiple fused rings.Sequence Homology, Amino Acid: The degree of similarity between sequences of amino acids. This information is useful for the analyzing genetic relatedness of proteins and species.Amino Acid Sequence: The order of amino acids as they occur in a polypeptide chain. This is referred to as the primary structure of proteins. It is of fundamental importance in determining PROTEIN CONFORMATION.Amino Acid Substitution: The naturally occurring or experimentally induced replacement of one or more AMINO ACIDS in a protein with another. If a functionally equivalent amino acid is substituted, the protein may retain wild-type activity. Substitution may also diminish, enhance, or eliminate protein function. Experimentally induced substitution is often used to study enzyme activities and binding site properties.Cloning, Molecular: The insertion of recombinant DNA molecules from prokaryotic and/or eukaryotic sources into a replicating vehicle, such as a plasmid or virus vector, and the introduction of the resultant hybrid molecules into recipient cells without altering the viability of those cells.Amino Acids, Essential: Amino acids that are not synthesized by the human body in amounts sufficient to carry out physiological functions. They are obtained from dietary foodstuffs.Amino Acid Transport Systems: Cellular proteins and protein complexes that transport amino acids across biological membranes.Sequence Alignment: The arrangement of two or more amino acid or base sequences from an organism or organisms in such a way as to align areas of the sequences sharing common properties. The degree of relatedness or homology between the sequences is predicted computationally or statistically based on weights assigned to the elements aligned between the sequences. This in turn can serve as a potential indicator of the genetic relatedness between the organisms.Molecular Sequence Data: Descriptions of specific amino acid, carbohydrate, or nucleotide sequences which have appeared in the published literature and/or are deposited in and maintained by databanks such as GENBANK, European Molecular Biology Laboratory (EMBL), National Biomedical Research Foundation (NBRF), or other sequence repositories.Base Sequence: The sequence of PURINES and PYRIMIDINES in nucleic acids and polynucleotides. It is also called nucleotide sequence.Escherichia coli: A species of gram-negative, facultatively anaerobic, rod-shaped bacteria (GRAM-NEGATIVE FACULTATIVELY ANAEROBIC RODS) commonly found in the lower part of the intestine of warm-blooded animals. It is usually nonpathogenic, but some strains are known to produce DIARRHEA and pyogenic infections. Pathogenic strains (virotypes) are classified by their specific pathogenic mechanisms such as toxins (ENTEROTOXIGENIC ESCHERICHIA COLI), etc.Binding Sites: The parts of a macromolecule that directly participate in its specific combination with another molecule.Amino Acid Motifs: Commonly observed structural components of proteins formed by simple combinations of adjacent secondary structures. A commonly observed structure may be composed of a CONSERVED SEQUENCE which can be represented by a CONSENSUS SEQUENCE.Mutation: Any detectable and heritable change in the genetic material that causes a change in the GENOTYPE and which is transmitted to daughter cells and to succeeding generations.Models, Molecular: Models used experimentally or theoretically to study molecular shape, electronic properties, or interactions; includes analogous molecules, computer-generated graphics, and mechanical structures.Kinetics: The rate dynamics in chemical or physical systems.Mutagenesis, Site-Directed: Genetically engineered MUTAGENESIS at a specific site in the DNA molecule that introduces a base substitution, or an insertion or deletion.Protein Conformation: The characteristic 3-dimensional shape of a protein, including the secondary, supersecondary (motifs), tertiary (domains) and quaternary structure of the peptide chain. PROTEIN STRUCTURE, QUATERNARY describes the conformation assumed by multimeric proteins (aggregates of more than one polypeptide chain).Structure-Activity Relationship: The relationship between the chemical structure of a compound and its biological or pharmacological activity. Compounds are often classed together because they have structural characteristics in common including shape, size, stereochemical arrangement, and distribution of functional groups.Polycyclic Compounds: Compounds consisting of two or more fused ring structures.Substrate Specificity: A characteristic feature of enzyme activity in relation to the kind of substrate on which the enzyme or catalytic molecule reacts.Recombinant Proteins: Proteins prepared by recombinant DNA technology.Amino Acids, Branched-Chain: Amino acids which have a branched carbon chain.Phenylalanine: An essential aromatic amino acid that is a precursor of MELANIN; DOPAMINE; noradrenalin (NOREPINEPHRINE), and THYROXINE.DNA, Complementary: Single-stranded complementary DNA synthesized from an RNA template by the action of RNA-dependent DNA polymerase. cDNA (i.e., complementary DNA, not circular DNA, not C-DNA) is used in a variety of molecular cloning experiments as well as serving as a specific hybridization probe.Sequence Homology, Nucleic Acid: The sequential correspondence of nucleotides in one nucleic acid molecule with those of another nucleic acid molecule. Sequence homology is an indication of the genetic relatedness of different organisms and gene function.Peptide Fragments: Partial proteins formed by partial hydrolysis of complete proteins or generated through PROTEIN ENGINEERING techniques.Protein Structure, Tertiary: The level of protein structure in which combinations of secondary protein structures (alpha helices, beta sheets, loop regions, and motifs) pack together to form folded shapes called domains. Disulfide bridges between cysteines in two different parts of the polypeptide chain along with other interactions between the chains play a role in the formation and stabilization of tertiary structure. Small proteins usually consist of only one domain but larger proteins may contain a number of domains connected by segments of polypeptide chain which lack regular secondary structure.Leucine: An essential branched-chain amino acid important for hemoglobin formation.Protein Binding: The process in which substances, either endogenous or exogenous, bind to proteins, peptides, enzymes, protein precursors, or allied compounds. Specific protein-binding measures are often used as assays in diagnostic assessments.Amino Acids, SulfurBacterial Proteins: Proteins found in any species of bacterium.DNA: A deoxyribonucleotide polymer that is the primary genetic material of all cells. Eukaryotic and prokaryotic organisms normally contain DNA in a double-stranded state, yet several important biological processes transiently involve single-stranded regions. DNA, which consists of a polysugar-phosphate backbone possessing projections of purines (adenine and guanine) and pyrimidines (thymine and cytosine), forms a double helix that is held together by hydrogen bonds between these purines and pyrimidines (adenine to thymine and guanine to cytosine).Peptides: Members of the class of compounds composed of AMINO ACIDS joined together by peptide bonds between adjacent amino acids into linear, branched or cyclical structures. OLIGOPEPTIDES are composed of approximately 2-12 amino acids. Polypeptides are composed of approximately 13 or more amino acids. PROTEINS are linear polypeptides that are normally synthesized on RIBOSOMES.Molecular Weight: The sum of the weight of all the atoms in a molecule.Tryptophan: An essential amino acid that is necessary for normal growth in infants and for NITROGEN balance in adults. It is a precursor of INDOLE ALKALOIDS in plants. It is a precursor of SEROTONIN (hence its use as an antidepressant and sleep aid). It can be a precursor to NIACIN, albeit inefficiently, in mammals.Phylogeny: The relationships of groups of organisms as reflected by their genetic makeup.Protein Structure, Secondary: The level of protein structure in which regular hydrogen-bond interactions within contiguous stretches of polypeptide chain give rise to alpha helices, beta strands (which align to form beta sheets) or other types of coils. This is the first folding level of protein conformation.Genes, Bacterial: The functional hereditary units of BACTERIA.Restriction Mapping: Use of restriction endonucleases to analyze and generate a physical map of genomes, genes, or other segments of DNA.Sequence Analysis, DNA: A multistage process that includes cloning, physical mapping, subcloning, determination of the DNA SEQUENCE, and information analysis.Chromatography, High Pressure Liquid: Liquid chromatographic techniques which feature high inlet pressures, high sensitivity, and high speed.Cell Line: Established cell cultures that have the potential to propagate indefinitely.Alanine: A non-essential amino acid that occurs in high levels in its free state in plasma. It is produced from pyruvate by transamination. It is involved in sugar and acid metabolism, increases IMMUNITY, and provides energy for muscle tissue, BRAIN, and the CENTRAL NERVOUS SYSTEM.Proteins: Linear POLYPEPTIDES that are synthesized on RIBOSOMES and may be further modified, crosslinked, cleaved, or assembled into complex proteins with several subunits. The specific sequence of AMINO ACIDS determines the shape the polypeptide will take, during PROTEIN FOLDING, and the function of the protein.Recombinant Fusion Proteins: Recombinant proteins produced by the GENETIC TRANSLATION of fused genes formed by the combination of NUCLEIC ACID REGULATORY SEQUENCES of one or more genes with the protein coding sequences of one or more genes.Electrophoresis, Polyacrylamide Gel: Electrophoresis in which a polyacrylamide gel is used as the diffusion medium.Conserved Sequence: A sequence of amino acids in a polypeptide or of nucleotides in DNA or RNA that is similar across multiple species. A known set of conserved sequences is represented by a CONSENSUS SEQUENCE. AMINO ACID MOTIFS are often composed of conserved sequences.RNA, Messenger: RNA sequences that serve as templates for protein synthesis. Bacterial mRNAs are generally primary transcripts in that they do not require post-transcriptional processing. Eukaryotic mRNA is synthesized in the nucleus and must be exported to the cytoplasm for translation. Most eukaryotic mRNAs have a sequence of polyadenylic acid at the 3' end, referred to as the poly(A) tail. The function of this tail is not known for certain, but it may play a role in the export of mature mRNA from the nucleus as well as in helping stabilize some mRNA molecules by retarding their degradation in the cytoplasm.Species Specificity: The restriction of a characteristic behavior, anatomical structure or physical system, such as immune response; metabolic response, or gene or gene variant to the members of one species. It refers to that property which differentiates one species from another but it is also used for phylogenetic levels higher or lower than the species.Lysine: An essential amino acid. It is often added to animal feed.DNA Primers: Short sequences (generally about 10 base pairs) of DNA that are complementary to sequences of messenger RNA and allow reverse transcriptases to start copying the adjacent sequences of mRNA. Primers are used extensively in genetic and molecular biology techniques.Isoleucine: An essential branched-chain aliphatic amino acid found in many proteins. It is an isomer of LEUCINE. It is important in hemoglobin synthesis and regulation of blood sugar and energy levels.Biodegradation, Environmental: Elimination of ENVIRONMENTAL POLLUTANTS; PESTICIDES and other waste using living organisms, usually involving intervention of environmental or sanitation engineers.Amino Acid Transport Systems, Basic: Amino acid transporter systems capable of transporting basic amino acids (AMINO ACIDS, BASIC).Molecular Structure: The location of the atoms, groups or ions relative to one another in a molecule, as well as the number, type and location of covalent bonds.Carrier Proteins: Transport proteins that carry specific substances in the blood or across cell membranes.Magnetic Resonance Spectroscopy: Spectroscopic method of measuring the magnetic moment of elementary particles such as atomic nuclei, protons or electrons. It is employed in clinical applications such as NMR Tomography (MAGNETIC RESONANCE IMAGING).Plasmids: Extrachromosomal, usually CIRCULAR DNA molecules that are self-replicating and transferable from one organism to another. They are found in a variety of bacterial, archaeal, fungal, algal, and plant species. They are used in GENETIC ENGINEERING as CLONING VECTORS.Pyrenes: A group of condensed ring hydrocarbons.Cattle: Domesticated bovine animals of the genus Bos, usually kept on a farm or ranch and used for the production of meat or dairy products or for heavy labor.Amino Acids, Basic: Amino acids with side chains that are positively charged at physiological pH.Biological Transport: The movement of materials (including biochemical substances and drugs) through a biological system at the cellular level. The transport can be across cell membranes and epithelial layers. It also can occur within intracellular compartments and extracellular compartments.Glycine: A non-essential amino acid. It is found primarily in gelatin and silk fibroin and used therapeutically as a nutrient. It is also a fast inhibitory neurotransmitter.Codon: A set of three nucleotides in a protein coding sequence that specifies individual amino acids or a termination signal (CODON, TERMINATOR). Most codons are universal, but some organisms do not produce the transfer RNAs (RNA, TRANSFER) complementary to all codons. These codons are referred to as unassigned codons (CODONS, NONSENSE).Catalysis: The facilitation of a chemical reaction by material (catalyst) that is not consumed by the reaction.Tyrosine: A non-essential amino acid. In animals it is synthesized from PHENYLALANINE. It is also the precursor of EPINEPHRINE; THYROID HORMONES; and melanin.Open Reading Frames: A sequence of successive nucleotide triplets that are read as CODONS specifying AMINO ACIDS and begin with an INITIATOR CODON and end with a stop codon (CODON, TERMINATOR).Trypsin: A serine endopeptidase that is formed from TRYPSINOGEN in the pancreas. It is converted into its active form by ENTEROPEPTIDASE in the small intestine. It catalyzes hydrolysis of the carboxyl group of either arginine or lysine. EC 22.214.171.124.Mutagenesis: Process of generating a genetic MUTATION. It may occur spontaneously or be induced by MUTAGENS.Arginine: An essential amino acid that is physiologically active in the L-form.Genes: A category of nucleic acid sequences that function as units of heredity and which code for the basic instructions for the development, reproduction, and maintenance of organisms.Glutamine: A non-essential amino acid present abundantly throughout the body and is involved in many metabolic processes. It is synthesized from GLUTAMIC ACID and AMMONIA. It is the principal carrier of NITROGEN in the body and is an important energy source for many cells.Cyanogen Bromide: Cyanogen bromide (CNBr). A compound used in molecular biology to digest some proteins and as a coupling reagent for phosphoroamidate or pyrophosphate internucleotide bonds in DNA duplexes.Valine: A branched-chain essential amino acid that has stimulant activity. It promotes muscle growth and tissue repair. It is a precursor in the penicillin biosynthetic pathway.Saccharomyces cerevisiae: A species of the genus SACCHAROMYCES, family Saccharomycetaceae, order Saccharomycetales, known as "baker's" or "brewer's" yeast. The dried form is used as a dietary supplement.Polymerase Chain Reaction: In vitro method for producing large amounts of specific DNA or RNA fragments of defined length and sequence from small amounts of short oligonucleotide flanking sequences (primers). The essential steps include thermal denaturation of the double-stranded target molecules, annealing of the primers to their complementary sequences, and extension of the annealed primers by enzymatic synthesis with DNA polymerase. The reaction is efficient, specific, and extremely sensitive. Uses for the reaction include disease diagnosis, detection of difficult-to-isolate pathogens, mutation analysis, genetic testing, DNA sequencing, and analyzing evolutionary relationships.Nitrogen: An element with the atomic symbol N, atomic number 7, and atomic weight [14.00643; 14.00728]. Nitrogen exists as a diatomic gas and makes up about 78% of the earth's atmosphere by volume. It is a constituent of proteins and nucleic acids and found in all living cells.DNA, Bacterial: Deoxyribonucleic acid that makes up the genetic material of bacteria.Methionine: A sulfur-containing essential L-amino acid that is important in many body functions.Point Mutation: A mutation caused by the substitution of one nucleotide for another. This results in the DNA molecule having a change in a single base pair.Aspartic Acid: One of the non-essential amino acids commonly occurring in the L-form. It is found in animals and plants, especially in sugar cane and sugar beets. It may be a neurotransmitter.Stereoisomerism: The phenomenon whereby compounds whose molecules have the same number and kind of atoms and the same atomic arrangement, but differ in their spatial relationships. (From McGraw-Hill Dictionary of Scientific and Technical Terms, 5th ed)Amino Acids, DiaminoAmines: A group of compounds derived from ammonia by substituting organic radicals for the hydrogens. (From Grant & Hackh's Chemical Dictionary, 5th ed)Sequence Analysis: A multistage process that includes the determination of a sequence (protein, carbohydrate, etc.), its fragmentation and analysis, and the interpretation of the resulting sequence information.Protein Biosynthesis: The biosynthesis of PEPTIDES and PROTEINS on RIBOSOMES, directed by MESSENGER RNA, via TRANSFER RNA that is charged with standard proteinogenic AMINO ACIDS.Proline: A non-essential amino acid that is synthesized from GLUTAMIC ACID. It is an essential component of COLLAGEN and is important for proper functioning of joints and tendons.Gene Library: A large collection of DNA fragments cloned (CLONING, MOLECULAR) from a given organism, tissue, organ, or cell type. It may contain complete genomic sequences (GENOMIC LIBRARY) or complementary DNA sequences, the latter being formed from messenger RNA and lacking intron sequences.Sequence Deletion: Deletion of sequences of nucleic acids from the genetic material of an individual.Cysteine: A thiol-containing non-essential amino acid that is oxidized to form CYSTINE.HydrocarbonsExcitatory Amino Acids: Endogenous amino acids released by neurons as excitatory neurotransmitters. Glutamic acid is the most common excitatory neurotransmitter in the brain. Aspartic acid has been regarded as an excitatory transmitter for many years, but the extent of its role as a transmitter is unclear.Transcription, Genetic: The biosynthesis of RNA carried out on a template of DNA. The biosynthesis of DNA from an RNA template is called REVERSE TRANSCRIPTION.Benzo(a)pyrene: A potent mutagen and carcinogen. It is a public health concern because of its possible effects on industrial workers, as an environmental pollutant, an as a component of tobacco smoke.Membrane Proteins: Proteins which are found in membranes including cellular and intracellular membranes. They consist of two types, peripheral and integral proteins. They include most membrane-associated enzymes, antigenic proteins, transport proteins, and drug, hormone, and lectin receptors.Transfection: The uptake of naked or purified DNA by CELLS, usually meaning the process as it occurs in eukaryotic cells. It is analogous to bacterial transformation (TRANSFORMATION, BACTERIAL) and both are routinely employed in GENE TRANSFER TECHNIQUES.Crystallography, X-Ray: The study of crystal structure using X-RAY DIFFRACTION techniques. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)Oxidation-Reduction: A chemical reaction in which an electron is transferred from one molecule to another. The electron-donating molecule is the reducing agent or reductant; the electron-accepting molecule is the oxidizing agent or oxidant. Reducing and oxidizing agents function as conjugate reductant-oxidant pairs or redox pairs (Lehninger, Principles of Biochemistry, 1982, p471).Circular Dichroism: A change from planar to elliptic polarization when an initially plane-polarized light wave traverses an optically active medium. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)Liver: A large lobed glandular organ in the abdomen of vertebrates that is responsible for detoxification, metabolism, synthesis and storage of various substances.Macromolecular Substances: Compounds and molecular complexes that consist of very large numbers of atoms and are generally over 500 kDa in size. In biological systems macromolecular substances usually can be visualized using ELECTRON MICROSCOPY and are distinguished from ORGANELLES by the lack of a membrane structure.Amino Acid Transport System A: A sodium-dependent neutral amino acid transporter that accounts for most of the sodium-dependent neutral amino acid uptake by mammalian cells. The preferred substrates for this transporter system include ALANINE; SERINE; and GLUTAMINE.Amino Acids, Neutral: Amino acids with uncharged R groups or side chains.Multigene Family: A set of genes descended by duplication and variation from some ancestral gene. Such genes may be clustered together on the same chromosome or dispersed on different chromosomes. Examples of multigene families include those that encode the hemoglobins, immunoglobulins, histocompatibility antigens, actins, tubulins, keratins, collagens, heat shock proteins, salivary glue proteins, chorion proteins, cuticle proteins, yolk proteins, and phaseolins, as well as histones, ribosomal RNA, and transfer RNA genes. The latter three are examples of reiterated genes, where hundreds of identical genes are present in a tandem array. (King & Stanfield, A Dictionary of Genetics, 4th ed)Hydrolysis: The process of cleaving a chemical compound by the addition of a molecule of water.Hydrogen-Ion Concentration: The normality of a solution with respect to HYDROGEN ions; H+. It is related to acidity measurements in most cases by pH = log 1/2[1/(H+)], where (H+) is the hydrogen ion concentration in gram equivalents per liter of solution. (McGraw-Hill Dictionary of Scientific and Technical Terms, 6th ed)Gene Expression: The phenotypic manifestation of a gene or genes by the processes of GENETIC TRANSCRIPTION and GENETIC TRANSLATION.Evolution, Molecular: The process of cumulative change at the level of DNA; RNA; and PROTEINS, over successive generations.Transaminases: A subclass of enzymes of the transferase class that catalyze the transfer of an amino group from a donor (generally an amino acid) to an acceptor (generally a 2-keto acid). Most of these enzymes are pyridoxyl phosphate proteins. (Dorland, 28th ed) EC 2.6.1.Plant Proteins: Proteins found in plants (flowers, herbs, shrubs, trees, etc.). The concept does not include proteins found in vegetables for which VEGETABLE PROTEINS is available.Temperature: The property of objects that determines the direction of heat flow when they are placed in direct thermal contact. The temperature is the energy of microscopic motions (vibrational and translational) of the particles of atoms.Blotting, Northern: Detection of RNA that has been electrophoretically separated and immobilized by blotting on nitrocellulose or other type of paper or nylon membrane followed by hybridization with labeled NUCLEIC ACID PROBES.Viral Proteins: Proteins found in any species of virus.Chromatography, Gel: Chromatography on non-ionic gels without regard to the mechanism of solute discrimination.Mass Spectrometry: An analytical method used in determining the identity of a chemical based on its mass using mass analyzers/mass spectrometers.Sequence Analysis, Protein: A process that includes the determination of AMINO ACID SEQUENCE of a protein (or peptide, oligopeptide or peptide fragment) and the information analysis of the sequence.Shikimic Acid: A tri-hydroxy cyclohexene carboxylic acid important in biosynthesis of so many compounds that the shikimate pathway is named after it.Rabbits: The species Oryctolagus cuniculus, in the family Leporidae, order LAGOMORPHA. Rabbits are born in burrows, furless, and with eyes and ears closed. In contrast with HARES, rabbits have 22 chromosome pairs.Amino Acids, Cyclic: A class of amino acids characterized by a closed ring structure.Threonine: An essential amino acid occurring naturally in the L-form, which is the active form. It is found in eggs, milk, gelatin, and other proteins.Chemistry: A basic science concerned with the composition, structure, and properties of matter; and the reactions that occur between substances and the associated energy exchange.Chromatography, Ion Exchange: Separation technique in which the stationary phase consists of ion exchange resins. The resins contain loosely held small ions that easily exchange places with other small ions of like charge present in solutions washed over the resins.Enzyme Stability: The extent to which an enzyme retains its structural conformation or its activity when subjected to storage, isolation, and purification or various other physical or chemical manipulations, including proteolytic enzymes and heat.Swine: Any of various animals that constitute the family Suidae and comprise stout-bodied, short-legged omnivorous mammals with thick skin, usually covered with coarse bristles, a rather long mobile snout, and small tail. Included are the genera Babyrousa, Phacochoerus (wart hogs), and Sus, the latter containing the domestic pig (see SUS SCROFA).Chymotrypsin: A serine endopeptidase secreted by the pancreas as its zymogen, CHYMOTRYPSINOGEN and carried in the pancreatic juice to the duodenum where it is activated by TRYPSIN. It selectively cleaves aromatic amino acids on the carboxyl side.DNA-Binding Proteins: Proteins which bind to DNA. The family includes proteins which bind to both double- and single-stranded DNA and also includes specific DNA binding proteins in serum which can be used as markers for malignant diseases.Cricetinae: A subfamily in the family MURIDAE, comprising the hamsters. Four of the more common genera are Cricetus, CRICETULUS; MESOCRICETUS; and PHODOPUS.DNA Adducts: The products of chemical reactions that result in the addition of extraneous chemical groups to DNA.Chemical Phenomena: The composition, conformation, and properties of atoms and molecules, and their reaction and interaction processes.Benzene DerivativesCarbon Isotopes: Stable carbon atoms that have the same atomic number as the element carbon, but differ in atomic weight. C-13 is a stable carbon isotope.COS Cells: CELL LINES derived from the CV-1 cell line by transformation with a replication origin defective mutant of SV40 VIRUS, which codes for wild type large T antigen (ANTIGENS, POLYOMAVIRUS TRANSFORMING). They are used for transfection and cloning. (The CV-1 cell line was derived from the kidney of an adult male African green monkey (CERCOPITHECUS AETHIOPS).)Cell Membrane: The lipid- and protein-containing, selectively permeable membrane that surrounds the cytoplasm in prokaryotic and eukaryotic cells.Epitopes: Sites on an antigen that interact with specific antibodies.Genetic Complementation Test: A test used to determine whether or not complementation (compensation in the form of dominance) will occur in a cell with a given mutant phenotype when another mutant genome, encoding the same mutant phenotype, is introduced into that cell.Pseudomonas: A genus of gram-negative, aerobic, rod-shaped bacteria widely distributed in nature. Some species are pathogenic for humans, animals, and plants.Models, Chemical: Theoretical representations that simulate the behavior or activity of chemical processes or phenomena; includes the use of mathematical equations, computers, and other electronic equipment.Endopeptidases: A subclass of PEPTIDE HYDROLASES that catalyze the internal cleavage of PEPTIDES or PROTEINS.Serine: A non-essential amino acid occurring in natural form as the L-isomer. It is synthesized from GLYCINE or THREONINE. It is involved in the biosynthesis of PURINES; PYRIMIDINES; and other amino acids.Dietary Proteins: Proteins obtained from foods. They are the main source of the ESSENTIAL AMINO ACIDS.Repetitive Sequences, Amino Acid: A sequential pattern of amino acids occurring more than once in the same protein sequence.Receptors, Amino Acid: Cell surface proteins that bind amino acids and trigger changes which influence the behavior of cells. Glutamate receptors are the most common receptors for fast excitatory synaptic transmission in the vertebrate central nervous system, and GAMMA-AMINOBUTYRIC ACID and glycine receptors are the most common receptors for fast inhibition.Chickens: Common name for the species Gallus gallus, the domestic fowl, in the family Phasianidae, order GALLIFORMES. It is descended from the red jungle fowl of SOUTHEAST ASIA.Catalytic Domain: The region of an enzyme that interacts with its substrate to cause the enzymatic reaction.Protein Sorting Signals: Amino acid sequences found in transported proteins that selectively guide the distribution of the proteins to specific cellular compartments.Escherichia coli Proteins: Proteins obtained from ESCHERICHIA COLI.Fungal Proteins: Proteins found in any species of fungus.Oxygenases: Oxidases that specifically introduce DIOXYGEN-derived oxygen atoms into a variety of organic molecules.Histidine: An essential amino acid that is required for the production of HISTAMINE.DNA Mutational Analysis: Biochemical identification of mutational changes in a nucleotide sequence.Protein PrecursorsCulture Media: Any liquid or solid preparation made specifically for the growth, storage, or transport of microorganisms or other types of cells. The variety of media that exist allow for the culturing of specific microorganisms and cell types, such as differential media, selective media, test media, and defined media. Solid media consist of liquid media that have been solidified with an agent such as AGAR or GELATIN.Protein Processing, Post-Translational: Any of various enzymatically catalyzed post-translational modifications of PEPTIDES or PROTEINS in the cell of origin. These modifications include carboxylation; HYDROXYLATION; ACETYLATION; PHOSPHORYLATION; METHYLATION; GLYCOSYLATION; ubiquitination; oxidation; proteolysis; and crosslinking and result in changes in molecular weight and electrophoretic motility.Glutamic Acid: A non-essential amino acid naturally occurring in the L-form. Glutamic acid is the most common excitatory neurotransmitter in the CENTRAL NERVOUS SYSTEM.Peptide Mapping: Analysis of PEPTIDES that are generated from the digestion or fragmentation of a protein or mixture of PROTEINS, by ELECTROPHORESIS; CHROMATOGRAPHY; or MASS SPECTROMETRY. The resulting peptide fingerprints are analyzed for a variety of purposes including the identification of the proteins in a sample, GENETIC POLYMORPHISMS, patterns of gene expression, and patterns diagnostic for diseases.Pseudomonas putida: A species of gram-negative, aerobic bacteria isolated from soil and water as well as clinical specimens. Occasionally it is an opportunistic pathogen.Oligopeptides: Peptides composed of between two and twelve amino acids.Blotting, Southern: A method (first developed by E.M. Southern) for detection of DNA that has been electrophoretically separated and immobilized by blotting on nitrocellulose or other type of paper or nylon membrane followed by hybridization with labeled NUCLEIC ACID PROBES.Gene Expression Regulation, Bacterial: Any of the processes by which cytoplasmic or intercellular factors influence the differential control of gene action in bacteria.Aminoisobutyric Acids: A group of compounds that are derivatives of the amino acid 2-amino-2-methylpropanoic acid.Protein Folding: Processes involved in the formation of TERTIARY PROTEIN STRUCTURE.Isoenzymes: Structurally related forms of an enzyme. Each isoenzyme has the same mechanism and classification, but differs in its chemical, physical, or immunological characteristics.Cells, Cultured: Cells propagated in vitro in special media conducive to their growth. Cultured cells are used to study developmental, morphologic, metabolic, physiologic, and genetic processes, among others.Chromosome Mapping: Any method used for determining the location of and relative distances between genes on a chromosome.Amino Acyl-tRNA Synthetases: A subclass of enzymes that aminoacylate AMINO ACID-SPECIFIC TRANSFER RNA with their corresponding AMINO ACIDS.Hydrophobic and Hydrophilic Interactions: The thermodynamic interaction between a substance and WATER.Dioxygenases: Non-heme iron-containing enzymes that incorporate two atoms of OXYGEN into the substrate. They are important in biosynthesis of FLAVONOIDS; GIBBERELLINS; and HYOSCYAMINE; and for degradation of AROMATIC HYDROCARBONS.Phenotype: The outward appearance of the individual. It is the product of interactions between genes, and between the GENOTYPE and the environment.Soil Pollutants: Substances which pollute the soil. Use for soil pollutants in general or for which there is no specific heading.Transcription Factors: Endogenous substances, usually proteins, which are effective in the initiation, stimulation, or termination of the genetic transcription process.PhenanthrenesGenetic Variation: Genotypic differences observed among individuals in a population.Carbohydrates: The largest class of organic compounds, including STARCH; GLYCOGEN; CELLULOSE; POLYSACCHARIDES; and simple MONOSACCHARIDES. Carbohydrates are composed of carbon, hydrogen, and oxygen in a ratio of Cn(H2O)n.Sequence Homology: The degree of similarity between sequences. Studies of AMINO ACID SEQUENCE HOMOLOGY and NUCLEIC ACID SEQUENCE HOMOLOGY provide useful information about the genetic relatedness of genes, gene products, and species.Ligands: A molecule that binds to another molecule, used especially to refer to a small molecule that binds specifically to a larger molecule, e.g., an antigen binding to an antibody, a hormone or neurotransmitter binding to a receptor, or a substrate or allosteric effector binding to an enzyme. Ligands are also molecules that donate or accept a pair of electrons to form a coordinate covalent bond with the central metal atom of a coordination complex. (From Dorland, 27th ed)Large Neutral Amino Acid-Transporter 1: A CD98 antigen light chain that when heterodimerized with CD98 antigen heavy chain (ANTIGENS, CD98 HEAVY CHAIN) forms a protein that mediates sodium-independent L-type amino acid transport.Oligodeoxyribonucleotides: A group of deoxyribonucleotides (up to 12) in which the phosphate residues of each deoxyribonucleotide act as bridges in forming diester linkages between the deoxyribose moieties.Genes, Fungal: The functional hereditary units of FUNGI.Benzoates: Derivatives of BENZOIC ACID. Included under this heading are a broad variety of acid forms, salts, esters, and amides that contain the carboxybenzene structure.Thermodynamics: A rigorously mathematical analysis of energy relationships (heat, work, temperature, and equilibrium). It describes systems whose states are determined by thermal parameters, such as temperature, in addition to mechanical and electromagnetic parameters. (From Hawley's Condensed Chemical Dictionary, 12th ed)Oligonucleotide Probes: Synthetic or natural oligonucleotides used in hybridization studies in order to identify and study specific nucleic acid fragments, e.g., DNA segments near or within a specific gene locus or gene. The probe hybridizes with a specific mRNA, if present. Conventional techniques used for testing for the hybridization product include dot blot assays, Southern blot assays, and DNA:RNA hybrid-specific antibody tests. Conventional labels for the probe include the radioisotope labels 32P and 125I and the chemical label biotin.Cystine: A covalently linked dimeric nonessential amino acid formed by the oxidation of CYSTEINE. Two molecules of cysteine are joined together by a disulfide bridge to form cystine.Molecular Conformation: The characteristic three-dimensional shape of a molecule.Dimerization: The process by which two molecules of the same chemical composition form a condensation product or polymer.Oxidoreductases: The class of all enzymes catalyzing oxidoreduction reactions. The substrate that is oxidized is regarded as a hydrogen donor. The systematic name is based on donor:acceptor oxidoreductase. The recommended name will be dehydrogenase, wherever this is possible; as an alternative, reductase can be used. Oxidase is only used in cases where O2 is the acceptor. (Enzyme Nomenclature, 1992, p9)Gas Chromatography-Mass Spectrometry: A microanalytical technique combining mass spectrometry and gas chromatography for the qualitative as well as quantitative determinations of compounds.Asparagine: A non-essential amino acid that is involved in the metabolic control of cell functions in nerve and brain tissue. It is biosynthesized from ASPARTIC ACID and AMMONIA by asparagine synthetase. (From Concise Encyclopedia Biochemistry and Molecular Biology, 3rd ed)Phenol: An antiseptic and disinfectant aromatic alcohol.Solubility: The ability of a substance to be dissolved, i.e. to form a solution with another substance. (From McGraw-Hill Dictionary of Scientific and Technical Terms, 6th ed)Serine Endopeptidases: Any member of the group of ENDOPEPTIDASES containing at the active site a serine residue involved in catalysis.Time Factors: Elements of limited time intervals, contributing to particular results or situations.Toluene: A widely used industrial solvent.Plants: Multicellular, eukaryotic life forms of kingdom Plantae (sensu lato), comprising the VIRIDIPLANTAE; RHODOPHYTA; and GLAUCOPHYTA; all of which acquired chloroplasts by direct endosymbiosis of CYANOBACTERIA. They are characterized by a mainly photosynthetic mode of nutrition; essentially unlimited growth at localized regions of cell divisions (MERISTEMS); cellulose within cells providing rigidity; the absence of organs of locomotion; absence of nervous and sensory systems; and an alternation of haploid and diploid generations.Glutathione Transferase: A transferase that catalyzes the addition of aliphatic, aromatic, or heterocyclic FREE RADICALS as well as EPOXIDES and arene oxides to GLUTATHIONE. Addition takes place at the SULFUR. It also catalyzes the reduction of polyol nitrate by glutathione to polyol and nitrite.Carcinogens: Substances that increase the risk of NEOPLASMS in humans or animals. Both genotoxic chemicals, which affect DNA directly, and nongenotoxic chemicals, which induce neoplasms by other mechanism, are included.Tissue Distribution: Accumulation of a drug or chemical substance in various organs (including those not relevant to its pharmacologic or therapeutic action). This distribution depends on the blood flow or perfusion rate of the organ, the ability of the drug to penetrate organ membranes, tissue specificity, protein binding. The distribution is usually expressed as tissue to plasma ratios.Mutation, Missense: A mutation in which a codon is mutated to one directing the incorporation of a different amino acid. This substitution may result in an inactive or unstable product. (From A Dictionary of Genetics, King & Stansfield, 5th ed)Exons: The parts of a transcript of a split GENE remaining after the INTRONS are removed. They are spliced together to become a MESSENGER RNA or other functional RNA.<|endoftext|>
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## Basic Functions and Why You Should Know About Them
No, I did not say “bodily functions”. That is discussed in another blog. We’re talking math today.
So, my son was doing his homework the other night and yelled out from his room:”Daaaadddyyyy!!! Do you know what a parabola is?” For those of you who do not have teenage children this is code for “can you help me with my homework”. After reliving a few high school memories that came along with the word “parabola” I wondered over to his room to see what the latest homework challenge was going to be…
When helping my kids with their homework, I often think of how important and still relevant some of the basic math is we learnt in high school. I would like to talk a little about basic functions and how they are still used well after you have handed in your last math homework assignment.
Many (most?) scientific laws are expressed as relations between two or more variables – often physical quantities. Next comes the chicken or the egg conundrum. Were the results from an experiment used to formulate “empirical laws” or did we use existing knowledge and math to come up with new theories – that we will invariably later have to test. Welcome to the world of research!
If two variables are related in such a way that one of them (the dependent or response variable) is determined when the other is known (the independent or explanatory variable), then there exists what is termed a functional relationship between the variables.
y = f(x)
For example the relationship of height to weight in humans. In general, the taller we are the heavier we get. This results in what is called a straight-line relationship.
But not all relationships are linear. How about if we were to throw a ball up into the air and measure it’s trajectory? It would look a little like the picture on the left.
Although initially the value of the height of the ball increases with time, there comes a point when the ball stops rising and starts to fall back down to earth. The resulting curve is called – you guessed it – a parabola.
The math functions for the parabola and that of the straight line are actually related. Yes, I am serious! They both belong to the family of math functions called polynomials. In my next posts I will talk a little about how we describe these functions and how we can put them to work for us in the world of medical research.
For now, decompress watching this hilarious movie trailer Biloxi Blues which is all about basic training (you can now relate) and…
… I’ll see you in the blogosphere,
Pascal Tyrrell<|endoftext|>
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Proteins are essential nutrients for the human body. They are one of the building blocks of body tissue and can also serve as a fuel source. As a fuel, proteins provide as much energy density as carbohydrates. Unlike carbohydrates and fat, your body does not store protein, so it has no reservoir to draw from when you’re running low.
What is protein?
Protein is found throughout the body in muscle, bone, skin, hair, and virtually every other body part or tissue. It makes up the enzymes that power many chemical reactions.
Chemically, protein is composed of amino acids, which are organic compounds made of carbon, hydrogen, nitrogen, oxygen or sulfur. Amino acids are the building blocks of proteins, and proteins are the building blocks of muscle mass. When protein is broken down in the body it helps to fuel muscle mass, which helps metabolism.
Benefits of a Protein Rich Diet
- Growth and Maintenance: Under normal circumstances, your body breaks down the same amount of protein that it uses to build and repair tissues. Other times, it breaks down more protein than it can create, thus increasing your body’s needs. This typically happens in periods of illness, during pregnancy and while breastfeeding. People recovering from an injury or surgery, older adults and athletes require more protein as well.
- Regulating body processes: Body processes are influenced by hormones proteins that act as chemical messengers to help cells, tissues and organs communicate. For example, they signal the uptake of glucose into a cell, stimulate the growth of tissue and bone, signal the kidneys to reabsorb water and aid in almost all facets of metabolism in your body.
- Provide structural component: These proteins provide structure and support for cells. On a larger scale, they also allow the body to move. Contractual protein is responsible for muscle contraction and movement. Structural protein assures that your cells maintain their shape and resist deformity. Keratin is a structural protein found in your hair, nails and skin. Collagen is another structural protein that provides the framework for the ligaments that hold your bones together, in addition to the tendons that attach muscles to those bones.
- Maintains proper pH: Protein plays a vital role in regulating the concentrations of acids and bases in your blood and other bodily fluids. A variety of buffering systems allows your bodily fluids to maintain normal pH ranges. A constant pH is necessary, as even a slight change in pH can be harmful or potentially deadly. One way your body regulates pH is with proteins. Hemoglobin binds small amounts of acid, helping to maintain the normal pH value of your blood.
Reported by Dr. Himani<|endoftext|>
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Down to Earth: How space science gave us... airbags
Physicist Stuart Higgins looks at how technologies and breakthroughs made originally for the space race are now helping to make life better for us here on the ground, and this week... airbags.
Stuart - If you want to go to space, you need to get off the planet and you need a lot of energy to overcome gravity’s pull. The technology of choice for years has been the rocket. Take NASA’s space shuttle, for example; it had a big orange fuel tank attached to its belly containing liquid rocket fuel, and two thin rockets strapped to its sides. It was these two rocket motors that provided most of the thrust to get the shuttle off the ground. Each was filled with over 450 tons of solid rocket fuel. Each rocket was heavier than a fully loaded jumbo jet and produced 12 million newtons of thrust.
But how do you set off a lump of rocket fuel that's over 15 metres tall? You need a pyrotechnic initiator; it’s a gadget that starts off a chemical reaction. In NASA rockets, an electrical current is passed through a wire which heats up and ignites a small explosive charge. This in turn sets off a reaction in a chemical called lead azide; it’s a molecule made up of the metal lead and three nitrogen items, that’s the azide part. These nitrogen atoms react to produce heat and nitrogen gas which trigger another more powerful explosive that in turn sets off the rocket and… lift off.
It turns out that this rocket science is used in lifesaving technology here on Earth… the airbag.
Airbags, like rockets, contain a pyrotechnic initiator which causes the rapid breakdown of a chemical producing lots of gas. In fact, one rocket manufacturer in Japan directly used its technology to help a car manufacturer make their airbags.
Nowadays, airbags tend to use chemicals that react more efficiently and cleanly, but many were originally based on sodium azide which, like the lead azide used by NASA, reacts producing nitrogen gas. Two molecule of sodium azide reacts to give three molecules of nitrogen gas, and the gas occupies a volume far greater than the original solid. In fact, only about 130 grams of sodium azide is needed for a 70 litre airbag.
During a crash, an electrical circuit triggers the reaction filling the airbag in less than 30 milliseconds, much faster than the blink of an eye. The airbag increases the time it takes you to slow down during a crash - the longer time taken, the lower the force applied to the body. This is particularly important for delicate organs such as the brain which can strike the inside of the skull.
So that’s how launching space rockets has led to airbags that save thousands of lives each year.<|endoftext|>
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|Authors:||Stuart Reges, Marty Stepp|
|Chapter:||Introduction To Java Programming|
Write a complete Java program that prints the following output:
A "quoted" String is
'much' better if you learn
the rules of "escape sequences."
Also, "" represents an empty String.
Don't forget: use \" instead of " !
'' is not the same as "
Extend your hand to students and help them to get better grades.<|endoftext|>
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Glacial Lake Tonawanda was created with the retreat of the last Wisconsin Glacier. The lake was located east of the Niagara River. It covered the area between most of western New York and Rochester.
Although the lake was large in area, it had shallow water levels. The water along the eastern shore at Rochester, New York was only four feet deep.
At first Lake Tonawanda’s only water outlet was the same as Lake Lundy, at Rome New York. Due to rising land in the east, this outlet was cutoff, forcing waters to seek other outlets.
Lake Tonawanda had a total of five water outlets over the 644 kilometer (400 mile) Niagara Escarpment. These outlets were located in Holley, Medina, Gasport, Lockport and Lewiston, all in New York State.
Only the outlet which became the main spillway was at Queenston – Lewiston. The draining waters flowed over the Niagara Escarpment and this is where the water falls of Niagara were born. This water course continues to be the main outlet which exists today.
Lake Erie was still very large, but the width of the Niagara River was much wider than it is today. The entire area where the Falls are today was under water. The water depth was 9 – 12 meters (30 -40 feet) and the slope, formed by glacial moraine, was on the west edge of Queen Victoria Park. Queen Victoria Park is where both the Skylon Tower and Minolta Tower are now situated. This was a river bank about 12,000 years ago before the gorge was created.
The retreating glacier was the main cause of the reduction in size of the much wider and much deeper glacial river known as the St. David’s River. This transformed the St. David’s River into what is now a smaller and shallower river known as the Niagara River.
As the glacier retreated north, the water flowed to the much lower land which uncovered new land. This caused the draining of the Lake Erie basin and Lake Tonawanda. As these waters drained, the river became much smaller. This resulted in the discovery of land such as Niagara Falls, New York, Grand Island, Three Sisters Islands and Queen Victoria Park in Niagara Falls, Ontario.<|endoftext|>
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Tooth sensitivity — also known as dentine hypersensitivity — affects the tooth via exposed root surfaces. This occurs when the enamel that protects the teeth wears away, or when gum recession occurs, exposing the underlying surface, the dentine, thus, reducing the protection the enamel and gums provide to the tooth and root.
If hot, cold, sweet or very acidic foods and drinks, breathing in cold air, or touching the affected surface makes your teeth or a tooth sensitive, then you may have dentine hypersensitivity. Tooth sensitivity can come and go over time.
There are many causes of tooth sensitivity, including:
- Worn tooth enamel from using a hard toothbrush or brushing too aggressively
- Tooth erosion due to highly acidic foods and beverages
- Tooth erosion due to bulimia or gastroesophageal reflux disease (GORD)
- Gum recession that leaves your root surface exposed
Proper oral hygiene is the key to preventing gums from receding and causing sensitive-tooth pain. If you brush your teeth incorrectly, or over-brush, your teeth may become sensitive. Ask your dentist if you have any questions about your daily oral hygiene routine.
Brushing properly twice daily for 2 minutes with a soft toothbrush with a toothpaste that does not have high levels of abrasives, and flossing once a day, can help reduce the chance of tooth sensitivity. A diet low in acidic foods and drinks also helps prevent tooth sensitivity.
In addition to recommending a soft toothbrush and a toothpaste without high levels of abrasives, your dentist may suggest an arginine and calcium toothpaste or a fluoride rinse, or a high fluoride level toothpaste specially formulated to make your teeth less sensitive and provide extra protection against decay. Other treatments — such as fluoride varnishes — can be painted onto the teeth to provide added protection.
There are several conditions which can cause pain, but which are not tooth sensitivity:
- Dental caries
- A cracked or chipped tooth
- Leakage around restorations<|endoftext|>
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The United States’ involvement in the Vietnam War was a divisive chapter in American history. Lending economic and military support to the South Vietnamese government against the Communist North, Washington’s participation in the conflict lasted from the early 1950s to 1973. While in the beginning there was general public acceptance of the war, by 1965 opposition to American involvement in Vietnam grew due to the increasing deployment of troops and the rising number of casualties. In July of 1965, the U.S. government doubled the number of draftees to 35,000 each month. Graphic news footage of the fighting also contributed to the public’s disapproval. Opposition took the form of anti-war demonstrations and draft resistance, and protests broke out on university campuses across the country. Although the U.S. officially withdrew from Vietnam in 1973, it left an indelible mark on the lives of veterans, local communities, and American society.
During and after the United States’ involvement in the conflict in Vietnam, citizens reacted to the war through varied artistic expression. Art became a powerful form of protest and activism, as it was used to raise awareness of social issues and inspire Americans to join the movement against the war. Additionally, once the U.S. began to withdraw troops, people used art to commemorate the war and the loss of life, as well as to consider U.S. involvement overseas. The creative response demonstrates how people participated in American society and civic life, as well as how they contributed to a growing social movement during the 1960s and 1970s.
The collections available in Archives & Special Collections allow us to examine a variety of artistic responses to U.S. engagement in the Vietnam War:
- Poras Collection of Vietnam War Memorabilia: This collection includes a wide variety of materials from the Vietnam War era, including buttons, photographs, fliers, booklets, posters, stamps, flags, audio recordings, and comic books. The collection contains pamphlets and flyers with artistic illustrations in protest of the war, as well as official propaganda in support of the war. The finding aid is available at https://collections.ctdigitalarchive.org/islandora/object/20002%3A860115772.
- First Casualty Press Records: This collection is comprised of poetry and fiction submitted to First Casualty Press to be considered for publication. The works were written by Vietnam War veterans concerning their experiences of the war. The collection also contains correspondence between the First Casualty Press and authors, publishers, and readers, as well as materials related to the publication process. The finding aid is available at https://collections.ctdigitalarchive.org/islandora/object/20002%3A860138522.
- Adam Nadel Photography Collection: This collection consists of thirteen large photographs of Cambodian and Vietnamese people who were affected in some way by the Vietnam War. Recognized internationally for his work, Adam Nadel completed a project on war and its consequences in 2010. Many of the individuals featured in the photographs of this collection are war veterans, both male and female. The finding aid is available at https://collections.ctdigitalarchive.org/islandora/object/20002%3A860114426.
- Bread and Puppet Theater Collection: Founded in 1963, the Bread and Puppet Theater was made up of an experimental theater troupe whose performances combined puppets, masks, and dance. Performances focused on political and social issues, including protesting the United States’ involvement in the Vietnam War. The collection consists of illustrated story scripts, handbills, and performance programs, including a small newspaper from 1967 with illustrations from the theater’s story script about the violence in Vietnam. The finding aid is available at https://collections.ctdigitalarchive.org/islandora/object/20002%3A860130712.
- Storrs Draft Information Committee Records: The Storrs Draft Information Committee was a counseling center on the University of Connecticut’s campus that was established to help men of draft age during the Vietnam War. This collection includes information associated with draft counseling, draft resistance, and protest movement groups at UConn. In particular, the collection contains information on how to renounce U.S. citizenship, documents detailing draft law, and American deserter and draft resistance newspapers from Canada. Some of these documents contain unique illustrations and photographs. The finding aid is available at https://collections.ctdigitalarchive.org/islandora/object/20002%3A860124336.
- Alternative Press Collection (APC): Founded by students in the late 1960s, the APC includes newspapers, books, pamphlets, and artifacts covering activism for social and political change. This includes multiple volumes of a bulletin called the “Viet Report” from 1965-1986. While the “Viet Report” primarily consists of articles from a variety of perspectives on the war and the state of Vietnam, artwork in the form of illustrations and photographs are also included. These reports can be found in our digital repository beginning here: https://archives.lib.uconn.edu/islandora/object/20002%3A01641656.
We invite you to view these collections in the reading room in Archives & Special Collections at the Thomas J. Dodd Research Center if you need resources on the artistic response to the Vietnam War. Our staff is happy to assist you in accessing these and other collections in the archives.
This post was written by Alexandra Borkowski, a UConn PhD student and student assistant in Archives & Special Collections.<|endoftext|>
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# Numerical Integration: Gaussian Methods in 1D
The Gaussian integration, known also as the method of gaussian quadrature, is a numerical approximation of a definite integral of a function in a general interval . We will apply this approach to the computation of the integral of a polynomial function (which, for sufficiently big n, will give an exact result) and of general functions (approximate result).
Gauss quadrature formula is defined for the reference interval as a weighted sume of the values of at appropiately chosen points:
.
We need to choose both points and weights . The points are the zeros of Legendre polinomials . This is a family of polynomial functions represented in the following figure according to the degree (only the first ones). See more information at this Wikipedia link.
For i =1,..,5
The values and weights for these points in the Gauss quadrature formulas are shown in the following table
For example, if we choose n=3 the Gauss approximation formula is
.
## Example:
Now we show an example of the different steps needed to compute the Gauss aprimation value of an integral. As you will see, due to that is a polynomial function, the result is exact when we choose the proper number of Gauss points.
#### Function to integrate
As an example, consider the polynomial function , defined in the reference interval [-1,1]
First we defined the function as an inline function using Matlab
f=@(x) x.^4-3*x.^2+1; %defined as an inline function
#### Order of integration (number of Gauss points)
When you fix a number of points n, the Gauss integration formula is exact for polynomials of degree 2n-1. In our example the polynomial degree is 4, then if we choose only two points n=2 the integral is still not exact. Therefore, the proper choose is n=3.
n=3;
% because our function is a polynomial of order 4 we have to choose n=3
switch (n) %switch executes only one piece of the code according to the value of n
case 1 %if n=1
w=2; pG=0;
case 2 %if n=2
w=[1,1]; pG=[-1/sqrt(3), 1/sqrt(3)];
case 3 %if n=3
w=[5/9, 8/9, 5/9]; pG=[-sqrt(3/5), 0, sqrt(3/5)];
otherwise
error('No data are defined for this value');
end
fprintf('n=%d , pG = %f %f %f ',n,pG)
n=3 , pG = -0.774597 0.000000 0.774597
#### The final Gauss formula
You can use a sequential formula
sumat = 0;
for i=1:size(pG,2)
sumat = sumat + w(i)*f(pG(i));
end
intFseq = sumat
intFseq = 0.4000
or equivalently, a more compact form of the same procedure
intFcompact = w*f(pG)'
intFcompact = 0.4000
#### Check the error
Here, because it is a simple example, we can compute the primitive of the function and solve the integral using Barrow's rule. Then, we compare the result with the numerical value coming from the Gauss formula.
primitiveF=@(x) x.^5/5-x.^3+x; %just to check the error
barrowRule=primitiveF(1)-primitiveF(-1);
errorInt = abs(barrowRule - intFcompact)
errorInt = 2.2204e-16
## Generalization to integration over a general interval [a,b]
When the integration interval is not [-1,1] you have to make a change of variables in the intergral in order to apply the previous formulas to the new interval.
being and .
This means that the Gauss points, initially defined in the interval , must be transformed into points on the new integration interval
#### Example:
Compute the integral of the same function, but now in the interval . So we have to use the change of variables for integrals.
a=2;
b=3;
pGnew = a+(b-a)*(1+pG)/2; %we have already the value of pG
intFcompactNew = (w*f(pGnew)')*(b-a)/2 %the weight values are the same
intFcompactNew = 24.2000
## Exercise 1:
According to the table included above, you have to use the function [w,pt] = gaussValues1D(n), included at the end of this script, to see how the approximation of the integral is improved (you must safe the gaussValues1D function in a new file named gaussValues1D.m).
f=@(x) x.^6-x.^4-3*x.^2+1; %defined as an inline function
primitiveF=@(x) x.^7/7-x.^5/5-x.^3+x; %just to check the error
barrowRuleNew=primitiveF(3)-primitiveF(-2);
a=-2;
b=3;
for n=1:5 %for n=6 must return an error
[w,pG]=gaussValues1D(n); %function to be implemented
pGnew = a+(b-a)*(1+pG)/2; %we have already the value of pG
intFcompactNew = (w*f(pGnew)')*(b-a)/2; %the weight values are the same
errorInt = abs(barrowRuleNew - intFcompactNew);
fprintf('for n= %d the error is =%e \n',n,errorInt)
end
for n= 1 the error is =2.446987e+02 for n= 2 the error is =1.769180e+02 for n= 3 the error is =2.790179e+01 for n= 4 the error is =2.842171e-13 for n= 5 the error is =4.263256e-13
## Exercise 2:
Using the function gaussValues1D, approximate the value of the integral in of the function . Use different values of n and show the errors comparing with the true integral value
Sol:
n, error
1 3.1706e-01
2 7.1183e-03
3 6.1578e-05
4 2.8092e-07
5 7.9140e-10
## Function gaussValues 1D
function [w,pt]=gaussValues1D(n)
switch (n)
case 1
w=2; pt=0;
case 2
w=[1,1]; pt=[-1/sqrt(3), 1/sqrt(3)];
case 3
w=[5/9, 8/9, 5/9]; pt=[-sqrt(3/5), 0, sqrt(3/5)];
case 4
w=[(18+sqrt(30))/36, (18+sqrt(30))/36, (18-sqrt(30))/36, (18-sqrt(30))/36];
pt=[-sqrt(3/7-2/7*(sqrt(6/5))), sqrt(3/7-2/7*(sqrt(6/5))),...
-sqrt(3/7+2/7*(sqrt(6/5))), sqrt(3/7+2/7*(sqrt(6/5)))];
case 5
w=[(322+13*sqrt(70))/900, (322+13*sqrt(70))/900, 128/225,...
(322-13*sqrt(70))/900,(322-13*sqrt(70))/900];
pt=[-1/3*sqrt(5-2*sqrt(10/7)), 1/3*sqrt(5-2*sqrt(10/7)),0,...
-1/3*sqrt(5+2*sqrt(10/7)), 1/3*sqrt(5+2*sqrt(10/7))];
otherwise
error('No data are defined for this value');
end
end
(c) Numerical Factory 2023<|endoftext|>
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by Texas Science Gal
5th – 8th Grade
In this download there are 5 different natural disaster vocabulary words and QR codes. This download focuses on the natural disasters that have strong impacts on the ecosystems including tornadoes, hurricanes, tsunamis, earthquakes, and volcanoes.
To use this, simply print these out for a group of 2-4 and have students sort out all of the cards. Once students have sorted the codes, you can give them the guided questions to help with comprehension. This helps assess what they have learned and put it into writing. There is also an answer key provided. I find that it makes a great way to integrate technology, get students hands on, and to reinforce the content knowledge.
and get THOUSANDS OF PAGE VIEWS for your TpT products!
Go to http://www.pinterest.com/TheBestofTPT/ for even more free products!<|endoftext|>
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Angela Duckworth’s "grit" has captured the imagination of educators, youth program leaders, and policymakers alike, leading many to agree that we should seek to cultivate grit in our youth. According to The Character Lab, grit is correlated with success and defined as "perseverance and passion for long-term goals”. It was further popularized by author Paul Tough (How Children Succeed: Grit, Curiosity, and the Hidden Power of Character, 2012) and others.
Like others, we have written a lot about grit in our LIAS blog. But others have called on us to look more closely at the notion of grit and how it intersects with issues of bias, poverty, inequality, deficit thinking, and race.
"Educational outcome disparities are not the result of deficiencies in marginalized communities' cultures, mindsets, or grittiness, but rather of inequities."
- Paul Gorski in Reaching and Teaching Students in Poverty
We believe that we can think more deeply about grit by reviewing these writings below:
- Rejecting “Grit” While Embracing Effort, Engagement
- What's Missing When We Talk About Grit
- Forget Grit. Focus on Inequality.
- Examining the cultural narrative around these ideologies
- What’s the Relationship Among Grit, Poverty, and Racism?
- 'Grit Is in Our DNA': Why Teaching Grit Is Inherently Anti-Black
- Emphasis on 'Grit' Is Unfair to Some Students, Critics Say
- Is 'Grit' Racist?
- The problem with teaching ‘grit’ to poor kids? They already have it. Here’s what they really need.
- Black and brown boys don’t need to learn “grit,” they need schools to stop being racist<|endoftext|>
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10 Virtual Tools for the Math Classroom
It is no secret that many students are not passionate about math. Students feel disconnected from what is taught in class, unsure of the benefits of math and reluctant to pursue careers in the field. Edtech is trying to change these attitudes by providing them with new ways to engage with numbers. Many companies have developed virtual tools for math, which allow students to learn, practice, and have fun with different math concepts. We will discuss ten of the best on the market.
- Stepping Stones 2.0: Comprehensive Mathematics– from ORIGO Education integrates print and digital resources to give teachers flexibility in how they teach K-6 math. SS 2.0 is loaded with additional practice, effective strategies, visual models, and teacher supports. Slatecast lets the teacher broadcast a resource onto the class whiteboard to emphasize or reteach a concept. Kathy Beach, a teacher in North Thurston Public Schools, says about State cast, “What a great way to practice facts and have everyone on the computer.”
- Geometry Pad– This virtual graph paper allows students to draw shapes, charts, and other geometric features. Students can change the properties of shapes, zoom in, save their work and add written notes on the side. Geometry Pad is a great application that can be used with students of any age and across mathematical disciplines.
- Pattern Shapes– Understanding the properties of shapes, fractions and creating precise figures is easy with Pattern Shapes. Students can use the virtual protractor to measure angles, change the dimensions and color of forms and annotate answers. It is ideal for elementary and middle school students, and the bright colored shapes can inspire creative design.
- Globaloria– Learning math through games is a great educational tool. Globaloria allows students to create games that test STEM subjects. With a gallery full of games, students can explore creations that were made by their peers. This application aims to promote STEM subjects on a global level through games and social networking.
- MathsPlayground– This collection of math-based games is perfect for younger students. Aligned with Common Core standards the games are separated by grade and topic. Students will enjoy learning while playing interesting games. The games test timetables, fractions, and other mathematical concepts. Combining education with easy to play games is what makes MathsPlayground ideal for young students.
- FluidMath– FluidMath is the first “pen-centric “platform that works on iPads and interactive whiteboards. Students and teachers can write, in their own handwriting, as they solve problems and engage with difficult concepts. FluidMath has won many awards, and its many features make it a great tool for both teachers and students in any math classroom.
- GetTheMath– The aim of this tool is to relate algebra to the real world. Through topics like “Math in Music” and “Math in Fashion,” students can learn how math is an integral part of everyday life. There are videos, exercises and other ways that students can engage with algebra in its real world setting. GetTheMath is an excellent way to combine theory with application.
- Dragon Box– This learner-based approach to math claims that 83% of children learn the basics of algebra in an hour. Through interactive games and explanations, students as young as five are introduced to algebra and how variables work. Students have no idea they are engaging with academic content, and the graphics are colorful and cute.
- Academy of Math– Aimed at children struggling with math in school, Academy of Math is a comprehensive tool that helps students get results. Videos and ongoing assessments tools put students in the driver’s seat of their own education. There are various topics to choose from, and educators can implement the resources on this platform into their teaching.
- Studygeek– Mathematical vocabulary is fundamental to understanding math. Study Geek is a great learning tool that has an alphabetical glossary of thousands of math vocabulary words. There is also a selection of informative videos that cover everything from geometry to algebra. The games aim is to test math vocabulary retention, and students will enjoy playing a game and learning at the same time.
So, there you have it. All of these tools push students towards self-exploration and allow them to see how math is an integral part of the world they live in. Through the use of these tools, students can also take control of their academic achievement, and foster a positive relationship with a subject that previously felt ambivalent about.<|endoftext|>
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Q:
# How many diagonals does a nonagon have?
A:
A nonagon, or enneagon, is a polygon with nine sides and nine vertices, and it has 27 distinct diagonals. The formula for determining the number of diagonals of an n-sided polygon is n(n - 3)/2; thus, a nonagon has 9(9 - 3)/2 = 9(6)/2 = 54/2 = 27 diagonals.
## Keep Learning
The diagonal of a polygon is any line segment joining two nonadjacent vertices. A nonagon has nine vertices, and each vertex has six other vertices that are not adjacent; thus, each vertex forms six diagonals. For each vertex, there are three fewer diagonals than there are vertices. This is the first part of the procedure for finding the number of diagonals of a polygon: n(n - 3) = 9 x 6 = 54. Each diagonal has two end points, so in order to not count a duplicate diagonal, the final step is to divide 54 by 2, which results in 27 diagonals.
Sources:
## Related Questions
• A:
A person draws a hexagon by drawing a closed polygon with six sides and six vertices. A regular hexagon has six equal sides and six equal angles. A person can draw a regular hexagon using a geometric construction.
Filed Under:
• A:
The polygon with 1,000 sides, 1,000 vertices and 1,000 angles is called a chiliagon or a 1,000-gon. The term "chiliagon," with a Greek-derived prefix meaning “thousand,” is a proposed name for the 1,000-sided, two-dimensional shape, but it is seldom used.
Filed Under:
• A:
A nine-sided shape is called a nonagon or enneagon. A nonagon is a type of polygon. Polygons are closed-plane figures that are made of line segments.<|endoftext|>
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# Spectrum Math Grade 5 Chapter 6 Lesson 6 Answer Key Dividing Whole Numbers by Fractions
Practice with the help of Spectrum Math Grade 5 Answer Key Chapter 6 Lesson 6.6 Dividing Whole Numbers by Fractions regularly and improve your accuracy in solving questions.
## Spectrum Math Grade 5 Chapter 6 Lesson 6.6 Dividing Whole Numbers by Fractions Answers Key
To divide a whole number by a fraction, first write the whole number as a fraction. Then, multiply by the reciprocal of the divisor.
Divide. Write answers in simplest form.
Question 1.
a. 5 ÷ $$\frac{1}{3}$$ = _______________
15
Explanation:
Given,
5 ÷ $$\frac{1}{3}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{5}{1}$$ ÷ $$\frac{1}{3}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{5}{1}$$ × $$\frac{3}{1}$$
Multiply across numerators and denominators.
$$\frac{5 \times 3}{1 \times 1}$$ = 15
b. 6 ÷ $$\frac{1}{8}$$ = _______________
48
Explanation:
Given,
6 ÷ $$\frac{1}{8}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{6}{1}$$ ÷ $$\frac{1}{8}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{6}{1}$$ × $$\frac{8}{1}$$
Multiply across numerators and denominators.
$$\frac{6 \times 8}{1 \times 1}$$ = 48
c. 2 ÷ $$\frac{1}{5}$$ = _______________
10
Explanation:
Given,
2 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{2}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{2}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{2 \times 5}{1 \times 1}$$ = 10
d. 8 ÷ $$\frac{1}{7}$$ = _______________
56
Explanation:
Given,
8 ÷ $$\frac{1}{7}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{8}{1}$$ ÷ $$\frac{1}{7}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{8}{1}$$ × $$\frac{7}{1}$$
Multiply across numerators and denominators.
$$\frac{8 \times 7}{1 \times 1}$$ = 56
Question 2.
a. 9 ÷ $$\frac{1}{4}$$ = _______________
36
Explanation:
Given,
9 ÷ $$\frac{1}{4}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{9}{1}$$ ÷ $$\frac{1}{4}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{9}{1}$$ × $$\frac{4}{1}$$
Multiply across numerators and denominators.
$$\frac{9 \times 4}{1 \times 1}$$ = 36
b. 10 ÷ $$\frac{1}{6}$$ = _______________
60
Explanation:
Given,
10 ÷ $$\frac{1}{6}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{10}{1}$$ ÷ $$\frac{1}{6}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{10}{1}$$ × $$\frac{6}{1}$$
Multiply across numerators and denominators.
$$\frac{10 \times 6}{1 \times 1}$$ = 60
c. 15 ÷ $$\frac{1}{5}$$ = _______________
75
Explanation:
Given,
15 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{15}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{15}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{15 \times 5}{1 \times 1}$$ = 75
d. 4 ÷ $$\frac{1}{8}$$ = _______________
32
Explanation:
Given,
4 ÷ $$\frac{1}{8}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{4}{1}$$ ÷ $$\frac{1}{8}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{4}{1}$$ × $$\frac{8}{1}$$
Multiply across numerators and denominators.
$$\frac{4 \times 8}{1 \times 1}$$ = 32
Question 3.
a. 4 ÷ $$\frac{1}{5}$$ = _______________
20
Explanation:
Given,
4 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{4}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{4}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{4 \times 5}{1 \times 1}$$ = 20
b. 5 ÷ $$\frac{1}{9}$$ = _______________
45
Explanation:
Given,
5 ÷ $$\frac{1}{9}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{5}{1}$$ ÷ $$\frac{1}{9}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{5}{1}$$ × $$\frac{9}{1}$$
Multiply across numerators and denominators.
$$\frac{5 \times 9}{1 \times 1}$$ = 45
c. 5 ÷ $$\frac{1}{5}$$ = _______________
25
Explanation:
Given,
5 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{5}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{5}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{5 \times 5}{1 \times 1}$$ = 25
d. 10 ÷ $$\frac{1}{11}$$ = _______________
110
Explanation:
Given,
10 ÷ $$\frac{1}{11}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{10}{1}$$ ÷ $$\frac{1}{11}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{10}{1}$$ × $$\frac{11}{1}$$
Multiply across numerators and denominators.
$$\frac{10 \times 11}{1 \times 1}$$ = 110
Question 4.
a. 4 ÷ $$\frac{1}{12}$$ = _______________
48
Explanation:
Given,
4 ÷ $$\frac{1}{12}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{4}{1}$$ ÷ $$\frac{1}{12}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{4}{1}$$ × $$\frac{12}{1}$$
Multiply across numerators and denominators.
$$\frac{4 \times 12}{1 \times 1}$$ = 48
b. 6 ÷ $$\frac{1}{9}$$ = _______________
54
Explanation:
Given,
6 ÷ $$\frac{1}{9}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{6}{1}$$ ÷ $$\frac{1}{9}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{6}{1}$$ × $$\frac{9}{1}$$
Multiply across numerators and denominators.
$$\frac{6 \times 9}{1 \times 1}$$ = 54
c. 3 ÷ $$\frac{1}{7}$$ = _______________
21
Explanation:
Given,
3 ÷ $$\frac{1}{7}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{3}{1}$$ ÷ $$\frac{1}{7}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{3}{1}$$ × $$\frac{7}{1}$$
Multiply across numerators and denominators.
$$\frac{3 \times 7}{1 \times 1}$$ = 21
d. 5 ÷ $$\frac{1}{12}$$ = _______________
35
Explanation:
Given,
5 ÷ $$\frac{1}{12}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{5}{1}$$ ÷ $$\frac{1}{12}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{5}{1}$$ × $$\frac{12}{1}$$
Multiply across numerators and denominators.
$$\frac{5 \times 12}{1 \times 1}$$ = 60
Divide. Write answers in simplest form.
Question 1.
a. 4 ÷ $$\frac{1}{3}$$ = _______________
12
Explanation:
Given,
4 ÷ $$\frac{1}{3}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{4}{1}$$ ÷ $$\frac{1}{3}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{4}{1}$$ × $$\frac{3}{1}$$
Multiply across numerators and denominators.
$$\frac{4 \times 3}{1 \times 1}$$ = 12
b. 12 ÷ $$\frac{1}{5}$$ = _______________
60
Explanation:
Given,
12 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{12}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{12}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{12 \times 5}{1 \times 1}$$ = 60
c. 19 ÷ $$\frac{1}{6}$$ = _______________
114
Explanation:
Given,
19 ÷ $$\frac{1}{6}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{19}{1}$$ ÷ $$\frac{1}{6}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{19}{1}$$ × $$\frac{6}{1}$$
Multiply across numerators and denominators.
$$\frac{19 \times 6}{1 \times 1}$$ = 114
d. 10 ÷ $$\frac{1}{6}$$ = _______________
60
Explanation:
Given,
10 ÷ $$\frac{1}{6}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{10}{1}$$ ÷ $$\frac{1}{6}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{10}{1}$$ × $$\frac{6}{1}$$
Multiply across numerators and denominators.
$$\frac{10 \times 6}{1 \times 1}$$ = 60
Question 2.
a. 17 ÷ $$\frac{1}{4}$$ = _______________
68
Explanation:
Given,
17 ÷ $$\frac{1}{4}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{17}{1}$$ ÷ $$\frac{1}{4}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{17}{1}$$ × $$\frac{4}{1}$$
Multiply across numerators and denominators.
$$\frac{17 \times 4}{1 \times 1}$$ = 68
b. 16 ÷ $$\frac{1}{9}$$ = _______________
144
Explanation:
Given,
16 ÷ $$\frac{1}{9}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{16}{1}$$ ÷ $$\frac{1}{9}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{16}{1}$$ × $$\frac{9}{1}$$
Multiply across numerators and denominators.
$$\frac{16 \times 9}{1 \times 1}$$ = 144
c. 9 ÷ $$\frac{1}{6}$$ = _______________
54
Explanation:
Given,
9 ÷ $$\frac{1}{6}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{9}{1}$$ ÷ $$\frac{1}{6}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{9}{1}$$ × $$\frac{6}{1}$$
Multiply across numerators and denominators.
$$\frac{9 \times 6}{1 \times 1}$$ = 54
d. 7 ÷ $$\frac{1}{2}$$ = _______________
14
Explanation:
Given,
7 ÷ $$\frac{1}{2}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{7}{1}$$ ÷ $$\frac{1}{2}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{7}{1}$$ × $$\frac{2}{1}$$
Multiply across numerators and denominators.
$$\frac{7 \times 2}{1 \times 1}$$ = 14
Question 3.
a. 2 ÷ $$\frac{1}{5}$$ = _______________
10
Explanation:
Given,
2 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{2}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{2}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{2 \times 5}{1 \times 1}$$ = 10
b. 14 ÷ $$\frac{1}{5}$$ = _______________
70
Explanation:
Given,
14 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{14}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{14}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{14 \times 5}{1 \times 1}$$ = 70
c. 4 ÷ $$\frac{1}{10}$$ = _______________
40
Explanation:
Given,
4 ÷ $$\frac{1}{10}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{4}{1}$$ ÷ $$\frac{1}{10}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{4}{1}$$ × $$\frac{10}{1}$$
Multiply across numerators and denominators.
$$\frac{4 \times 10}{1 \times 1}$$ = 40
d. 8 ÷ $$\frac{1}{8}$$ = _______________
64
Explanation:
Given,
8 ÷ $$\frac{1}{8}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{8}{1}$$ ÷ $$\frac{1}{8}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{8}{1}$$ × $$\frac{8}{1}$$
Multiply across numerators and denominators.
$$\frac{8 \times 8}{1 \times 1}$$ = 64
Question 4.
a. 2 ÷ $$\frac{1}{7}$$ = _______________
14
Explanation:
Given,
2 ÷ $$\frac{1}{7}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{2}{1}$$ ÷ $$\frac{1}{7}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{2}{1}$$ × $$\frac{7}{1}$$
Multiply across numerators and denominators.
$$\frac{2 \times 7}{1 \times 1}$$ = 14
b. 16 ÷ $$\frac{1}{5}$$ = _______________
80
Explanation:
Given,
16 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{16}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{16}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{16 \times 5}{1 \times 1}$$ = 80
c. 13 ÷ $$\frac{1}{5}$$ = _______________
65
Explanation:
Given,
13 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{13}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{13}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{13 \times 5}{1 \times 1}$$ = 65
d. 12 ÷ $$\frac{1}{3}$$ = _______________
36
Explanation:
Given,
12 ÷ $$\frac{1}{3}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{12}{1}$$ ÷ $$\frac{1}{3}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{12}{1}$$ × $$\frac{3}{1}$$
Multiply across numerators and denominators.
$$\frac{12 \times 3}{1 \times 1}$$ = 36
Question 5.
a. 5 ÷ $$\frac{1}{7}$$ = _______________
35
Explanation:
Given,
5 ÷ $$\frac{1}{7}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{5}{1}$$ ÷ $$\frac{1}{7}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{5}{1}$$ × $$\frac{7}{1}$$
Multiply across numerators and denominators.
$$\frac{5 \times 7}{1 \times 1}$$ = 35
b. 3 ÷ $$\frac{1}{9}$$ = _______________
27
Explanation:
Given,
3 ÷ $$\frac{1}{9}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{3}{1}$$ ÷ $$\frac{1}{9}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{3}{1}$$ × $$\frac{9}{1}$$
Multiply across numerators and denominators.
$$\frac{3 \times 9}{1 \times 1}$$ = 27
c. 15 ÷ $$\frac{1}{8}$$ = _______________
120
Explanation:
Given,
15 ÷ $$\frac{1}{8}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{15}{1}$$ ÷ $$\frac{1}{8}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{15}{1}$$ × $$\frac{8}{1}$$
Multiply across numerators and denominators.
$$\frac{15 \times 8}{1 \times 1}$$ = 120
d. 6 ÷ $$\frac{1}{7}$$ = _______________
42
Explanation:
Given,
6 ÷ $$\frac{1}{7}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{6}{1}$$ ÷ $$\frac{1}{7}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{6}{1}$$ × $$\frac{7}{1}$$
Multiply across numerators and denominators.
$$\frac{6 \times 7}{1 \times 1}$$ = 42
Question 6.
a. 11 ÷ $$\frac{1}{2}$$ = _______________
22
Explanation:
Given,
11 ÷ $$\frac{1}{2}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{11}{1}$$ ÷ $$\frac{1}{2}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{11}{1}$$ × $$\frac{2}{1}$$
Multiply across numerators and denominators.
$$\frac{11 \times 2}{1 \times 1}$$ = 22
b. 19 ÷ $$\frac{1}{3}$$ = _______________
57
Explanation:
Given,
19 ÷ $$\frac{1}{3}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{19}{1}$$ ÷ $$\frac{1}{3}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{19}{1}$$ × $$\frac{3}{1}$$
Multiply across numerators and denominators.
$$\frac{19 \times 3}{1 \times 1}$$ = 57
c. 8 ÷ $$\frac{1}{9}$$ = _______________
72
Explanation:
Given,
8 ÷ $$\frac{1}{9}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{8}{1}$$ ÷ $$\frac{1}{9}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{8}{1}$$ × $$\frac{9}{1}$$
Multiply across numerators and denominators.
$$\frac{8 \times 9}{1 \times 1}$$ = 72
d. 18 ÷ $$\frac{1}{5}$$ = _______________
90
Explanation:
Given,
18 ÷ $$\frac{1}{5}$$
To divide a whole number by a fraction,
first write the whole number as a fraction.
$$\frac{18}{1}$$ ÷ $$\frac{1}{5}$$
Then, multiply by the reciprocal of the divisor.
$$\frac{18}{1}$$ × $$\frac{5}{1}$$
Multiply across numerators and denominators.
$$\frac{18 \times 5}{1 \times 1}$$ = 90
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Scroll to Top<|endoftext|>
| 4.84375 |
466 |
# Area of Parallelograms
Warmup
Activity #1
`Find the Area of a Parallelogram.`
Although it is impossible to fit a whole number of unit squares into a rectangle that is 5 3/4 units long and √7 units wide, for example, we declare, nonetheless, that the area of such a rectangle is the product of these two numbers.
• Move the slider below the parallelogram back and forth to change the value of the base.
• Leave the slider at the right end.
• Move the vertical slider up and down the scale to change the value of the height.
• Leave the slider at the top.
• Move the Hint slider back and forth to toggle between a parallelogram and a square.
• Answer the questions below.
Activity #2
`Explore the Maximum Area of a Rectangle Given a Perimeter.`
In the following activity, we are going to be working with different dimensions of a rectangle with a fixed perimeter. Our goal is to find out which dimension will give us the greatest area. This is an important application to land developers. Follow the instructions below.
• Move the slider to adjust the rectangles width.
• Record the widths & areas for w = 15, 30, 45, 60, 75, 90, 105, 120, 135.
• Observe when the areas are the same.
• Observe the maximum area.
• Answer the questions below.
(1.) How is the length being determined?
(2.) Describe the rectangle that has the maximum area.
Tyler was trying to find the area of the parallelogram below.
• Move the slider to see how he did it.
Elena was also trying to find the area of a parallelogram.
• Move the slider to see how she did it.
• Answer the questions below.
(1.) How are the two strategies for finding the area of a parallelogram the same?
(2.) How are they different?
Quiz Time
The quiz below can be completed at www.ixl.com. After a trial practice, click on our subscription link in the video, to see how much you’ll save when you signup from our website.<|endoftext|>
| 4.40625 |
790 |
Algebra Tutorials!
Try the Free Math Solver or Scroll down to Tutorials!
Depdendent Variable
Number of equations to solve: 23456789
Equ. #1:
Equ. #2:
Equ. #3:
Equ. #4:
Equ. #5:
Equ. #6:
Equ. #7:
Equ. #8:
Equ. #9:
Solve for:
Dependent Variable
Number of inequalities to solve: 23456789
Ineq. #1:
Ineq. #2:
Ineq. #3:
Ineq. #4:
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Ineq. #6:
Ineq. #7:
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Solve for:
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8.1 Definitions and Roots
▪ Need To Know
▪ How to find the root of a number
▪ Categorizing roots
▪ How to find the root of an expression
▪ How to solve application problems
involving roots
Idea of Square Roots
In mathematics, once we learn an operation,
we also learn the reverse of that operation.
For real numbers x and y,
If y = x2 then x = ___
Don’t Forget
Definitions and Roots
Definition – If x is any positive real number, then
is the square root of x and
is the square root of x.
Examples:
Name the two square roots of 81.
Find the root of
Simplify
Evaluate
Recall Number Sets
▪ Rational Numbers
are the numbers of
the form a/b where
b is not zero.
▪ Rational Numbers is
the set of all fractions
Recall Number Sets
▪ Irrational Numbers
are the numbers
that can not be
written as fractions
▪ Real Numbers are
the collection of all
Rational and
Irrational Numbers.
Categorize Roots
Square Roots & Absolute Value
Consider and
Both come out positive.
Recall: |-7| = 7 and |7| = 7.
For all real numbers A, ___________________
Examples:
Application
The speed of sound in feet per second, V, traveling
through air with a temperature of t is given by the
formula below. Find the speed of sound when the
temperature is 5° C.
8.1 Conclusion
▪ The square root is the number that
“undoes” the square.
▪ Square roots can be positive or negative.
▪ The square root of zero is zero.
▪ The square root of a negative number is
not a Real Number
▪ A square root results in a
Rational, Irrational or Non-Real Number
▪ Need To Know
1. With numbers
2. With variables
▪ If A and B are real numbers (> 0), then
______________________________
▪ Simplify: Perfect Square
▪ Simplify: Perfect Square
▪ Simplify:
Square roots
undo squares
▪ Simplify:
end
▪ Need To Know
▪ Quotient Rule for Square Roots
1. With Fractions
2. By Rationalizing the Denominator
Quotient Rule for Square Roots
▪ If A and B are real numbers (B ≠ 0),
then
__________________________________
▪ Simplify:
The Idea of a Simplified Radical
Which fraction is the simplest?
Rationalizing Denominators
▪ Goal: Change the fraction to make the
denominator come out “nice”.
Guidelines for Simplification
1. Remove ____________________________
2. Remove ____________________________
3. Remove ____________________________
▪ Examples:
Guidelines for Simplification<|endoftext|>
| 4.4375 |
486 |
BY BOB MCNITT AND DR. WILLIAM W. MILLER
Understanding How Deer See Will Affect How You Hunt Them
Anyone who hunts deer quickly learns that the animal’s primary survival tool is its remarkable ability to accurately detect and identify even the most subtle smells. Let a deer pick up your scent and it’s game over. And benefiting from its mobile, oversized ears, it can quickly detect the slightest of sounds — such as clothing brushing against a small branch as a hunter moves, or the click of a gun’s safety being released. Given, the deer’s sense of smell and hearing (to a lesser extent) is far superior to ours, but what about its eyesight? How does it really see the world, especially when compared to what we see? Can deer see orange? Are they color blind? Do they see in just black and white? Is their vision good or poor?
Realtree asked one of the nation’s foremost experts on animal ophthalmology — the study and treatment of animal eye diseases and traits — Dr. William W. Miller, professor of veterinary ophthalmology at Mississippi State University. Miller is also an avid deer hunter.
DEER EYE ANATOMY
“To understand what deer see you have to know a little bit about the anatomy of their eyes,” Miller says. “Let’s begin with the front of the eye and work our way to the back. The front of the eye is the cornea. The cornea is clear and serves as the window through which light enters the eye. Like most herbivores (cows, elk, horses) deer have a large cornea that allows a maximum amount of light to enter the eye. Its large size provides for a wide field of view, giving optimum peripheral vision.
“The next aid to the deer’s vision is the pupil, the opening in the iris through which light passes to reach the retina. Since deer are herbivores, they graze or browse. The pupil of the deer, like those of cattle, elk, sheep, and caribou, for example, are oval or rectangular with the long axis of the pupil parallel to the horizon. You have to remember deer and the other examples are prey species, or designed to be eaten by predators.”<|endoftext|>
| 3.71875 |
1,482 |
# Exterior angle of a triangle
🏆Practice parts of a triangle
The exterior angle of a triangle is the one that is found between the original side and the extension of the side.
The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.
It is defined as follows:
$α=∢A+∢B$
## Test yourself on parts of a triangle!
Is the straight line in the figure the height of the triangle?
## Exterior angle of a triangle
Until today we have dealt with internal angles, perhaps also with adjacent angles, but we have not talked about external angles. Don't worry, the topic of the exterior angle of a triangle is very easy to understand and its property can be very useful for solving exercises more quickly.
Shall we start?
## What is the exterior angle of a triangle?
The exterior angle of a triangle is the one that is found between the original side and the extension of the side.
Join Over 30,000 Students Excelling in Math!
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## What does the continuation of the side mean?
Imagine someone draws a triangle and falls asleep as they are finishing it.
Without realizing it, they continue drawing one side a little more...
and Bam! An exterior angle is created.
The exterior angle is outside of the triangle and is found between the original side and the side they continued drawing while asleep (the continuation of the side).
### Let's look at an example
Observe: An exterior angle is the one that is found between an original side of the triangle and the extension of the side and not between two extensions.
Note:
Whenever the angle is outside the triangle and is found between an original side of the triangle and the extension of another side of the triangle, it will be considered an exterior angle of the triangle.
Do you know what the answer is?
### Examples of Exterior Angles
Great! Now that we have understood what an exterior angle is and that we can recognize it from a distance, we can move on to the property of the exterior angle of a triangle.
Property of the exterior angle of a triangle
The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.
Given that:
$∢A=80$
$∢B=20$
How much does the exterior angle measure?
Solution:
Let's denote the exterior angle with $α$:
According to the property of the exterior angle of the triangle, the exterior angle $α$ must be equal to the sum of the two interior angles of the triangle that are not adjacent to it.
That is, $∢A+∢B$
Therefore, all we have to do is add them up and find out the exterior angle:
$α=80+20$
$α=100$
Look, we could have found the value of the exterior angle in another way!
We know that the sum of the interior angles of a triangle is $180$.
Therefore, $∢ACB=180-20-80$
$∢ACB=80$
$∢ABC$ is the angle adjacent to $α$, the exterior angle of the triangle that we need to find out.
We also know that the sum of the adjacent angles is $180$.
Therefore we can determine that:
$∢80+α=80$
$α=100$
Look, In certain cases you will not be explicitly asked for the value of the exterior angle.
They might ask you, for example, about some interior angle of the triangle that you could figure out through the exterior angle.
### Let's look at an example
Given the following triangle:
Data:
$∢A=90$
$α=110$
Find the value of $∢B$
Solution:
We can solve the problem in two ways:
The first is based on the Exterior Angle Theorem of a triangle and understand that $α$ is an exterior angle of the triangle and is equal to the sum of the two interior angles that are not adjacent to it. That is, $∢A+∢B$
Then,
the equation would be:
$110=90+∢B$
$∢B=20$
The second way to solve the problem is to remember that the sum of the adjacent angles equals $180$, then $∢ACB$ is equal to $70$.
$180-110=70$
Now, let's remember that the sum of the interior angles of a triangle is $180$
and we can find $∢B$
$∢B=180-90-70$
$∢B=20$
Notice that we have arrived at the same result, but solving through the property of the exterior angle of a triangle has been faster to reach it.
Useful Information:
The sum of the three exterior angles of a triangle equals $360$ degrees.
In conclusion, it is important and really worth knowing the property of the exterior angle of a triangle to solve problems easily and quickly, although in several cases you will be able to manage without this magnificent theorem.
## Examples and exercises with solutions of an exterior angle of a triangle
### examples.example_title
Which of the following is the height in triangle ABC?
### examples.explanation_title
Let's remember the definition of height:
A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.
Therefore, the one that forms a 90-degree angle is side AB with side BC
AB
### examples.example_title
Given the isosceles triangle ABC,
The side AD is the height in the triangle ABC
and inside it, EF is drawn:
AF=5 AB=17
What is the perimeter of the trapezoid EFBC?
### examples.explanation_title
To find the perimeter of the trapezoid, all its sides must be added:
We will focus on finding the bases.
To find GF we use the Pythagorean theorem: $A^2+B^2=C^2$in the triangle AFG
We replace
$3^2+GF^2=5^2$
We isolate GF and solve:
$9+GF^2=25$
$GF^2=25-9=16$
$GF=4$
We perform the same process with the side DB of the triangle ABD:
$8^2+DB^2=17^2$
$64+DB^2=289$
$DB^2=289-64=225$
$DB=15$
We start by finding FB:
$FB=AB-AF=17-5=12$
Now we reveal EF and CB:
$GF=GE=4$
$DB=DC=15$
This is because in an isosceles triangle, the height divides the base into two equal parts so:
$EF=GF\times2=4\times2=8$
$CB=DB\times2=15\times2=30$
All that's left is to calculate:
$30+8+12\times2=30+8+24=62$
62<|endoftext|>
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Jump to a New ChapterIntroduction to the SAT IIIntroduction to SAT II Math ICStrategies for SAT II Math ICMath IC FundamentalsAlgebraPlane GeometrySolid GeometryCoordinate GeometryTrigonometryFunctionsStatisticsMiscellaneous MathPractice Tests Are Your Best Friends
11.1 Statistical Analysis 11.2 Probability 11.3 Permutations and Combinations 11.4 Group Questions
11.5 Sets 11.6 Key Formulas 11.7 Review Questions 11.8 Explanations
Sets
Already in this chapter we’ve covered how to analyze the data in a set and how to deal with two sets that have overlapping members. For the Math IC, there are two more concepts concerning sets that you need to understand: union and intersection.
Union
The union of two or more sets is the set that contains all of the elements of the two original sets. The union of two sets A and B is symbolized this way:
For example, the union of the sets A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} is
This set contains every element that is in either set. If x is an element of then it must be an element of A, or of B, or of both.
Intersection
The intersection of two sets is the set of their overlapping elements. The intersection of the two sets A and B is symbolized as
The intersection of the sets A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, for example, is = {4,5}. If x is an element of then x must be an element of both A and B.
Jump to a New ChapterIntroduction to the SAT IIIntroduction to SAT II Math ICStrategies for SAT II Math ICMath IC FundamentalsAlgebraPlane GeometrySolid GeometryCoordinate GeometryTrigonometryFunctionsStatisticsMiscellaneous MathPractice Tests Are Your Best Friends
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# Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT
ISBN: 9780132296540 301
## Solution for problem 27 Chapter 2.3
Elementary Linear Algebra with Applications | 9th Edition
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Problem 27
Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT is column equivalent to
Step-by-Step Solution:
Step 1 of 3
Math246 Lecture 6: Second Order Differential Equations a y +b y cy=h(t) Now, we will be looking at second order differential equations of the form . Of particular interest is when h(t)=0. Let’s look at some examples. General Problems y +25y=0 1. What function essentially doesn’t change when you take its derivative Yes, the x exponential function e . So now all we need to do is get the exponent correctly. One e5t 25e 3t solution is . If you take the derivative twice, you end up with . Another 5t −5t solution is e−5t . A more general solution is y (t=ae +be . 2. Let’s now solve the same problem with some initial conditions. y(0)=5 and y'(0)=20 . So we need to take the derivative of the general solution first. ' 5t −5t y t =5ae −5be Now we plug in the initial conditions. 20=5a−5b 5=a+b a=5−b b So and we can plug that into the first equation and solve for . 20=25−5b−5b=25−10 b 1
Step 2 of 3
Step 3 of 3
## Discover and learn what students are asking
Calculus: Early Transcendental Functions : Inverse Trigonometric Functions: Integration
?In Exercises 1-20, find the indefinite integral. $$\int \frac{12}{1+9 x^{2}} d x$$
Statistics: Informed Decisions Using Data : Properties of the Normal Distribution
?In Problems 25–28, the graph of a normal curve is given. Use the graph to identify the values of ? and ?.
Statistics: Informed Decisions Using Data : Estimating a Population Mean
?The procedure for constructing a t-interval is robust. Explain what this means.
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Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT<|endoftext|>
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Many Western European languages use the same alphabet as English, with one significant difference. There can be diacritics (or accents) above certain characters.
For a student of a foreign language, it’s important to place those accents correctly. Sometimes these tiny small markers can be forgotten, for example when conjugating a verb. If the accents are forgotten, the verb conjugation may even fail.
To check that the accents are placed correctly, have a look at the reverse conjugator. There you can write the infinitive without accents and the reverse conjugator tells, whether accents should be added or not.
Check for instance the Spanish verb ‘reir’. (Note! I misspelled it on purpose)
Verb conjugation is the process of forming all the verbal forms from the dictionary lookup word. (generally infinitive). Reverse conjugation means getting the infinitive from any conjugated verb form.
In linguistics revers conjugation would rather be named morphological analysis. In morphological analysis the different parts of word are analyzed: the stem and the modifiers. The modifiers are those parts of a verb that denote mood, tense, number, person, etc.
In Verbix the reverse conjugation (or morphological analysis) is made simple. The user simply enters any verb form, and Verbix tells if it’s a verb or not. If it’s a verb, then Verbix returns the infinitives that can be conjugated.
The English language has a limited number of irregular verbs. Once you learn them, it’s pretty straight-forward to conjugate all English verbs.
There are, however, a number of verbs that are regular but undergo orthographical changes. Orthographical changes are there to ensure that the pronunciation is preserved in the different forms of a verb.
’To panic’ is one of these verbs. An automated verb conjugator might conjugate the verb in past ’paniced’, but that’s wrong. The correct past is ’panicked’, so there is a ’k’ attached to the stem to preserve the pronunciation.
The dictionary of Verbix knows a half dozen of verbs that are conjugated like panic. And the past forms are marked in blue to denote the orthographic change.
In addition the built-in rules of Verbix also know orthographic rules. So, don’t panic! Verbix knows how to conjugate verbs ending in ’c’.
Links to go:
Somewhen in the past I got familiar with a site called Ethnologue. It contains information about all languages in the world.
One thing that I was missing though is the lack of maps that would better show where the languages are spoken. At that time there wasn’t any other site either that would contain many languages plotted on the map.
So where is Muna spoken for instance?
Verb conjugation is the central part of the sentence in most languages.
Verb conjugation is the creation of derived forms of a verb from its principal parts by inflection. Principal parts is sometimes the infinitive like “cantar” in Spanish, but it can also be verb theme like “skriva – skriver – skrev -skrivit” in Swedish.
In Spanish it’s enough to know the infinitive of a verb to get all the conjugated forms; in the case of regular verbs all the conjugated forms are formed with the same set of endings. Unfortunately there is a big amount of irregular verbs that don’t follow the regular verb conjugation patters.
As shown in the example above, in Swedish verb conjugation there’s a verb theme consisting of four verb forms. All Swedish verb forms are formed by applying the same set of endings to the theme. The theme itself must be memorized, because it contains information about irregularities, if any.<|endoftext|>
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# Hypothesis Testing: An Overview
1. A Level Maths Topics
2. Statistics Topics
3. Hypothesis Testing
Hypothesis testing is an important part of statistical analysis and a key component of A Level maths and statistics. It is a tool used to draw conclusions about a population based on a sample of data. By testing hypotheses, we can make decisions about whether a hypothesis is true or false, and use this knowledge to inform our understanding of the population. Hypothesis testing is a powerful tool for discovering and understanding relationships between variables. It helps us to evaluate the reliability of our data and to identify potential issues or biases in our results.
Hypothesis testing can also help us to understand the causes behind a particular phenomenon, and to gain insights into the behaviour of a population. In this article, we will explore the basics of hypothesis testing. We'll look at how to formulate hypotheses, select appropriate tests, interpret results, and draw conclusions. We'll also discuss some of the limitations of hypothesis testing and how it can be used in practice.
#### Hypothesis testing
is an important concept in A Level Maths and Statistics. It involves using statistical methods to test a hypothesis about a population parameter.
Hypothesis testing is used to make decisions about whether a particular hypothesis is true or false. It is also used to compare groups of data in order to determine if there are any significant differences between them. The basic process of hypothesis testing involves formulating an initial hypothesis, gathering data, performing the necessary calculations and then interpreting the results. The types of hypotheses that can be tested using hypothesis testing include null hypotheses, which assume that there is no difference between two variables, and alternative hypotheses, which state that there is a difference between two variables. There are also directional hypotheses, which predict the direction of the difference (e.g., if variable A is greater than variable B).When performing a hypothesis test, it is important to make certain assumptions about the data.
The most common assumption is that the data follows a normal distribution. It is also important to assume that the sample size is large enough to accurately represent the population and that the data is independent and identically distributed. Calculating a hypothesis test involves several steps. First, the null hypothesis must be tested against the alternative hypothesis. The next step is to calculate the test statistic, which is used to determine whether or not the null hypothesis should be rejected or accepted.
Common tests include the t-test, chi-square test and ANOVA. After calculating the test statistic, it must be compared to a critical value based on the level of significance that was set before beginning the hypothesis test. Interpreting the results of a hypothesis test can be tricky. If the test statistic is greater than the critical value, then the null hypothesis can be rejected and the alternative hypothesis accepted. However, it is also important to take into account any errors that may have been made in formulating the hypothesis or collecting and analyzing the data.
Common errors include Type I errors (false positives) and Type II errors (false negatives).Finally, there are several tips for performing a successful hypothesis test. It is important to ensure that all assumptions are met, such as having an adequate sample size and ensuring that the data follows a normal distribution. It is also important to clearly define all hypotheses before beginning the test and to set an appropriate level of significance. Finally, it is important to interpret the results of a hypothesis test cautiously and take into account any errors that may have been made.
## The Basics of Hypothesis Testing
Hypothesis testing is an important concept in A Level Maths and Statistics.
In order to do this, it is necessary to understand the basics of hypothesis testing. The process of hypothesis testing involves making an initial statement (the hypothesis) about a population parameter. This statement is then tested using statistical methods to determine whether it is likely to be true or false. If the hypothesis is likely to be true, it is accepted; if it is unlikely to be true, it is rejected.
There are two types of hypotheses that can be tested: one-tailed and two-tailed. A one-tailed test looks at whether the population parameter is greater than or less than a certain value, while a two-tailed test looks at whether the population parameter is equal to a certain value. In order for a hypothesis test to be valid, certain assumptions must be made. These include assumptions about the sample size, the distribution of the population, and the level of significance.
It is important to ensure that these assumptions are met in order for the results of the test to be reliable.
## Tips for Successful Hypothesis Testing
When performing hypothesis testing, it is important to be aware of the different types of errors that may occur. There are two main types of errors: type I errors and type II errors. A type I error occurs when the null hypothesis is rejected even though it is actually true, while a type II error occurs when the null hypothesis is accepted even though it is false. It is important to be aware of these errors and to understand how to avoid them when performing a hypothesis test. Another important factor to consider when performing a hypothesis test is the level of significance.
This is the probability that an observed result would occur if the null hypothesis was true. Generally, the lower the level of significance, the more reliable the results will be. When selecting a level of significance, it is important to consider the practical implications of the results. Before beginning a hypothesis test, it is also important to determine the appropriate sample size. The sample size should be large enough to provide reliable results, but not so large that it becomes cost-prohibitive or time-consuming.
Choosing an appropriate sample size will help ensure that the results of the hypothesis test are valid. Finally, it is important to remember that hypothesis testing is an iterative process. After performing a test, it is important to analyze and interpret the results, and then decide if further tests are needed. By following these steps, it is possible to conduct successful hypothesis testing.
## Calculating Hypothesis Tests
Hypothesis testing involves using a variety of statistical tests to draw conclusions about a population parameter. In this section, we will look at how to calculate specific tests such as the t-test, chi-square test, and ANOVA. The t-test is used to compare the means of two independent samples.
It can be used to determine if there is a statistically significant difference between the two samples. To calculate the t-test, you need to calculate the difference between the two sample means, and then divide this by the standard error of the difference. The resulting statistic can then be compared to a critical value for significance. The chi-square test is used to compare observed frequencies with expected frequencies. It is often used in hypothesis testing to determine if two categorical variables are related.
To calculate the chi-square test, you need to calculate the difference between the observed and expected frequencies and then divide this by the expected frequency. The resulting statistic can then be compared to a critical value for significance. The ANOVA (Analysis of Variance) test is used to compare the means of three or more independent samples. It can be used to determine if there is a statistically significant difference between the means of the samples. To calculate the ANOVA, you need to calculate the difference between the sample means and then divide this by the variance of the differences.
The resulting statistic can then be compared to a critical value for significance. In conclusion, hypothesis testing involves using a variety of statistical tests to draw conclusions about a population parameter. The t-test, chi-square test, and ANOVA are all important tests that are used in hypothesis testing and can be calculated using the methods outlined above.
## What is Hypothesis Testing?
Hypothesis testing is a statistical process used to determine whether a particular hypothesis about a population parameter is true or false. It involves the use of data and statistical methods to compare the results of an experiment or study to what would be expected if the hypothesis were true. Hypothesis testing is an important concept in A Level Maths and Statistics, as it can be used to determine the accuracy of a hypothesis and the validity of results.
The basic idea behind hypothesis testing is that an experimenter must decide whether or not the results of an experiment support or reject a particular hypothesis. This process requires the use of data and statistical methods to compare the results of an experiment to what would be expected if the hypothesis were true. If the results of an experiment differ significantly from what would be expected if the hypothesis were true, then the hypothesis can be rejected. Hypothesis testing is important because it allows researchers to determine the validity of their findings.
By testing a hypothesis, researchers can ensure that their findings are reliable and accurate. This process can also help researchers identify potential sources of error, which can then be addressed in future experiments.
## Interpreting Results
Interpreting the results of a hypothesis test can be tricky, as there is the potential for errors that could lead to incorrect conclusions. It is important to understand the different types of errors that can occur when interpreting the results of a hypothesis test, as well as how to identify them.
#### Type 1 Error:
A type 1 error, also known as a false positive, occurs when an experiment rejects a null hypothesis that should not have been rejected.
This means that the experimenter incorrectly concludes that a difference exists when in reality it does not. For example, if a researcher was testing whether or not a new drug was effective in treating a certain condition, a type 1 error would occur if the researcher concluded that the drug was effective when in fact it was not.
#### Type 2 Error:
A type 2 error, also known as a false negative, occurs when an experiment fails to reject a null hypothesis that should have been rejected. This means that the experimenter incorrectly concludes that no difference exists when in reality one does. For example, if a researcher was testing whether or not a new drug was effective in treating a certain condition, a type 2 error would occur if the researcher concluded that the drug was not effective when in fact it was.
#### Interpreting Results:
To correctly interpret the results of a hypothesis test, it is important to understand both types of errors and how to identify them.
If the test results indicate that there is no difference between the observed data and what would be expected under the null hypothesis, then it is likely that no difference exists between the two groups. However, if the test results indicate that there is a difference between the observed data and what would be expected under the null hypothesis, then it is likely that there is an actual difference between the two groups. It is important to keep in mind that there is still a possibility of making an error when interpreting the results of a hypothesis test. This article has provided an overview of hypothesis testing, including the basics, how to calculate it, and more. It has also provided some tips on how to perform a successful hypothesis test.
Hypothesis testing is an important concept in A Level Maths and Statistics and understanding it can help you better understand the data you are working with. In summary, hypothesis testing is a powerful tool for making decisions about data sets. Understanding the basic concepts and calculations involved can help you confidently and accurately interpret your results. With the right techniques and approach, you can use hypothesis testing to draw meaningful conclusions from your data.<|endoftext|>
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Writing style is the manner of expressing thought in language characteristic of an individual, period, school, or nation… Learn how to write a letter in formal and informal ways.
How to Write A Letter in English
Rules for writing Informal letters:
- Write your full name and address even if it is an informal letter.
- Divide your letter in small paragraphs.
- Keep your writing simple.
- Make a good choice of words especially if you are writing an apology letter or a letter to express your condolences in case of a death.
- Most people close the letter with phrases like ‘Yours affectionately/With love/All the best/Take care’ etc.
Rules for writing Formal letters:
Let us understand a few ground rules while writing formal letters:
- You need to write your full name, address and date before you begin the letter
- Address the person you are writing the letter to with correct name and designation.
- It is always advisable to start the letter with ‘Respected Sir/Madam’ or ‘Dear Sir/Madam’ and then mention the name and the address.
- Before beginning to write the letter you must state the purpose of the letter in one line titled ‘Subject’.
- Your letter should be very crisp giving out only that information which is required.
How to Write A Letter in English | Image<|endoftext|>
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# Formal Proof of the Infinitude of Primes
In the Infinitude of Primes post, we have shown intuitively that there are infinitely many primes. In this post, we use our intuitive proof to create a more formal proof. The proof was supposedly constructed by Euclid and was shown in his book, The Elements.
Euclid (via Wikimedia)
We are going to use the proof strategy called proof by contradiction. Our proof is summarized as follows:
1.) We assume that the opposite of our conjecture is true.
2.) From our assumption, deduce by logical arguments, and find a contradiction somewhere.
3.) Conclude that if our assumption has a contradiction, then our conjecture must be true.
Conjecture: There are infinitely many primes.
Proof:
1.) We assume that the opposite of our conjecture is true. Suppose there is a finite number primes. Assuming that there are only n of them, namely p1, p2, p3, …, pn.
2.) Let S be the set of all primes. Then S = {p1, p2, p3 ,…, pn}. Let N be the product of all primes added to 1. It follows that N = p1 p2 p3pn + 1.
Now, we know that N can either be prime or composite. On the one hand, if N is prime, then we found another prime number not in S. This contradicts our assumption that there are only n primes.
On the other hand, if N is composite, dividing it by any prime (or product of primes) in set S will leave a remainder of 1. Since N is composite, it must be a product of primes. This means that at least one of its factors is a prime not in S. In effect, we found another prime number not in set S. Again, this contradicts our assumption that there are only n primes.
3.) In any case, we found another prime not in S. Both contradict the assumption that there are only n primes. As a consequence, our assumption is false, which means that our conjecture is true.
The arguments above show that given any finite set of primes, we can always find a prime not on that set.
Therefore, there are infinitely many primes.<|endoftext|>
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Leopard Panthera pardus
The leopard is a member of the Felidae family with a wide range in some parts of sub-Saharan Africa, West Asia, the Middle East, South and Southeast Asia to Siberia. It is listed as Near Threatened on the IUCN Red List because it is declining in large parts of its range due to habitat loss and fragmentation, and hunting for trade and pest control.
The leopard is the smallest of the four “big cats” in the genus Panthera. Compared to other members of the Felidae, the leopard has relatively short legs and a long body with a large skull. It is similar in appearance to the jaguar, but is smaller and more slightly built. Its fur is marked with rosettes similar to those of the jaguar, but the leopard’s rosettes are smaller and more densely packed, and do not usually have central spots as the jaguars do. Both leopards and jaguars that are melanistic are known as black panthers.
The species’ success in the wild is in part due to its opportunistic hunting behavior, its adaptability to habitats, its ability to run at speeds approaching 58 kilometres per hour (36 mph), its unequaled ability to climb trees even when carrying a heavy carcass, and its notorious ability for stealth. The leopard consumes virtually any animal that it can hunt down and catch. Its habitat ranges from rainforest to desert terrains.
This image was captured on the Chobe River, near Kasane, northern Botswana, Southern Africa.<|endoftext|>
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# Can you always use synthetic division when dividing polynomials?
Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.
No, if the degree of the denominator is not 1, then you cannot use synthetic division. If the degree of the denominator is greater than 1, then you must use polynomial long division.
Also Know, how do you divide polynomials with synthetics? Synthetic division is another way to divide a polynomial by the binomial x – c , where c is a constant.
1. Step 1: Set up the synthetic division.
2. Step 2: Bring down the leading coefficient to the bottom row.
3. Step 3: Multiply c by the value just written on the bottom row.
4. Step 4: Add the column created in step 3.
Also Know, when can you use synthetic division?
Synthetic division is a shortcut that can be used when the divisor is a binomial in the form x – k. In synthetic division, only the coefficients are used in the division process.
Can you use synthetic division instead of long division?
Synthetic Division. Synthetic division is another method of dividing polynomials. It is a shorthand of long division that only works when you are dividing by a polynomial of degree 1.
### What is synthetic division and examples?
Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1. To illustrate the process, recall the example at the beginning of the section. Divide 2×3−3×2+4x+5 2 x 3 − 3 x 2 + 4 x + 5 by x+2 using the long division algorithm.
### Can you do synthetic division with a fraction?
Since you are dividing by a polynomial of degree 1, the degree of the solution will be 1 less than the degree of the dividend. For this problem, the answer starts with a power of 2, then a power of 1, then a power of 0 (the constant). The last value in the bottom row is the remainder and is written as a fraction.
### How do you teach synthetic division?
?Using Synthetic Division Step 2: Set the denominator equal to 0 and solve to find the number to put as the divisor. Step 3: Set up the problem using only the coefficients of each term in the numerator. Step 4: Bring down the first coefficient. Step 5: Multiply the divisor by the number you brought down.
### How do you divide polynomials with long division?
Divide the first term of the numerator by the first term of the denominator, and put that in the answer. Multiply the denominator by that answer, put that below the numerator. Subtract to create a new polynomial.
### Why is synthetic division important?
Synthetic division. but the method generalizes to division by any monic polynomial, and to any polynomial. The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division.
### What is synthetic division method?
Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.
### What is the divisor in synthetic division?
5 is called the divisor, 47 is the dividend, 9 is the quotient, and 2 is the remainder. First, to use synthetic division, the divisor must be of the first degree and must have the form x − a. In this example, the divisor is x − 2, with a = 2.<|endoftext|>
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## Calculus with Applications (10th Edition)
$$y = - \frac{3}{4}x - \frac{9}{4}$$
\eqalign{ & y = \frac{3}{{x - 1}},\,\,\,\,\,x = - 1 \cr & \cr & {\text{find the derivative of }}y \cr & \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\frac{3}{{x - 1}}} \right] \cr & \frac{{dy}}{{dx}} = - \frac{3}{{{{\left( {x - 1} \right)}^2}}} \cr & {\text{Find the slope of the tangent line at }}x = - 1 \cr & m = {\left. {\frac{{dy}}{{dx}}} \right|_{x = - 1}} \cr & m = - \frac{3}{{{{\left( { - 1 - 1} \right)}^2}}} \cr & m = - \frac{3}{4} \cr & \cr & {\text{Evaluate the function at }}x = - 1 \cr & y\left( { - 1} \right) = \frac{3}{{ - 1 - 1}} \cr & y\left( { - 1} \right) = - \frac{3}{2}{\text{ }} \cr & {\text{we know the point }}\left( { - 1, - \frac{3}{2}} \right){\text{ and the slope }}m = - \frac{3}{4} \cr & {\text{find the equation of the tangent line using the point - slope form of a line}} \cr & y - {y_1} = m\left( {x - {x_1}} \right) \cr & y - \left( { - \frac{3}{2}} \right) = - \frac{3}{4}\left( {x + 1} \right) \cr & {\text{simplifying}} \cr & y + \frac{3}{2} = - \frac{3}{4}x - \frac{3}{4} \cr & y + \frac{3}{2} = - \frac{3}{4}x - \frac{3}{4} - \frac{3}{2} \cr & y = - \frac{3}{4}x - \frac{9}{4} \cr}<|endoftext|>
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# 2.8: Find Multiples and Factors (Part 2)
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## Identify Prime and Composite Numbers
Some numbers, like $$72$$, have many factors. Other numbers, such as $$7$$, have only two factors: $$1$$ and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number $$1$$ is neither prime nor composite. It has only one factor, itself.
Definition: Prime Numbers and Composite Numbers
A prime number is a counting number greater than $$1$$ whose only factors are $$1$$ and itself.
A composite number is a counting number that is not prime.
Figure $$\PageIndex{5}$$ lists the counting numbers from $$2$$ through $$20$$ along with their factors. The highlighted numbers are prime, since each has only two factors.
Figure $$\PageIndex{5}$$: Factors of the counting numbers from 2 through 20, with prime numbers highlighted
The prime numbers less than $$20$$ are $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, $$17$$, and $$19$$. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than $$1$$ and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.
HOW TO: DETERMINE IF A NUMBER IS PRIME.
• Step 1. Test each of the primes, in order, to see if it is a factor of the number.
• Step 2. Start with $$2$$ and stop when the quotient is smaller than the divisor or when a prime factor is found.
• Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
Example $$\PageIndex{8}$$: prime or composite
Identify each number as prime or composite:
1. $$83$$
2. $$77$$
Solution
1. Test each prime, in order, to see if it is a factor of $$83$$, starting with $$2$$, as shown. We will stop when the quotient is smaller than the divisor.
Prime Test Factor of 83?
2 Last digit of 83 is not 0, 2, 4, 6, or 8. No.
3 8 + 3 = 11, and 11 is not divisible by 3. No.
5 The last digit of 83 is not 5 or 0. No.
7 83 ÷ 7 = 11.857…. No.
11 83 ÷ 11 = 7.545… No.
We can stop when we get to $$11$$ because the quotient ($$7.545…$$) is less than the divisor. We did not find any prime numbers that are factors of $$83$$, so we know $$83$$ is prime.
1. Test each prime, in order, to see if it is a factor of $$77$$.
Prime Test Factor of 77?
2 Last digit is not 0, 2, 4, 6, or 8. No.
3 7 + 7 = 14, and 14 is not divisible by 3. No.
5 The last digit is not 5 or 0. No.
7 77 ÷ 11 = 7 Yes.
Since $$77$$ is divisible by $$7$$, we know it is not a prime number. It is composite.
Exercise $$\PageIndex{15}$$
Identify the number as prime or composite: $$91$$
composite
Exercise $$\PageIndex{16}$$
Identify the number as prime or composite: $$137$$
prime
## Key Concepts
Divisibility Tests
A number is divisible by
2 if the last digit is 0, 2, 4, 6, or 8
3 if the sum of the digits is divisible by 3
5 if the last digit is 5 or 0
6 if divisible by both 2 and 3
10 if the last digit is 0
• Factors If $$a\cdot b = m$$, then $$a$$ and $$b$$ are factors of $$m$$, and $$m$$ is the product of $$a$$ and $$b$$.
• Find all the factors of a counting number.
• Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
• If the quotient is a counting number, the divisor and quotient are a pair of factors.
• If the quotient is not a counting number, the divisor is not a factor.
• List all the factor pairs.
• Write all the factors in order from smallest to largest.
• Determine if a number is prime.
• Test each of the primes, in order, to see if it is a factor of the number.
• Start with $$2$$ and stop when the quotient is smaller than the divisor or when a prime factor is found.
• If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
## Glossary
multiple of a number
A number is a multiple of $$n$$ if it is the product of a counting number and $$n$$
divisibility
If a number $$m$$ is a multiple of $$n$$, then we say that $$m$$ is divisible by $$n$$.
prime number
A prime number is a counting number greater than $$1$$ whose only factors are $$1$$ and itself.
composite number
A composite number is a counting number that is not prime.
## Practice Makes Perfect
### Identify Multiples of Numbers
In the following exercises, list all the multiples less than 50 for the given number.
1. 2
2. 3
3. 4
4. 5
5. 6
6. 7
7. 8
8. 9
9. 10
10. 12
### Use Common Divisibility Tests
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 4, 5, 6, and 10.
1. 84
2. 96
3. 75
4. 78
5. 168
6. 264
7. 900
8. 800
9. 896
10. 942
11. 375
12. 750
13. 350
14. 550
15. 1430
16. 1080
17. 22,335
18. 39,075
### Find All the Factors of a Number
In the following exercises, find all the factors of the given number.
1. 36
2. 42
3. 60
4. 48
5. 144
6. 200
7. 588
8. 576
### Identify Prime and Composite Numbers
In the following exercises, determine if the given number is prime or composite.
1. 43
2. 67
3. 39
4. 53
5. 71
6. 119
7. 481
8. 221
9. 209
10. 359
11. 667
12. 1771
## Everyday Math
1. Banking Frank’s grandmother gave him $100 at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added$15 to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.
Weeks after graduation Total number of dollars Frank put in the account Simplified Total
0 100 100
1 100 + 15 115
2 100 + 15 • 2 130
3 100 + 15 • 3
4 100 + 15 • []
5 100 + []
6
20
x
1. Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $75 to open the account. Every week, she added$20 to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.
Weeks after opening the account Total number of dollars Gina put in the account Simplified Total
0 75 75
1 75 + 20 95
2 75 + 20 • 2 115
3 75 + 20 • 3
4 75 + 20 • []
5 75 + []
6
20
x
## Writing Exercises
1. If a number is divisible by 2 and by 3, why is it also divisible by 6?
2. What is the difference between prime numbers and composite numbers?
## Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
## Contributors
• Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.<|endoftext|>
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While born in the same month, one hundred years apart, John Birch and Karl Marx could not have been more different.
Both men were born in May, but it is more than the 100 years between their births that separates the differences between the two. One was born of parents who faced poverty, while the other was the child of a successful lawyer. One had a father and a mother who took their faith in God very seriously, while the other had a father who ditched the faith of his own fathers, and was rather indifferent about his adopted religion. One’s Christian faith led him to missionary work in China, while the other’s lack of faith led him to denounce belief in God. One despised atheistic communism, while the other’s name will forever be associated with that evil system that has taken the lives of more than 100 million human beings.
The first man was John Morrison Birch (shown on left), born May 28, 1918, while the second was Karl Marx (shown on right), born May 5, 1818.
Marx was hired by the shadowy group known as The League of the Just (composed of discontented socialist revolutionaries who desired to unite all socialist revolutionaries into the Communist Party they were forming) to write the platform of the Communist Party. While the communists claimed to champion the cause of the poor working class, Marx was born in Trier, in the German Confederation, of well-to-do parents.
Marx’s father, Heinrich Marx, enjoyed a lucrative legal practice, with an income that placed him in the top five percent of the city. While many parents today read Dr. Seuss, Bible stories, or fairy tales to their young children, Heinrich read aloud to young Karl such radical Enlightenment thinkers as Rousseau and Voltaire. Later, Marx went to college at the University of Berlin where he was introduced to even more radical philosophies.
He quickly became a devotee of the philosophy of Georg Wilhelm Hegel, and Marx would later make use of Hegelian “dialectics” in constructing his “Scientific Socialism” that we know as Marxism or communism. He also fell under the direct radical influence of a professor of legal history, Edward Gans, an advocate of the Saint Simonians, early French socialists.
Marx became a dedicated atheist at the university, partly through the influence of Bruno Bauer, a professor, who argued that Jesus’ life was based on pious forgeries. In fact, Bauer said Jesus never even existed.
Because of his vocal atheism, Marx was denied a teaching position in Germany after completing his doctorate, and turned to journalism, which eventually led to The League of the Just to hire him to write infamous Communist Manifesto. Together with his other writings, such as Das Kapital, Marx inspired many of the worst tyrants of history, such as Lenin and Stalin in the Soviet Union, Mao Tse-tung in China, and Pol Pot in Cambodia.
John Birch, on the other hand, never wrote a book, but he did write a short essay about four months before his death, called “The War Weary Farmer,” which summarized his philosophy of life. “I want of government only protection against the violence and injustices of evil or selfish men,” Birch wrote in April 1945. The essay is a short, but eloquent statement of opposition to collectivism.
Birch was in China, where he had gone as a Baptist missionary, to reach the people of that country for Christ. While there, he saved the lives of Lt. Col. Jimmy Doolittle and his crew, who had bailed out in China, after the Tokyo bombing raid of April 14, 1942. Eventually, Birch enlisted to serve as an intelligence officer for the Army, building a vast intelligence network of Chinese informants, supplying General Claire Chennault (of the famous “Flying Tigers”) with information vital to the American war effort against Japan.
It was only a few days after victory over Japan had been achieved that young Birch was murdered by Chinese Communists. Robert Welch, a strongly anti-communist candy manufacturer (Sugar Daddies, Sugar Babies, and Junior Mints), after reading with indignation about Birch’s murder and its subsequent coverup by the U.S. government, proclaimed Birch’s killing the first American casualty of the Cold War. He later named his anti-Communist organization — The John Birch Society (JBS) — in honor of Birch.
While some have argued that Birch would not have approved of this use of his name, a review of Birch’s life, his philosophy as expressed in “The War Weary Farmer,” and the fact that his parents not only approved, but became life members of the JBS, would seem to indicate that Birch would have been deeply honored. The Birch Society’s motto, “Less Government, More Responsibility, and With God’s Help, a Better World,” aptly summarizes the life and words of John Birch.
The irony is great — Birch was murdered by atheistic Chinese Communists, inspired by the writings of the atheist communist Karl Marx, two men born one century apart.
The two men whose philosophy — one the belief in the liberty of the individual with a reliance on God, and the other who favored a collective dictatorship founded upon atheism — were even further apart than the century that marked their entrance into the world.<|endoftext|>
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# Derivatives of Composite Functions - Chain Rule
## Notes
To find the derivative of f, |
where f(x) = (2x + 1)^3
One way is to expand (2x + 1)3 using binomial theorem and find the derivative as a polynomial function as illustrated below.
d/(dx)f(x) = d/(dx) [(2x + 1)^3]
=d/(dx) (8x^3 + 12x^2 + 6x + 1)
= 24x^2 + 24x + 6
= 6 (2x + 1)^2
Now, observe that
f(x) = (h o g) (x)
where g(x) = 2x + 1 and h(x) = x^3.
Put t = g(x) = 2x + 1. Then f(x) = h(t) = t^3. Thus
(df)/(dx) = 6(2x + 1)^2 = 3(2x + 1)^2 . 2 = 3t^2 . 2 = (dh)/(dt) . (dt)/(dx)
The advantage with such observation is that it simplifies the calculation in finding the derivative of, say, (2x + 1)^100. We may formalise this observation in the following theorem called the chain rule.
## Theorem
Theorem: (Chain Rule) Let f be a real valued function which is a composite of two functions u and v; i.e., f = v o u. Suppose t = u(x) and if both (dt)/(dx) and (dv)/(dt) exist , we have
(df)/(dx) = (dv)/(dt) . (dt)/dx
We skip the proof of this theorem. Chain rule may be extended as follows. Suppose f is a real valued function which is a composite of three functions u, v and w; i.e., f = (w o u) o v. If t = v(x) and s = u(t), then
(df)/(dx) = (d("wou"))/(dt) . (dt)/(dx) = (dw)/(ds) . (ds)/(dt) . (dt)/(dx)
provided all the derivatives in the statement exist. Reader is invited to formulate chain rule for composite of more functions.
If you would like to contribute notes or other learning material, please submit them using the button below.
#### Video Tutorials
We have provided more than 1 series of video tutorials for some topics to help you get a better understanding of the topic.
Series 1
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### Shaalaa.com
Derivative of composite function-chain rule [00:16:26]
S
0%<|endoftext|>
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# Test: Unit Digits, Factorial Powers
## 10 Questions MCQ Test Quantitative Aptitude for GMAT | Test: Unit Digits, Factorial Powers
Description
Attempt Test: Unit Digits, Factorial Powers | 10 questions in 20 minutes | Mock test for GMAT preparation | Free important questions MCQ to study Quantitative Aptitude for GMAT for GMAT Exam | Download free PDF with solutions
QUESTION: 1
### 1727 has a units digit of:
Solution:
When raising a number to a power, the units digit is influenced only by the units digit of that number. For example 162 ends in a 6 because 62 ends in a 6.
1727 will end in the same units digit as 727.
The units digit of consecutive powers of 7 follows a distinct pattern:
The pattern repeats itself every four numbers so a power of 27 represents 6 full iterations of the pattern (6 × 4 = 24) with three left over. The "leftover three" leaves us back on a "3," the third member of the pattern 7, 9, 3, 1.
QUESTION: 2
### 11+22+33+...+1010 is divided by 5. What is the remainder?
Solution:
When a whole number is divided by 5, the remainder depends on the units digit of that number.
Thus, we need to determine the units digit of the number 11+22+33+...+1010. To do so, we need to first determine the units digit of each of the individual terms in the expression as follows:
To determine the units digit of the expression itself, we must find the sum of all the units digits of each of the individual terms:
1 + 4 + 7 + 6 + 5 + 6 + 3 + 6 + 9 = 47
Thus, 7 is the units digit of the number 11+22+33+...+1010. When an integer that ends in 7 is divided by 5, the remainder is 2. (Test this out on any integer ending in 7.)
Thus, the correct answer is C.
QUESTION: 3
### Given that p is a positive even integer with a positive units digit, if the units digit of p3 minus the units digit of p2 is equal to 0, what is the units digit of p + 3?
Solution:
The easiest way to approach this problem is to chart the possible units digits of the integer p. Since we know p is even, and that the units digit of p is positive, the only options are 2, 4, 6, or 8.
Only when the units digit of p is 6, is the units digit of p3p2 equal to 0.
The question asks for the units digit of p + 3. This is equal to 6 + 3, or 9.
QUESTION: 4
If x is a positive integer, what is the units digit of (24)(2x + 1)(33)(x + 1)(17)(x + 2)(9)(2x)?
Solution:
For problems that ask for the units digit of an expression, yet seem to require too much computation, remember the Last Digit Shortcut. Solve the problem step-by-step, but recognize that you only need to pay attention to the last digit of every intermediate product. Drop any other digits.
So, we can drop any other digits in the original expression, leaving us to find the units digit of:
(4)(2x + 1)(3)(x + 1)(7)(x + 2)(9)(2x)
This problem is still complicated by the fact that we don’t know the value of x. In such situations, it is often a good idea to look for patterns. Let's see what happens when we raise the bases 4, 3, 7, and 9 to various powers. For example: 31 = 3, 32 = 9, 33 = 27, 34 = 81, 35 = 243, and so on. The units digit of the powers of three follow a pattern that repeats every fourth power: 3, 9, 7, 1, 3, 9, 7, 1, and so on. The patterns for the other bases are shown in the table below:
The patterns repeat at least every fourth term, so let's find the units digit of (4)(2x + 1)(3)(x + 1)(7)(x + 2)(9)(2x) for at least four consecutive values of x:
x = 1: units digit of (43)(32)(73)(92) = units digit of (4)(9)(3)(1) = units digit of 108 = 8
x = 2: units digit of (45)(33)(74)(94) = units digit of (4)(7)(1)(1) = units digit of 28 = 8
x = 3: units digit of (47)(34)(75)(96) = units digit of (4)(1)(7)(1) = units digit of 28 = 8
x = 4: units digit of (49)(35)(76)(98) = units digit of (4)(3)(9)(1) = units digit of 108 = 8
The units digit of the expression in the question must be 8.
Alternatively, note that x is a positive integer, so 2x is always even, while 2x + 1 is always odd. Thus,
(4)(2x + 1) = (4)(odd), which always has a units digit of 4
(9)(2x) = (9)(even), which always has a units digit of 1
That leaves us to find the units digit of (3)(x + 1)(7)(x + 2). Rewriting, and dropping all but the units digit at each intermediate step,
(3)(x + 1)(7)(x + 2)
= (3)(x + 1)(7)(x + 1)(7)
= (3 × 7)(x + 1)(7)
= (21)(x + 1)(7)
= (1)(x + 1)(7) = 7, for any value of x.
So, the units digit of (4)(2x + 1)(3)(x + 1)(7)(x + 2)(9)(2x) is (4)(7)(1) = 28, then once again drop all but the units digit to get 8.
QUESTION: 5
If a and b are positive integers and x = 4a and y = 9b, which of the following is a possible units digit of xy?
Solution:
If a is a positive integer, then 4a will always have a units digit of 4 or 6. We can show this by listing the first few powers of 4:
41 = 4
42 = 16
43 = 64
44 = 256
The units digit of the powers of 4 alternates between 4 and 6. Since x = 4a, x will always have a units digit of 4 or 6.
Similarly, if b is a positive integer, then 9b will always have a units digit of 1 or 9. We can show this by listing the first few powers of 9:
91 = 9
92 = 81
93 = 729
94 = 6561
The units digit of the powers of 9 alternates between 1 and 9. Since y = 9b, y will always have a units digit of 1 or 9.
To determine the units digit of a product of numbers, we can simply multiply the units digits of the factors. The resulting units digit is the units digit of the product. For example, to find the units digit of (23)(39) we can take (3)(9) = 27. Thus, 7 is the units digit of (23)(39). So, the units digit of xy will simply be the units digit that results from multiplying the units digit of x by the units digit of y. Let's consider all the possible units digits of x and y in combination:
4 × 1 = 4, units digit = 4
4 × 9 = 36, units digit = 6
6 × 1 = 6, units digit = 6
6 × 9 = 54, units digit = 4
The units digit of xy will be 4 or 6.
QUESTION: 6
If x = 321 and y = 655, what is the remainder when xy is divided by 10?
Solution:
Since every multiple of 10 must end in zero, the remainder from dividing xy by 10 will be equal to the units’ digit of xy. In other words, the units’ digit will reflect by how much this number is greater than the nearest multiple of 10 and, thus, will be equal to the remainder from dividing by 10. Therefore, we can rephrase the question: “What is the units’ digit of xy?”
Next, let’s look for a pattern in the units’ digit of 321. Remember that the GMAT will not expect you to do sophisticated computations; therefore, if the exponent seems too large to compute, look for a shortcut by recognizing a pattern in the units' digits of the exponent:
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
As you can see, the pattern repeats every 4 terms, yielding the units digits of 3, 9, 7, and 1. Therefore, the exponents 31, 35, 39, 313, 317, and 321 will end in 3, and the units’ digit of 321 is 3.
Next, let’s determine the units’ digit of 655 by recognizing the pattern:
61 = 6
62 = 36
63 = 256
64 = 1,296
As shown above, all positive integer exponents of 6 have a units’ digit of 6. Therefore, the units' digit of 655 will also be 6.
Finally, since the units’ digit of 321 is 3 and the units’ digit of 655 is 6, the units' digit of 321 × 655 will be equal to 8, since 3 × 6 = 18. Therefore, when this product is divided by 10, the remainder will be 8.
QUESTION: 7
If x is a positive integer, what is the remainder when 712x+3 + 3 is divided by 5?
Solution:
To find the remainder when a number is divided by 5, all we need to know is the units digit, since every number that ends in a zero or a five is divisible by 5.
For example, 23457 has a remainder of 2 when divided by 5 since 23455 would be a multiple of 5, and 23457 = 23455 + 2.
Since we know that x is an integer, we can determine the units digit of the number 712x+3 + 3. The first thing to realize is that this expression is based on a power of 7. The units digit of any integer exponent of seven can be predicted since the units digit of base 7 values follows a patterned sequence:
We can see that the pattern repeats itself every 4 integer exponents.
The question is asking us about the 12x+3 power of 7. We can use our understanding of multiples of four (since the pattern repeats every four) to analyze the 12x+3 power.
12x is a multiple of 4 since x is an integer, so 712x would end in a 1, just like 74 or 78.
712x+3 would then correspond to 73 or 77 (multiple of 4 plus 3), and would therefore end in a 3.
If 712x+3 ends in a three, 712x+3 + 3 would end in a 3 + 3 = 6.
If a number ends in a 6, there is a remainder of 1 when that number is divided by 5.
QUESTION: 8
What is the units digit of (71)5(46)3(103)4 + (57)(1088)3 ?
Solution:
Since the question only asks about the units digit of the final solution, focus only on computing the units digit for each term. Thus, the question can be rewritten as follows:
(1)5(6)3(3)4 + (7)(8)3.
The units digit of 15 is 1.
The units digit of 63 is 6.
The units digit of 34 is 1.
The units digit of (1 × 6 × 1) is 6.
The units digit of 7 is 7.
The units digit of 83 is 2.
The units digit of (7 × 2) is 4.
The solution is equal to the units digit of (6 + 4), which is 0.
QUESTION: 9
if , what is the units digit of
Solution:
In order to answer this, we need to recognize a common GMAT pattern: the difference of two squares. In its simplest form, the difference of two squares can be factored as follows:
x2-y2= (x+y)(x-y)
Where, though, is the difference of two squares in the question above? It pays to recall that all even exponents are squares. For example.
x4= (x2)(x2)
x56= (x28)(x28)
Because the numerator in the expression in the question is the difference of two even exponents, we can factor it as the difference of two squares and simplify:
The units digit of the left side of the equation is equal to the units digit of the right side of the equation (which is what the question asks about). Thus, if we can determine the units digit of the expression on the left side of the equation, we can answer the question.
Since (13!) = 13 x 12 x 11 x 10 .... x 1, we know that 13! contains a factor of 10, so its units digit must be 0. Similarly, the units digit of (13!)will also have a units digit of 0. If we subtract 1 from this, we will be left with a number ending in 9.
Therefore, the units digit of is 9. The correct answer is E.
QUESTION: 10
What is the units digit of 17728 – 13323?
Solution:
Since the question asks only about the units digit, we can look for patterns in each of the numbers.
Let's begin with 17728:
Since this pattern will continue, the units digit of 17728 will be 1
Next, let's follow the same procedure with 13323:
Since this pattern will continue, the units digit of 13323 will be 7.
Therefore in calculating the expression 17728-13333 we can determine that the units digit of the solution will equal 1-7
Since, 7 is greater than 1, the subtraction here requires that we carry over from the tens place. Thus, we have 11-7, yielding the units digit 4.<|endoftext|>
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## College Algebra 7th Edition
$\displaystyle (1+ \frac{1}{x})^{6}=1+\frac{6}{x}+\frac{15}{x^{2}}+\frac{20}{x^{3}}+\frac{15}{x^{4}}+\frac{6}{x^{5}}+\frac{1}{x^{6}}$ Or written with negative exponents: $\displaystyle (1+ \frac{1}{x})^{6}=1+6x^{-1}+15x^{-2}+20x^{-3}+15x^{-4}+6x^{-5}+x^{-6}$
We expand using the Binomial Theorem: $\displaystyle (1+ \frac{1}{x})^{6}=\left(\begin{array}{l} 6\\ 0 \end{array}\right)1^{6}+\left(\begin{array}{l} 6\\ 1 \end{array}\right)1^{5}(\frac{1}{x})^1+\left(\begin{array}{l} 6\\ 2 \end{array}\right)1^{4}(\frac{1}{x})^{2}+\left(\begin{array}{l} 6\\ 3 \end{array}\right)1^{3}(\frac{1}{x})^{3}+\left(\begin{array}{l} 6\\ 4 \end{array}\right)1^{2}(\frac{1}{x})^{4}+\left(\begin{array}{l} 6\\ 5 \end{array}\right)1^{1}(\frac{1}{x})^{5}+\left(\begin{array}{l} 6\\ 6 \end{array}\right)(\frac{1}{x})^{6}=1+\frac{6}{x}+\frac{15}{x^{2}}+\frac{20}{x^{3}}+\frac{15}{x^{4}}+\frac{6}{x^{5}}+\frac{1}{x^{6}}$ Or written with negative exponents: $=1+6x^{-1}+15x^{-2}+20x^{-3}+15x^{-4}+6x^{-5}+x^{-6}$<|endoftext|>
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The gradient of a straight line is a measure of the slope of the line. The larger the value of the gradient, the steeper the slope.
If you know either the endpoints, or even just 2 different points on a straight line.
This is often referred to as the "slope formula".
It doesn’t matter which of the co-ordinates is (x1,y1) or (x2,y2) in calculations, the gradient result will be the same.
It is also the case that you don't have to use the End Points of a straight line, any 2 different points on the line in the formula will produce the gradient.
The nature of whether the gradient is positive or negative, also depends on the direction of the line.
A line with negative gradient, such as -4, slopes downward from left to right. \
A line with positive gradient, such as 4, slopes upward from left to right. /
Another way to work out the gradient of a straight line is with the "tan" button on a calculator.
When you know the size of the angle that a straight line makes with the positive direction of the x-axis.
The line L in the picture above, makes an angle of θ with the positive direction of the x-axis.
When you know the value of this angle, the gradient of a line such as L, can be given by m = tanθ.
The gradient of a horizontal line is always 0.
As it is a flat line with no slope.
The gradient of a vertical line is classed as undefined.
A vertical line isn’t flat, but as it's pointing straight up, it isn’t really sloping in any particular direction either.
In the image above, line A is Horizontal, and line B is vertical.
Lines that are parallel to each other have the same gradient, as they slope the same way.
Lines A and B in this example are both sloping the same way. So the gradient of each line is the same as the other.
If two lines are perpendicular, that is at a right angle to each other, then the gradients multiplied together will be equal to -1.<|endoftext|>
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The story of how the Starfighter was used to train future astronauts.
Being the first operational aircraft able to reach and maintain a speed of more than Mach 2.0, the Lockheed F-104 was a huge leap forward when strictly compared to the contemporary subsonic jets.
Thanks to its performance, the Starfighter was chosen to train test pilots destined to fly the X-15, a winged spacecraft that was air-launched by a B-52 Stratofortress, flew into space and then landed conventionally.
The idea to modify several F-104As to serve as “manned spacecraft transition trainers” is credited to astronaut Frank Borman who was both student and instructor at Edwards Air Force Base, California, home of the Air Force’s Aerospace Research Pilot School, later renamed U.S. Air Force Test Pilot School.
The major modifications to the Starfighters consisted in the addition of a 6,000 pound thrust rocket engine at the base of the vertical tail, reaction control thrusters in the nose and in each wing tip, a larger vertical tail, increased wing span, tanks to store the rocket propellants, provision for a full pressure suit, a cockpit hand controller to operate the reaction control thrusters, and modified cockpit instrumentation.
Moreover, the unnecessary equipment, like the gun, fire control system, tactical electronics, and auxiliary fuel tanks, was removed.
The Starfighters with these modifications were renamed NF-104s. They entered in service in 1963 and their pilots could zoom to more than 100,000 feet in a full pressure suit, experience zero “g”, and use reaction control to handle the aircraft.
Only about 35 students had the privilege to fly the NF-104 and each pilot had to be prepared for these “space flights” by using standard Starfighters. The first mission was a pressure suit familiarization flight, with the F-104 flown to high altitude with the cockpit depressurized allowing the student to experience a flight in a fully pressurized suit. To practice the zoom profile, the second flight was conducted in a two-seat F-104, with the instructor that showed to the student how reaching an altitude of 70-80,000 feet performing a 30 degree climb, while the last three missions were made in a single seat Starfighter increasing the climb angle to 45 degrees and reaching an altitude of 90,000 feet.
After these five preparation flights, the student finally performed the two programmed NF-104 missions.
As described by Steve Markman and Bill Holder in their book One Of A Kind Research Aircraft A History Of In Flight Simulators, Testbeds & Prototypes, the typical flight syllabus started with taking off on jet power, climb to 30-40,000 feet, and accelerate to Mach 1.7-1.9. Then the pilot ignited the rocket engine and pitched the nose up to start the steep climb.
After two minutes the Starfighter passed through 80,000 feet, the jet engine flamed out, the rocket engine ran out of fuel and the pilot began a parabolic arc to the peak altitude.
It was during the parabolic arc that the pilot experienced “weightlessness” for about one minute and used the side stick to fire the reaction control rockets to control the aircraft’s pitch, roll and yaw motions.
Once at a lower altitude, the pilot restarted the jet engine and made a conventional landing: the whole mission lasted about 35 minutes from taxi to landing and was performed in a full pressure suit.
One NF-104 was destroyed on Dec. 10 1963. The plane was piloted by legendary Col. Chuck Yeager at that time the Aerospace Research Pilot School Commander. Yeager was attempting to reach an altitude record and after a 60 degree climb, while he was at 101,595 feet, the Starfighter experienced an uncontrollable yawing and rolling motion.
Yeager wasn’t able to recover the plane and was forced to eject at 8,500 feet.
During the separation from the ejection seat the rocket nozzle hit his face shield breaking it, while the combination of the red hot nozzle and oxygen in his helmet produced a flame that burned his face and set several parachute cords on fire.
Yeager was able to extinguish the flames with his glove hands and after the accident was hospitalized for two weeks.
The accident was depicted in the book (and film of the same name) “The Right Stuff”.
Another NF-104 flight almost ended in disaster on June 15, 1971, when Capt. Howard Thompson experienced a rocket engine explosion while trying to lit it at 35,000 feet and Mach 1,15: luckily Thompson made a safe lading to Edwards AFB using the normal jet engine.
The program was terminated when it was decided that the aerospace training mission would be performed by NASA and the last NF-104 flight was performed in December 1971.
During its service with the U.S. Air Force the highest altitude reached by an NF-104 was 121,800 feet, achieved by Maj. Robert Smith during acceptance testing.
Today the last of the NF-104s is on static display in front of the Air Force Test Pilot School at Edwards AFB.
Image credit: U.S. Air Force<|endoftext|>
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# 19.2: Construtible numbers
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In the classical compass-and-ruler constructions initial configuration can be completely described by a finite number of points; each line is defined by two points on it and each circle is described by its center and a point on it (equivalently, you may describe a circle by three points on it).
The same way the result of construction can be described by a finite collection of points.
We may always assume that the initial configuration has at least two points; if not add one or two points to the configuration. Moreover, applying a scaling to the whole plane, we can assume that the first two points in the initial configuration lie at distance 1 from each other.
In this case we can choose a coordinate system, such that one of the initial points is the origin $$(0,0)$$ and yet another initial point has the coordinates $$(1,0)$$. In this coordinate system, the initial configuration of $$n$$ points is described by $$2\cdot n-4$$ numbers — their coordinates $$x_3,y_3,\dots,x_n,y_n$$.
It turns out that the coordinates of any point constructed with a compass and ruler can be written thru the numbers $$x_3,y_3,\dots,x_n,y_n$$ using the four arithmetic operations "$$+$$", "$$-$$", "$$\cdot$$", "$$/$$" and the square root "$$\sqrt{\phantom{a}}$$".
For example, assume we want to find the points $$X_1=(x_1,y_1)$$ and $$X_2=(x_2,y_2)$$ of the intersections of a line passing thru $$A=(x_A,y_A)$$ and $$B=(x_B,y_B)$$ and the circle with center $$O=(x_O,y_O)$$ that passes thru the point $$W=(x_W,y_W)$$. Let us write the equations of the circle and the line in the coordinates $$(x,y)$$:
\left\{ \begin{aligned} (x-x_O)^2+(y-y_O)^2&=(x_W-x_O)^2+(y_W-y_O)^2, \\ (x-x_A)\cdot(y_B-y_A)&=(y-y_A)\cdot(x_B-x_A). \end{aligned} \right.
Note that coordinates $$(x_1,y_1)$$ and $$(x_2,y_2)$$ of the points $$X_1$$ and $$X_2$$ are solutions of this system. Expressing $$y$$ from the second equation and substituting the result in the first one, gives us a quadratic equation in $$x$$, which can be solved using "$$+$$", "$$-$$", "$$\cdot$$", "$$/$$" and "$$\sqrt{\phantom{a}}$$" only.
The same can be performed for the intersection of two circles. The intersection of two lines is even simpler; it is described as a solution of two linear equations and can be expressed using only four arithmetic operations; the square root "$$\sqrt{\phantom{a}}$$" is not needed.
On the other hand, it is easy to make compass-and-ruler constructions that produce segments of the lengths $$a+b$$ and $$a-b$$ from two given segments of lengths $$a>b$$.
To perform "$$\cdot$$", "$$/$$" and "$$\sqrt{\phantom{a}}$$" consider the following diagram: let $$[AB]$$ be a diameter of a circle; fix a point $$C$$ on the circle and let $$D$$ be the foot point of $$C$$ on $$[AB]$$. By Corollary 9.3.1, the angle $$ACB$$ is right. Therefore
$$\triangle ABC\sim\triangle ACD\sim \triangle CBD.$$
It follows that $$AD\cdot BD=CD^2$$.
Using this diagram, one should guess a solution to the following exercise.
Exercise $$\PageIndex{1}$$
Given two line segments with lengths $$a$$ and $$b$$, give a ruler-and-compass construction of a segments with lengths $$\tfrac {a^2}b$$ and $$\sqrt{a\cdot b}$$.
Hint
To construct $$\sqrt{a \cdot b}$$: (1) construct points $$A, B$$, and $$D \in [AB]$$ such that $$AD = a$$ and $$BD =b$$; (2) construct the circle $$\Gamma$$ on the diameter $$[AB]$$; (3) draw the line $$\ell$$ thru $$D$$ perpendicular to $$(AB)$$; (4) let $$C$$ be an intersection of $$\Gamma$$ and $$\ell$$. Then $$DC = \sqrt{a \cdot b}$$.
The construction of $$\dfrac{a^2}{b}$$ is analogous.
Taking $$1$$ for $$a$$ or $$b$$ above, we can produce $$\sqrt a$$, $$a^2$$, $$\tfrac1b$$. Combining these constructions we can produce $$a\cdot b=(\sqrt{a\cdot b})^2$$, $$\tfrac ab=a\cdot\tfrac 1b$$. In other words we produced a compass-and-ruler calculator which can do "$$+$$", "$$-$$", "$$\cdot$$", "$$/$$", and "$$\sqrt{\phantom{a}}$$".
The discussion above gives a sketch of the proof of the following theorem:
Theorem $$\PageIndex{1}$$
Assume that the initial configuration of geometric construction is given by the points $$A_1=(0,0)$$, $$A_2=(1,0)$$, $$A_3=(x_3,y_3),\dots,A_n=(x_n,y_n)$$. Then a point $$X=(x,y)$$ can be constructed using a compass-and-ruler construction if and only if both coordinates $$x$$ and $$y$$ can be expressed from the integer numbers and $$x_3$$, $$y_3$$, $$x_4$$, $$y_4,\dots,x_n,y_n$$ using the arithmetic operations "$$+$$", "$$-$$", "$$\cdot$$", "$$/$$", and the square root "$$\sqrt{\phantom{a}}$$".
The numbers that can be expressed from the given numbers using the arithmetic operations and the square root “$$\sqrt{\phantom{a}}$$” are called constructible; if the list of given numbers is not given, then we can only use the integers.
The theorem above translates any compass-and-ruler construction problem into a purely algebraic language. For example:
• The impossibility of a solution for doubling the cube problem states that $$\sqrt[3]{2}$$ is not a constructible number. That is $$\sqrt[3]{2}$$ cannot be expressed thru integers using "$$+$$", "$$-$$", "$$\cdot$$", "$$/$$", and "$$\sqrt{\phantom{a}}$$".
• The impossibility of a solution for squaring the circle states that $$\sqrt{\pi}$$, or equivalently $$\pi$$, is not a constructible number.
• The Gauss–Wantzel theorem says for which integers $$n$$ the number $$\cos\tfrac{2\cdot\pi}n$$ is constructible.
Some of these statements might look evident, but rigorous proofs require some knowledge of abstract algebra (namely, field theory) which is out of the scope of this book.
In the next section, we discuss similar but simpler examples of impossible constructions with an unusual tool.
Exercise $$\PageIndex{2}$$
1. Show that diagonal or regular pentagon is $$\tfrac{1+\sqrt5}2$$ times larger than its side.
2. Use (a) to make a compass-and-ruler construction of a regular pentagon.
Hint
(a) Look at the diagram of regular pentagon; show that the angles marked the same way have the same angle measue. Conclude the that $$XC =BC$$ and $$\triangle ABC \sim \triangle AXB$$. Therefore
$$\dfrac{AB}{AC} = \dfrac{AX}{AB} = \dfrac{AC - AB}{AB} = \dfrac{AC}{AB} - 1$$.
It remains to solve for $$\dfrac{AC}{AB}$$.
(b) Choose two points $$P$$ and $$Q$$ and use the compass-and-ruler calculator to construct two points $$V$$ and $$W$$ such that $$VW = \dfrac{1 + \sqrt{5}}{2} \cdot PQ$$. Then construct a pentagon with the sides $$PQ$$ and diagonals $$VW$$.<|endoftext|>
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Here’s some drone news worth buzzing about. Researchers have developed insect-sized flying robots that have the ability to perch on objects using static electricity in order to conserve energy and extend flight times.
In a report in the University of Washington’s UW Today, the work of a team of Harvard roboticists and a UW mechanical engineer is detailed. The tiny drones, nicknamed RoboBees, employ a switchable electroadhesive that allows them to land on materials such as glass, wood or a leaf.
“Think of perching as landing without a runway. Birds, bats and insects do it,” says a video showcasing the drones in Science Magazine.
The electroadhesion patch attached to the RoboBees works similarly to the way your hair sticks to a balloon. But this means the drones must be very light in order to stick to a surface. So the engineers’ drones weigh in at around 100 mg, which is equal to the weight of a real honeybee.
The patch requires about 1,000 times less power to perch than it does to hover, and this can dramatically extend the operation life of the robot for missions ranging from providing views of disaster areas, detecting hazardous chemicals or providing communication in remote regions.
“One of the biggest difficulties with building insect-sized robots is that the physics change as you go that small. A lot of technologies that have been deployed successfully on larger robots become impractical on a centimeter-sized robot,” co-author Sawyer Fuller, UW assistant professor of mechanical engineering, said in UW Today. “We take inspiration from flying insects because they’ve already found solutions for these challenges.”<|endoftext|>
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Writing Goal: Students will become writers who know how to employ a wide range of strategies as they write and to use different writing process elements appropriately to communicate with different audiences for a variety of purposes.
Critical Reading and Analysis Goal: Students will become accomplished, active readers who value ambiguity and complexity, and who can demonstrate a wide range of strategies for understanding texts, including interpretations with an awareness of, attentiveness to, and curiosity toward other perspectives.
History and Theory Goal: Students will develop a comprehensive knowledge of the variety of texts in diverse time periods and in diverse locations, as well as know the critical and historical principles behind the construction of literary, linguistic, and cultural histories, in order to demonstrate an active participation in scholarship.
Research Goal: Students will be able to follow a research process from proposal, research, drafts, to final projects.
Collaborative Learning Goal: Students will learn that the ability to communicate their ideas to a larger audience is as important as having the ideas themselves, and that sharing and coordinating ideas sustains and develops the larger intellectual sphere, of which they are a part. Students will understand the connection between collaborative learning and their intended professional field(s), including but not limited to their future professional roles and responsibilities.
- Students can write texts informed by specific (as is appropriate for the discipline and course contexts) rhetorical strategies.
- Students can write in several modes and for different audiences and purposes, with an awareness of the social implications and theoretical issues that these shifts raise.
- Students can revise for content and edit for grammatical and stylistic clarity.
Critical Reading and Analysis Goal:
- Students can apply a wide range of strategies to comprehend, evaluate, and interpret texts. These strategies may include, but are not limited to: drawing on their prior experience, their interactions with other readers and writers, reflection, intertextuality, their knowledge of word meaning and of other texts, their word identification strategies, and their understanding of textual features (e.g., sound-letter correspondence, sentence structure, syntax, context, graphics, images).
- Students can evaluate the aesthetic and ethical value of texts.
- Students will demonstrate an ability to recognize how formal elements of language and genre shape meaning. They will recognize how writers can transgress or subvert generic expectations, as well as fulfill them.
History and Theory Goal:
- Students can demonstrate knowledge the terminology of literary and/or cultural periods in order to be active participants in a variety of literary and/or cultural fields.
- Students can identify and employ theoretical approaches to literary and/or cultural study (including, but not limited to, film studies, linguistics, and professional and technical writing).
- Students demonstrate an ability to read texts in relation to their historical and cultural contexts, in order to gain a richer understanding of both text and context, and to become more aware of themselves as situated historically and culturally.
- Students can identify and formulate questions for productive inquiry.
- Students can evaluate sources for credibility, bias, quality of evidence, and quality of reasoning.
- Students use citation methods and structures appropriate to their field of study.
Collaborative Learning Goal:
- Students can effectively peer review.
- Students can engage in thoughtful and critical debate.
- Students can produce quality collaborate projects.<|endoftext|>
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Math 7B, Lesson 6, Spring 2022, 3/20/2022
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Chapter 11.2 More Properties of Inequalities & Chapter 11.3 Simple Linear Inequalities
1. For any two numbers, and b, one and only one of the following relationships holds:
• a < b
• a = b
• a > b
2. If a < b and b < c, then a < c
3. When a number is added to or subtracted from both sides of an inequality, the inequality holds.
• If a < b, then a + k < b + k
4. When both sides of an inequality are multiplied by a non-zero number k, the inequality holds for k > 0, but the inequality sign is reversed for k < 0.
• if a < b and k >0, then ka < kb
• if a < b and k < 0, then ka > kb
5. We can apply the properties of inequalities learned in the previous sections to solve simple linear inequalities in one variable, such as
• 3x + 5 < 17
• 4x -9 > 7x + 8
• represents the solution on a number line
6. HW assignment
• handout: 1 page
• workbook: page 18:#7, #8, #9; page 19: #14, #15, #16, #17, #18
1. For any two numbers, and b, one and only one of the following relationships holds:
• a < b
• a = b
• a > b
2. If a < b and b < c, then a < c
3. When a number is added to or subtracted from both sides of an inequality, the inequality holds.
• If a < b, then a + k < b + k
4. When both sides of an inequality are multiplied by a non-zero number k, the inequality holds for k > 0, but the inequality sign is reversed for k < 0.
• if a < b and k >0, then ka < kb
• if a < b and k < 0, then ka > kb
5. We can apply the properties of inequalities learned in the previous sections to solve simple linear inequalities in one variable, such as
• 3x + 5 < 17
• 4x -9 > 7x + 8
• represents the solution on a number line
6. HW assignment
• handout: 1 page
• workbook: page 18:#7, #8, #9; page 19: #14, #15, #16, #17, #18<|endoftext|>
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May also be called: FXS, Martin-Bell Syndrome, Escalante’s Syndrome, Marker X Syndrome, FRAXA Syndrome
Fragile X syndrome is a genetic birth defect that causes mental impairment, ranging from learning disabilities to mental retardation, autistic behaviors, and problems with attention and hyperactivity.
More to Know
The genes that you inherit from your parents are like chemical blueprints affecting the way your body looks and functions. Fragile X syndrome happens when someone is born with a defect in a gene called Fragile X mental retardation 1 (FMR1). The FMR1 gene makes a protein that is important for normal brain development. When kids are born with Fragile X syndrome, the defect in the FMR1 gene means that their bodies don’t produce this protein. As a result, they have some degree of mental impairment, which can range from mild to severe.
Children with Fragile X have different features that include a long face, large ears, flat feet, and extremely flexible joints, especially fingers. Boys have the syndrome more often than girls and are more likely to have substantial intellectual disability rather than milder learning problems. Both boys and girls are likely to have emotional and behavioral problems.
Some states screen newborns for Fragile X syndrome, but many children with the condition are not identified until they are toddlers. Children with developmental delays may be referred for genetic testing that can identify Fragile X. Treatment for Fragile X syndrome depends on the severity of the mental impairment and the symptoms it causes. Kids with Fragile X syndrome often get therapy to learn important skills and medicine to help with some behavioral issues.
Keep in Mind
Early diagnosis and a treatment plan that includes parents, caregivers, teachers, and others close to them can help kids with Fragile X reach their full potential. There is no cure for Fragile X, although researchers are exploring ways to prevent it and deal with the complications.
All A to Z dictionary entries are regularly reviewed by KidsHealth medical experts.<|endoftext|>
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Miguasha : From water to land (The Miguasha National Park)
- English home
- Site map
The Devonian: Age of Fishes
- In search of our origins
- Records of past life
- The notion of geologic time
- The history of the Devonian System
- Witnesses to evolution
- A lost world
- Tectonic context
- A Devonian day
- Life in crisis
- The Late Devonian extinction event
- Records of past life
- The plant world
- The conquest of land
- The first forests
- The plants of Miguasha
- Spores by the millions
- The animal world
- Life in the sea
- The diversification of fish
- Toward the first tetrapods
- The animals of Miguasha
- A window through time
- Fossil quality
- Specimens by the thousands
- The food chain
- Of predators and prey
- Stomach contents
- The aquatic environment
- Land-based communities
- Jawless fish
- Jawed fish
- Jawless fish
- A window through time
- Osteolepiforms |
Since 1879, no less than 3,000 specimens of Eusthenopteron foordi have been excavated from the sedimentary rocks of the Miguasha cliffs, making this species one of the most common in the formation. (28 kb) The sheer abundance and excellent preservation of so many specimens of this ancient fish have allowed for numerous studies, leading to a level of recognition and fame that is usually only seen for living species.
(44 kb) Known around the world, Eusthenopteron is sometimes called the fish with legs, reflecting how similar its fin bones are to those of the tetrapods. Other traits that link it closely to tetrapods are the labyrinthodont teeth (teeth with folded sheets of dentine), characteristic of primitive tetrapods, and the presence of a choana in the palate, which enabled tetrapods to breathe air. Did the choana in Eusthenopteron confer the same ability? Impossible to know for certain, but it is quite likely that the animal had lungs, as did other groups like the dipnoi.
(84 kb) The median fins of Eusthenopteron are easily recognized by their pointed sail-like shape and are positioned far back on the body, which is typical in osteolepiform fish. These fins enabled the fish to accelerate rapidly and thus surprise its prey.
The head of Eusthenopteron displays a complex pattern of dermal bones. Small teeth adorn the edges of the upper and lower jaws, whereas pronounced fangs grew a little farther back in the mouth. It was evidently a predator, a fact that is also directly demonstrated by the presence of whole fish, sometimes even fellow members of its own species, still in the abdomen of some specimens.
(24 kb) The streamlined profiles of this fish can reach more than a metre long, but specimens come in all sizes, some only 2.7 cm long, which has allowed an exhaustive investigation into its growth. Studies have established that it underwent at least two types of growth spurts during its lifetime, during which the ossification of various parts of its skeleton was accelerated.
Nicknamed the Prince of Miguasha, Eusthenopteron foordi has been the sites ambassador to the world for more than a century.
The Prince of Miguasha
Japanese production of a 3D animated film showing an Eusthenopteron swimming underwater and feeding on fish and then raising its head above the water.
Note: For best viewing of this site, you will need these plugins:
Miguasha: A story written in stone
- The Gaspé Peninsula: A world of oceans and mountains
- The birth of the Appalachians
- The closing of an ocean
- A sea of fossils
- Faunal realms
- Devonian lands in the Gaspé Peninsula
- The Miguasha Group
- The Fleurant Formation
- The Escuminac Formation
- Geological characteristics
- Localization systems
- An ancient estuary
- An environment of exceptional preservation
- Of cliffs and men
- The Seigniory of Shoolbred
- A World Heritage Site
- Miguasha fossils around the world
- The geology craze of the 19th century
- The first discoveries
- Scientists come to Miguasha
- Links to Scotland
- Local fossil hunters
- Erik Jarvik and the Prince of Miguasha
- The birth of the Miguasha project
- The 1991 International Symposium
- Le Parc national de Miguasha
- Protecting a unique heritage
- Fossil digs and research
- The on-site museum
Site map | Feedback | Links | Sources | Credits
<< Callistiopterus | Elpistostegalians >>
© Miguasha National Park 2007
Title: Reconstruction of Eusthenopteron foordi
Author: Illustration by François Miville-Deschênes
Sources: Parc national de Miguasha
The Devonian osteolepiform Eusthenopteron foordi, a swift predator with a hydrodynamic body.
Title: Eusthenopteron foordi
Author: Jean-Pierre Sylvestre
Sources: Parc national de Miguasha
It was during the Devonian Period that sarcopterygian fish gave rise to the first terrestrial vertebrates. Eusthenopteron foordi (shown here) was long thought to be the transitional animal between fish and tetrapods, sharing features with both, but recent discoveries have shown that the elpistostegalians are even more closely related to four-legged vertebrates.
Title: Labyrinthodont tooth
Author: Moya Meredith Smith
Sources: Parc national de Miguasha
Cross-section through a tooth from Eusthenopteron foordi. The labyrinth pattern of infolded dentine inside the tooth is a characteristic feature shared by the first tetrapods.
Title: Bones of Eusthenopterons pectoral fin
Author: Parc national de Miguasha
Sources: Parc national de Miguasha
Plaster cast of the Erik Jraviks model of the bones in the pectoral fins of Eusthenopteron foordi. This bone structure was often compared to that of the tetrapods.
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|Name Translation||Robust Fin|
|Period||Devonian 380-360 Million Years Ago|
|Length||6.5 feet (2 meters) long|
Eusthenopteron was a tetrapod Lobe-finned fish in the Devonian Period 385 million years ago.
They’re one of the Devonian lob
e-finned prehistoric fishes that have limbs that are much like the first amphibian Ichthyostega and lived in the same period but not in the same time. Other lobe-finned fishes that are tetrapodomorphs were Panderichthys and Tiktaalik.
Eusthenopteron was very advanced in terms of fish from the Devonian. It was one of the few fishes that actually had strong, limb-like fins that could’ve helped to pull itself around when the water got shallow, making it that much closer to becoming and amphibian and therefore a land dweller. It likely evolved from earlier fish that had started to try and move into shallower water and therefore needed stronger fins to move through the more land-based environment. Eusthenopteron is significant in the fossil record because it’s sort of a “missing link” in the ev
olution of amphibians and land-based creatures. It shows how fish first evolved strong fins that were capable of pulling them around before gaining actual digits and lungs. Eusthenopteron is likely an ancestor of Tiktaalik, which is technically the first amphibian and therefore land vertebrate.
Eusthenopteron is a strange-looking fish. It was quite big, over 6.5 feet (2 meters) long, and was long and slenderly built. It had a large head with powerful jaws that could’ve been capable of delivering a nasty bite to any potential victims. It had several strong, pointy teeth
that were perfectly designed to keep a hold of its prey and tear it apart. It had several broad fins coming from all over its body, but its frontal and back fins were especially strange. They were very well-muscled and thick, unusual for a fish. Most scientists believe it used these fins to help pull itself around on the the ocean floor when it got shallow and that it was a sign to show that amphibians were close to evolving.
In Popular Culture
Eusthenopteron was featured in the documentary Animal Armageddon, where it was shown how to live until the Devonian Extinction occured. It was also in the hit Japanese movie Ponyo alongside several other ancient Devonian fish. Eusthenopterons also appeared in the 8th movie of The Land Before Time in the beginning in the ocean.
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- v.281(1782); 2014 May 7
The humerus of Eusthenopteron: a puzzling organization presaging the establishment of tetrapod limb bone marrow
Because of its close relationship to tetrapods, Eusthenopteron is an important taxon for understanding the establishment of the tetrapod body plan. Notably, it is one of the earliest sarcopterygians in which the humerus of the pectoral fin skeleton is preserved. The microanatomical and histological organization of this humerus provides important data for understanding the evolutionary steps that built up the distinctive architecture of tetrapod limb bones. Previous histological studies showed that Eusthenopteron‘s long-bone organization was established through typical tetrapod ossification modalities. Based on a three-dimensional reconstruction of the inner microstructure of Eusthenopteron‘s humerus, obtained from propagation phase-contrast X-ray synchrotron microtomography, we are now able to show that, despite ossification mechanisms and growth patterns similar to those of tetrapods, it also retains plesiomorphic characters such as a large medullary cavity, partly resulting from the perichondral ossification around a large cartilaginous bud as in actinopterygians. It also exhibits a distinctive tubular organization of bone-marrow processes. The connection between these processes and epiphyseal structures highlights their close functional relationship, suggesting that either bone marrow played a crucial role in the long-bone elongation processes or that trabecular bone resulting from the erosion of hypertrophied cartilage created a microenvironment for haematopoietic stem cell niches.
The Tetrapoda, predominantly terrestrial vertebrates with limbs rather than paired fins, are the most adaptively divergent group among the Sarcopterygii and arguably among the Osteichthyes as a whole. Extant tetrapods form a well-defined clade distinguished from their closest living relatives (the lungfishes) by numerous synapomorphies affecting all aspects of their biology. These synapomorphies must have arisen within the tetrapod stem group between the last common ancestor of tetrapods and lungfishes (where the tetrapod total group originated) and the last common ancestor of extant amphibians and amniotes (the tetrapod crown-group node). Many of the ‘key characters’ of tetrapods (e.g. limbs with digits, sacrum, fenestra ovalis, hyomandibula modified as stapes [ 1 ]) first appear over a relatively short segment of the stem group, approximately between the nodes subtending Tiktaalik and Acanthostega [ 2 , 3 ], and thus presumably evolved rapidly and in concert, but the ‘fish–tetrapod transition’ as a whole was a protracted process. Tetrapods that are unambiguously fully terrestrial do not appear in the fossil record until the Viséan (late Early Carboniferous), some 60 Myr after the oldest trackways with digits [ 4 , 5 ].
While fossils from the lower and upper ends of the tetrapod stem group are similar to other extant lobe-finned ‘fishes’ and to crown-group tetrapods, respectively, the middle segment of the stem group contains taxa with combinations of tetrapod synapomorphies and plesiomorphic characteristics that are not seen in any living vertebrate. Apomorphies relating to soft anatomy, physiology and behaviour were also being acquired step by step [ 6 – 8 ], but unfortunately we have very limited direct evidence for these changes.
One of the few palaeobiological data sources available to us is the microanatomy and histology of the bones. The limb bones are of particular interest here because of their functional role in the transition from water to land. The major elements of the paired appendage endoskeleton are conserved throughout the tetrapod stem and crown group, and can for the most part be homologized with endoskeletal fin elements in extant lungfishes and coelacanths [ 9 – 12 ]. Detailed homologies with the elements of actinopterygian fin skeletons are more difficult to establish, but the overall homology of the skeletons is uncontroversial [ 9 , 13 ]. However, patterns of growth and ossification in the appendage endoskeletons differ greatly between tetrapods and actinopterygians [ 14 , 15 ], as does the occurrence and nature of bone marrow. These differences probably reflect evolutionary innovations in the tetrapod stem group.
The few studies that have investigated the histology of fin skeletons of tetrapod stem group members [ 16 – 18 ] have all focused on Eusthenopteron, a relatively crownward form closely related to tetrapods [ 2 ]. Although Meunier & Laurin [ 17 ] concluded that tetrapod-like mechanisms of ossification already existed in Eusthenopteron long bones, Laurin et al. [ 16 ] noted that the compactness profile at mid-shaft was different from extant aquatic tetrapods and assumed that it would be characteristic of the primitively aquatic condition of Eusthenopteron. All these studies were based on two-dimensional examination of thin sections, a destructive technique that yields limited datasets because of the need to conserve the rare and precious specimens of fossil appendage bones.
Here, we present a non-destructive three-dimensional approach for a new microanatomical and palaeohistological analysis of stem tetrapods in an ontogenetic framework, using propagation phase-contrast X-ray synchrotron microtomography. We were able to image multiple specimens of the primitively aquatic sarcopterygian Eusthenopteron (juvenile and adult bones) to produce a more comprehensive palaeobiological dataset and draw more detailed conclusions than was hitherto possible. Subsequent papers will examine members of the tetrapod stem group with more derived character states, in order to cast light on the biology of the terrestrialization process and the evolution of tetrapod limbs.
2. Material and methods
Eusthenopteron occurs abundantly at the 380 Myr-old locality of Miguasha, Quebec, Canada (Frasnian, Late Devonian [ 19 ]). The abundance of fossil material makes it possible to investigate its ontogeny by means of size series that can be taken as approximate representations of growth series. We focused on the three-dimensionally preserved humeri of one small and two large individuals of Eusthenopteron from the collection of Naturhistoriska Riksmuseet in Stockholm. The small humerus (NRM P246c) is incompletely ossified and is interpreted as juvenile ( figure 1 a). One of the large humeri (NRM P248d) is preserved in articulation with the proximal end of the ulna ( figure 2 a) and is associated with more distal elements of the fin endoskeleton as well as the proximal ends of lepidotrichia. These elements were also scanned for comparative purposes.
Mid-shaft bone histology of the juvenile humerus of Eusthenopteron (NRM P246c). (a) Mesial view of the whole humerus showing the location of the high-resolution scan made at mid-shaft (voxel size: 0.678 μm). (b) Virtual thin section (made along the longitudinal axis) showing the primary bone deposit of cortical bone and its connection to the spongiosa. Some remnants of calcified cartilage are still preserved at the location of Katschenko’s line (chondrocyte lacunae) and within the spongiosa (Liesegang rings). (c) Transverse view of the three-dimensional organization of the vascular mesh embedded within the cortical bone and the underlying trabecular spongiosa. (d) Quantification of the volume of bone cells showing three recurrent periods of volume decrease (green layers pointed out with white arrows) interpreted as phases of decreased growth. (e) From left to right: top and longitudinal views of the vascular mesh showing the circular and radial alignment of the vascular canals (in pink). The osteocyte lacunae are represented in bright blue. c, cortical bone; cl, chondrocyte lacunae; Lr, Liesegang rings; ol, osteocyte lacunae; s, spongiosa; v, vascular mesh.
Mid-shaft bone histology of the adult humerus of Eusthenopteron (NRM P248d). (a) Mesial view of the whole humerus showing the location of the high-resolution scan made at mid-shaft (voxel size: 0.678 μm). (b) Transverse virtual thin section showing the primary bone deposit of cortical bone, the innermost part of which has been drastically eroded. Although X-ray tomography does not allow the nature of the bone matrix (i.e. the collagen fibre organization) to be determined, secondary bone can be distinguished from primary cortical bone because it is always demarcated by a resorption line. A secondary bone deposit of cellular endosteal bone is laid down on the inner surface of the primary cortex. The trabeculae of the spongiosa are also covered by a thin layer of cellular endosteal bone. (c) Transverse view of the three-dimensional organization of the vascular mesh embedded within the cortical bone and the underlying trabecular spongiosa. (d) Quantification of the volume of bone cells showing an obvious decrease of volume just under the surface of the primary bone and in the endosteal bone (blue) (cf. electronic supplementary material, figure S8 for detail). (e) From left to right: top and longitudinal views of the vascular mesh showing a non-oriented organization of the vascular canals (same colour code as for figure 1 ). c, cortical bone; pb, primary bone; rl, resorption line; s, spongiosa; sb, secondary bone; v, vascular mesh.
The specimens were imaged using propagation phase-contrast X-ray synchrotron radiation microtomography (PPC-SRμCT) at beamline ID19, European Synchrotron Radiation Facility (ESRF, Grenoble, France). A multiscale approach [ 20 , 21 ] was applied from 20.24 to 0.678 μm (see the electronic supplementary material for technical details).
A phase retrieval approach, based on a homogeneity assumption, was employed for reconstructing the data, using a modified version [ 21 ] of the algorithm developed by Paganin et al. [ 22 ]. Virtual thin sections were made using the protocol established by Tafforeau and Smith for virtual histology of teeth [ 20 , 23 ].
(a) Juvenile humerus
A transverse virtual thin section taken at mid-shaft in the juvenile humerus exhibits an extensive spongiosa (88% of the section area) surrounded by a 650–850 µm-thick layer of compact cortical bone (electronic supplementary material, figure S1). The spongiosa consists of numerous endochondral bone trabeculae, averaging 80 µm in thickness, that are densely and homogeneously distributed. A longitudinal thin section of the humerus shows several longitudinal tubular spaces within the trabecular mesh crossing the whole bone from the proximal epiphysis towards one of the distal epiphyses (electronic supplementary material, figure S1b).
At mid-shaft ( figure 1 a), the inner surface of the cortical bone is delimited by clusters of numerous large globular cell lacunae (cl, figure 1 b,d) that can be identified as chondrocyte lacunae of cartilage. This suggests that remnants of Katschenko’s line [ 14 , 24 – 26 ] are still present. Several stacks of Liesegang rings [ 24 ], typical of calcified cartilage, are also notable among the endochondral trabeculae (Lr, figure 1 b). In extant tetrapods, the spongiosa forms when chondroclasts create erosion bays in the cartilage that are then lined with a thin peripheral bone layer, and it is common for small remnants of calcified cartilage to be left behind by the process; spongiosa formation in Eusthenopteron appears to have been similar.
The compact cortical bone exhibits a uniform primary tissue ( figure 1 b) with numerous flattened osteocyte lacunae, ranging in volume between 100 and 340 µm3 ( figure 1 d). It contains two complete and one partial cycle of progressively increasing osteocyte volumes, each complete cycle measuring 350–450 µm in thickness ( figure 1 d). The bone cell lacunae are mostly aligned in parallel with the peripheral surface of the bone. They are evenly organized around a dense vascular mesh. These canals are obliquely radial and parallel with each other. They average 42 µm in diameter ( figure 1 c,e).
Towards the epiphysis (electronic supplementary material, figure S2a), the metaphyseal compact cortical bone, separated from the spongiosa by a cementing line, is made of primary bone tissue pierced with a dense vascularization (vc, electronic supplementary material, figure S2b) surrounded by numerous flattened osteocyte lacunae (ol, electronic supplementary material, figure S2b). The metaphyseal region shows numerous extrinsic fibres embedded in the bone matrix (ef, electronic supplementary material, figure S2b). The proximal ends of these fibres are cut off by an erosion surface lined with endosteal bone (sb, electronic supplementary material, figure S2b). Erosion and endosteal ossification have thus already started operating on the internal face of this very young cortex (sb, electronic supplementary material, figure S2c), but some remnants of calcified cartilage (Liesegang rings and chondrocyte lacunae) are still present (respectively, Lr and cl, electronic supplementary material, figure S2d,e).
(b) Adult humeri
At mid-shaft, the marrow spongiosa has spread to 96.5% of the total diameter due to internal erosion of the cortex, which now has an average thickness of only 290 µm (electronic supplementary material, figure S3b). Large bays of erosion, covered with a thin layer of endosteal bone (secondarily deposited and identified from the resorption line; figure 2 b), cut into the compact bone layer. The boundary between the compacta and spongiosa therefore remains sharp (rl, figure 2 b; electronic supplementary material, figure S3b). The spongiosa is less dense than in the juvenile and exhibits very thin endosteal trabeculae (electronic supplementary material, figures S1 and S3). It is dominated by longitudinal tubes, which cross the whole humerus between proximal and distal epiphyses ( figure 3 a,b; electronic supplementary material, figure S3). These tubes, 300 μm in diameter, end blindly at the articular surfaces of the epiphyses ( figure 3 c). They are anastomosed with smaller transverse tubules that fuse to the vascular mesh of the compact bone layer ( figure 3 c).
Organization of the spongiosa in the humerus of Eusthenopteron. (a) Virtual thin sections made in the same longitudinal plane in the three humeri of Eusthenopteron (from left to right: juvenile specimen NRM P246c, adult specimen NRM P248d, adult specimen NRM P248a). Based on a directional coloured light system, the longitudinal trabeculae appear in purple and the transverse trabeculae in green. The white arrows point out the longitudinal tubular structures. (b) Three-dimensional organization of the tubular structures within the spongiosa in mesial view and ventral view (adult Eusthenopteron NRM P248d). (c) Distal articular surfaces (adult Eusthenopteron, NRM P248d) showing a distinctive pattern of aborted channels, typical of the cartilage-bone junction in tetrapods, which produces bony septa in columns. (d) Images showing the connection of two tubular structures with the cortical vascular mesh (pointed out with white arrows) in the juvenile Eusthenopteron (NRM P246c).
The mid-shaft cortical bone shows numerous small (typically 170–200 μm3), oval, homogeneously distributed osteocyte lacunae ( figure 2 d,e). The osteocyte lacunae of the endosteal bone are notably smaller, typically 50–100 μm3 ( figure 2 d). Average osteocyte volume is slightly smaller in the adult cortex than in the juvenile (electronic supplementary material, figure S4). Two lines of arrested growth (LAGs) are visible in the outermost region of the cortex (thick white arrows, figure 2 b). The vascular mesh is mainly composed of radial canals, anastomosed at their bases with longitudinal small canals that parallel the inner surface of the cortex (v, figure 2 c). The great majority of the radial blood vessels are closed off at the level of the first LAG; only a few reach the surface [ 21 , 27 ].
Towards the epiphysis, at the location of the ossification notch (electronic supplementary material, figure S5a) [ 24 , 28 , 29 ], the compact cortical bone tissue presents the same cellular and vascular organization as at mid-shaft (electronic supplementary material, figure S5c). The trabeculae in the spongiosa are greatly remodelled and show no visible remnant of calcified cartilage (electronic supplementary material, figure S5b).
(c) Distal bones of adult fin
The proximal end of the ulna of NRM P248d contains longitudinal tubes identical to those in the associated humerus ( figure 3 b). The associated ulnare, which also contains such tubes, has a proportionately thicker cortex than the humerus ( figure 4 a). Four LAGs are preserved, the last two much more closely spaced than the inner three. Despite its greater thickness, this cortex, like that of the humerus, shows resorption bays on its inner surface and has thus been subject to remodelling linked to medullary expansion.
Skeletochronological analysis. (a) Longitudinal virtual thin sections made in the ulnare of Eusthenopteron (adult specimen NRM P248d) showing four LAGs (black arrows). Because the innermost region of the primary cortex was eroded, part of the LAG pattern is not visible anymore. A tightening of the peripheral LAGs is obvious. (b) Virtual thin section made in a basal lepidotrichia of the same adult specimen showing 11 LAGs (white arrows). The two last peripheral LAGs are slightly closer to each other. (c) Growth curves made from the measurement of the bone deposit accumulation of periosteal bone in the remaining cortex of the ulnare (electronic supplementary material, table S1) and in the lepidotrichia (electronic supplementary material, table S2).
Basal segments of lepidotrichia preserved in articulation with the fin endoskeleton contain 11 LAGs ( figure 4 b). These elements show no sign of internal resorption, and as they are known to ossify early in life in Eusthenopteron [ 30 ], they probably record the complete growth history of the specimen.
(a) A mosaic long-bone organization
In extant actinopterygians, the fin endoskeleton develops from a blastema that differentiates into cartilages [ 15 ]. Perichondral bone is deposited on the surface of the cartilages and continues to grow centrifugally as periosteal bone (electronic supplementary material, figure S6). Consequently, a typical metapterygial bone (electronic supplementary material, figure S6) is composed of a cartilaginous rod surrounded by a bony tube with cartilage projecting as condyles [ 15 ]. In older individuals, the rod of cartilage can be resorbed and sometimes replaced by endochondral ossification, resulting in a superficially ladder-like trabecular spongiosa [ 31 , 32 ] (electronic supplementary material, figure S6). Most of the space created by this process is filled with fatty tissue, nerves and blood vessels. There is no haematopoietic tissue [ 15 , 31 ].
In most extant tetrapods, by contrast, the cartilaginous rod only exists at an early stage of long-bone development [ 14 ]. It is rapidly covered with perichondral bone, and then periosteal bone, which thickens substantially in older individuals to form the cortical bone (electronic supplementary material, figure S7). At the articular extremities, an epiphyseal centre produces more cartilage, known as ‘growth cartilage’, consisting of longitudinal columns of aligned chondrocytes that become hypertrophied towards the diaphysis [ 33 , 34 ]. At mid-shaft, the cartilage template is progressively hollowed out, creating the medullary cavity [ 14 , 35 ]. When the erosion front reaches the base of the growth cartilage, some vascular channels and marrow processes begin to invade the columns of hypertrophied chondrocytes [ 34 , 36 ]. Endochondral bone is deposited along the cartilaginous septa of the growth plate and on the surface of remnants of cartilage in the diaphysis.
In the juvenile Eusthenopteron, the remnants of Katschenko’s line at mid-shaft coupled with the large diameter of the medulla (approx. 5.3 mm, or 55% of the complete shaft diameter of the adult humerus; electronic supplementary material, figures S1 and S3) show that the cartilaginous humerus had grown quite large before the onset of ossification. This resembles the actinopterygian condition. However, the intricate trabecular architecture and presence of Liesegang rings of calcified cartilage in the juvenile are tetrapod-like characteristics. Similarly, the bone surfaces at the location of articulations exhibit a distinctive pattern of aborted channels resembling the cartilage-bone junction in most tetrapods [ 37 – 39 ]. As in non-amniotic tetrapods (and some amniotes), there was no secondary ossification centre [ 25 , 26 , 40 ] in the epiphyses of Eusthenopteron.
In summary, the humerus of Eusthenopteron shows a late onset of ossification similar to that seen in an extant actinopterygian, but the subsequent processes of cartilage resorption, endochondral ossification and elongation growth are all tetrapod-like. The single exception to this pattern is the strong internal resorption of the cortex and expansion of the medullary spongiosa revealed by adult humeri (electronic supplementary material, figure S3), which differs from both extant tetrapods and actinopterygians [ 16 ]. This may have served to maximize the volume of bone marrow (see below). The proportionally thicker cortical bone of more distal elements (e.g. ulnare, figure 4 a; radius, ulna [ 16 ]) may reflect biomechanical requirements.
(b) Growth curve and life history
Growth biology and metabolic features are recorded in long-bone cortical microstructure [ 41 – 44 ]. All the bones studied here come from the same locality (Miguasha, Canada) and show no evidence of any peculiar taphonomic degradation. A comparison between the microstructures of the juvenile and adult bones should thus yield reliable life-history data for Eusthenopteron.
The adult endoskeletal bones and lepidotrichia all show repeated LAGs indicating a cyclical growth pattern (figures (figures22 and and4).4 ). In the juvenile humerus, there are no visible LAGs, but cyclical variations in volume of the osteocyte lacunae reflect cyclical bone growth (cf. arrows pointing out recurrent decreases of bone cell volume, figure 1 d). This is confirmed by small osteocyte lacunae at the location of LAGs in the adult ( figure 2 b,d; electronic supplementary material, figure S8). The cyclicity probably reflects an intrinsic biological cycle enhanced by environmental seasonality [ 28 ], which is annual in all extant tetrapods living in mild and warm seasonal climate conditions [ 28 ], like those of Miguasha during the Devonian [ 45 ]. In this case, the adult individual with preserved lepidotrichia was at least 11 years old at death ( figure 4 c). The last 2–3 years of life of the adult witnessed a dramatic slowing of growth, as evidenced in the ulnare and humerus by the prominent and closely spaced final two LAGs (figures (figures22 b and and44 c), the sealing off of the majority of cortical blood vessels by the first of these LAGs [ 21 , 27 ] and the greatly reduced osteocyte volume in the outermost bone layer ( figure 2 d; electronic supplementary material, figure S8).
Such a slow-down of bone deposition is well known in tetrapods [ 43 , 46 ] and probably reflects the onset of sexual maturity. In the lepidotrichia, which begin to ossify very early in life [ 30 ] and do not undergo internal resorption, 11 LAGs are present, and there is a notable slow-down between LAG 10 and 11 ( figure 4 b,c). These data thus suggest a pre-reproductive growth period of approximately 10–11 years for this individual of Eusthenopteron, which is considerably longer than in the majority of extant amphibians (pre-reproductive period of 5 years in average and rarely longer than 9 years for urodeles; 3 years in average for anurans) [ 28 ] and also longer than in the South American lungfish Lepidosiren (8 years maximum total life span; age of sexual maturity unknown) [ 47 ], but within the range of some sturgeons (e.g. shortnose sturgeon: pre-reproductive period of 2–11 years for males, 6–13 years for females, depending on population) [ 48 ] and slightly shorter than in the Australian lungfish Neoceratodus (pre-reproductive period of 15 years for males, 20 years for females) [ 49 ]. The size of the juvenile humerus of Eusthenopteron suggests that ossification of the pectoral fin endoskeleton began approximately halfway through this period. For comparison, the smallest complete individuals of Eusthenopteron from Miguasha with partly ossified humeri have a total body length of approximately 18.5–19 cm [ 30 ].
(c) The earliest evidence for functional bone marrow
The humerus of Eusthenopteron is largely composed of cancellous bone. The organization of the trabecular mesh reflects a longitudinal tubular configuration that crosses the long bone from the proximal to the distal epiphyses ( figure 3 a,b). This longitudinal mesh is slightly transversally anastomosed and is connected to the cortical vascular canals ( figure 3 d), strongly suggesting that it had a role related to blood supply. The configuration resembles the arterial organization in the metaphyseal region of young tetrapod long bones [ 50 , 51 ].
In tetrapod epiphyses, during early development, the cartilage is organized in columns of chondrocytes separated by longitudinal bony septa [ 34 ]. Blood vessels progressively invade these cartilaginous columns and release diffusible factors that play an important role in the apoptosis of chondrocytes and the establishment of endochondral ossification at the chondro-osseus junction during elongation growth [ 34 ]. This process results in the longitudinal orientation of the metaphyseal trabecular mesh [ 40 ]. Not only vascular channels (10–30 μm in humans [ 36 ]) but also larger marrow processes (30–70 μm in humans [ 36 ]) penetrate the hypertrophied cartilage. In adult tetrapods, this longitudinal configuration can no longer be observed because intense erosion and remodelling incorporates the vascular mesh into the medullary cavity [ 52 , 53 ].
In contrast to tetrapods, actinopterygian long bones have no haematopoietic bone marrow but only fatty tissues in the spaces created by the erosion of the cartilaginous rod [ 15 , 54 ]. Neither do they show any evidence of a growth plate with longitudinally oriented columns of chondrocytes [ 15 , 40 ]. The large longitudinal tubular mesh observed in Eusthenopteron humerus appears to constitute the earliest and phylogenetically deepest documented occurrence of a complex functional bone marrow in the tetrapod stem group. As the tubular channels in Eusthenopteron are obviously connected to the epiphyses, the appearance of a complex bone marrow seems to be related to the appearance of tetrapod-like epiphyseal structures and elongation growth. Eusthenopteron lacks the comprehensive remodelling and trabecular resorption that creates an open medullary cavity in the majority of extant tetrapods, but the reduction of trabeculae between the longitudinal tubes in the adult compared with the juvenile may represent an evolutionary precursor of this process.
Bone marrow has mostly been studied for its haematopoietic properties [ 55 – 57 ]. However, the marrow is also a source of both osteoblasts and osteoclasts [ 58 – 60 ]. Tetrapod bone marrow has been shown capable of degrading cartilage proteoglycans and inducing the initial stage of endochondral ossification [ 61 ]. Cumulative evidence has shown strong links between osteoblasts and haematopoietic components in long bones of living tetrapods [ 62 – 64 ]. It seems that the establishment of certain haematopoietic niches is regulated by osteoblasts and/or osteoclasts, whose appearance precedes the activity of haematopoietic stem cells (HSCs) during development [ 57 , 63 , 65 ]. Some HSCs are functionally dependent on their proximity to endosteal surfaces [ 55 ], and HSC niches are frequently found close to the endosteal [ 51 , 66 – 68 ] or epiphyseal surfaces of bones [ 69 ]. The organization of the marrow space in Eusthenopteron argues for a functional link to extension growth at the epiphyses, suggesting that the intimate relationship between hypertrophic cartilage remodelling, endochondral ossification and haematopoiesis seen in extant mammals [ 38 , 39 ] is primitive for tetrapods. By contrast, the absence of haematopoietic niches in the limb bones of some amphibians [ 51 ] or birds can be interpreted as a secondary simplification of long bones with no endochondral trabeculae in amphibians [ 26 , 40 ] and pneumatization in birds [ 70 ].
Eusthenopteron proves to possess a more distinctive combination of biological and life-history traits than previously thought [ 17 ]. Morphologically, it is a conventional predatory ‘fish’ with no obvious terrestrial adaptations [ 30 ]. Its humerus is similar in relative size and proportions to those of other lobe-finned sarcopterygians [ 9 ]. Ossification of the humerus began when the element was more than half adult size and the animal apparently several years old, whereas extant tetrapod limb bones ossify much earlier [ 25 , 28 , 71 ]. Eusthenopteron humerus contains an organized spongiosa that seems to have housed a functional bone marrow, and it underwent tetrapod-like extension growth. A marked slowing of growth in the adult probably indicates the onset of sexual maturity. If this interpretation is correct, the pre-reproductive growth period spanned a whole decade, considerably longer than in extant amphibians [ 28 ]. Unlike either extant ‘fishes’ or amphibians, Eusthenopteron eroded the inner face of the humeral cortex so vigorously that it actually grew thinner, and the spongiosa more extensive, as the animal approached adulthood.
The morphology and phylogenetic position of Eusthenopteron show that its tetrapod-like humeral characteristics are not terrestrial adaptations, a point that is also underscored by the remarkable thinning of the humeral cortex, which must have lessened the mechanical strength of the adult bone. This raises important questions about the original functional significance of the emplacement of marrow into the limb bones and the adoption of tetrapod-like extension growth, as well as about their possible role as enabling factors for terrestrialization. In order to address these questions and investigate life-history evolution across the ‘fish–tetrapod transition’, more comparative histological and microanatomical data from long bones of other lobe-finned sarcopterygians and early tetrapods will be studied.
We thank T. Mörs at Naturhistoriska Riksmuseet, Stockholm, J. O. R. Ebbestad at the Museum of Evolution, Uppsala, and M. Herbin at the Muséum national d’Histoire naturelle, Paris, for allowing the scans of material from their collections (respectively, NRM P246c and P248d, PMU 25739 and 25738, MNHN AZ AC 2005.72). We are very grateful to L. Zylberberg (UPMC, Paris), G. Clément (MNHN, Paris) and J.-S. Steyer (CNRS, Paris) for fruitful discussions; V. Dupret (Uppsala University) and D. Geffard-Kuryiama (MNHN, Paris) for their help during the scanning experiments; and the journal editors, M. Laurin (CNRS, Paris) and an anonymous reviewer for their helpful comments.
The funding of this scan project was assumed by the ESRF (Experiments EC203; and EC519) and ERC Advanced Investigator grant no. 233111 (P.E.A.).
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Our Memories are Stronger than we Thought
Anyone’s memory can fail. The question is, why? To explain how memories are stored, one model has dominated the neuroscience literature since the 1960s: the consolidation hypothesis, which over time has become a full-blown principle. This holds that information is not fixed in our memory instantly, but rather via a gradual process which takes place in long, complex stages. If this process is interrupted by the disruption of brain activity, the “recording” of a fresh memory is thought to be impaired. Our recent work1 shows that this dogma deserves serious reappraisal.
The dogma of memory consolidation
We reviewed all the publications that led to the consolidation model, starting with those from the 1960s. Many of them describe experiments conducted on rats and mice, in which treatment that disrupts brain function (for instance an electroshock or an anaesthetic), was administered to the animals just after they had been made to learn something, such as finding the right path in a maze. It turned out that the treatment caused amnesia, which was all the more severe if it was administered shortly after the learning process. However, if the period exceeded one or two hours, their memory was not affected at all. The proposed interpretation was that any memory is fragile for a period of one to two hours after its formation, and that its fixation can be jeopardised by treatment applied during this so-called “consolidation” phase.
Later, in the 1970s and 1980s, research focused on the biological basis of this consolidation model, in which neurons in the brain and molecules exchanged at the synapses (connections of varying strength between neurons) play a crucial role. Work at that time concluded that the molecular cascades that occur after learning lead to the development of a neural network which is widely distributed throughout the brain and stabilised through the formation of new proteins that enable the creation of new synaptic contacts. Consequently, this network, with synaptic contacts strengthened by consolidation, was thought to be the biological substrate of memories.
In the 2000s, studies subsequently suggested the existence of a similar process called reconsolidation. This takes place when old memories are recalled: a smell, the sight of a detail, a particular taste (such as that of a madeleine for Marcel Proust), etc. was thought to revive certain memories and allow them to be updated. This means that when an already consolidated memory is reactivated, it is supposed to become fragile and alterable again before being restabilised (reconsolidated) in the mind. According to the latter scenario, it might even be possible to use amnesic agents such as electroshocks or pharmaceutical substances to erase old memories. This led to new hopes for treatment, especially of pathological memories such as traumatic stress disorders.2
However, this consolidation/reconsolidation model is now being challenged. Our study, which summarizes and completes a number of previous findings, shows that the data in the scientific literature which serves as a basis for this hypothesis were not analysed correctly. These experiments also showed that administering a second time the treatment supposed to inhibit consolidation had unexpected results: amnesia was no longer observed! This second course was applied just before the animal was tested to see if it remembered what it had been taught. This means that recollection does in fact exist, but that to access it, the subject must be placed in the same state as it was when the memory was recorded. In other words, the disruptive treatment used (drugs, electroshock, etc) “forms part” of the memory, or put another way, it modifies an individual's state—and it is this state that is integrated into the memory. This is a well-known phenomenon called state dependency. It has long been established that information acquired under the influence of alcohol or a drug is better retained when the subject is once more under the influence of these substances than when they are not.
So why wasn’t this discovered sooner? In our paper, we explain that, although this hypothesis was proposed a very long time ago, it was disregarded because the consolidation scenario was so coherent and popular. Above all, it perfectly echoed early research work on the cellular and molecular bases of learning and on the synaptic plasticity model (long-term potentiation), discovered at around the same time. Another reason why this theory was not accepted is that, unlike studies on state dependency, in the case of amnesia the drug is administered after, and not before, learning.
It must therefore be admitted that one of the main characteristics of new memories is not their fragility but their malleability, i.e. their ability to integrate contemporary information about the event to be memorised. Like a drug-induced condition, some of these events are so significant that, in their absence, the subject is unable to recall the memory. What is extremely interesting is that the period of malleability that is seen at the time of reminiscence formation is also observed when an old souvenir is reactivated. It is through this process that we can update our memories, by adding new elements that do not erase the original information, but complement it.
This integration process based on the malleability of active memories (the state of memories during their initial formation and their reactivation) is a crucial characteristic that explains all the cases of memory alteration described in the literature, such as experimental amnesia, interference and false memories, and opens up new therapeutic avenues that we have successfully begun to explore.
Ultimately, the concept of integration that we support is a game changer. According to this principle, memory formation takes place very rapidly (probably in seconds rather than hours). It does not depend on the synthesis of new proteins. The synaptic alterations that accompany this process do not form the basis of the memory trace, but simply reflect the activity of the region. Memories are not fragile, and cannot be erased. However, old ones can be altered and made inaccessible. This concept re-endows the memory with the dynamic and flexible nature that characterizes brain function.
The analysis, views and opinions expressed in this section are those of the authors and do not necessarily reflect the position or policies of the CNRS.
- 1. P. Gisquet-Verrier, D. Riccio, "Memory integration: An alternative to the consolidation/reconsolidation hypothesis," Progress in neurobiology, available online since 18 October 2018.
- 2. These experiments work very well in animals but are not carried out on humans due to the toxicity of most treatments. Only propranolol (a beta-blocker) has been used with some success, and is being tested in the Paris MEM study on those involved in the Bataclan terrorist attack. Our team has also begun to explore other treatments in animals and humans, in collaboration with the psychiatry department at the Saint-Antoine Hospital in Paris, with very encouraging initial results.<|endoftext|>
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Engineering Calculus Notes 384
# Engineering Calculus Notes 384 - 372 CHAPTER 3. REAL-VALUED...
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Unformatted text preview: 372 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Differentiating with respect to x1 , this gives ∂f = 2a11 x1 + 2a12 x2 + 2a13 x3 ∂x1 = 2(a11 x1 + a12 x2 + a13 x3 ). Note that the quantity in parentheses is exactly the product of the first → → row of A = [Q] with [− ], or equivalently the first entry of A [− ]. For x x → − for the vector convenience, we will abuse notation, and write simply A x → whose coordinate column is A times the coordinate column of − : x → → [A− ] = A [− ] . x x You should check that the other two partials of f are the other coordinates → of A− , so x →→ −− → ∇ f ( u ) = 2A− . u If we also recall that the dot product of two vectors can be written in terms of their coordinate columns as →→ − · − = [− ]T [− ] →→ xy x y → → then the matrix form of f (− ) = Q(− ) becomes x x → →→ f (− ) = − · A− ; x x x we separate out this calculation as Remark 3.9.1. The gradient of a function of the form → →→ f (− ) = − · A− x x x is →→ −− → ∇ f ( x ) = 2A− . x Note that, while our calculation was for a 3 × 3 matrix, the analogous result holds for a 2 × 2 matrix as well. Now, the Lagrange multiplier condition for extrema of f on S 2 becomes → → A− = λ− . u u (3.31) → → Geometrically, this means that A− and − have the same direction (up to u u reversal, or possibly squashing to zero). Such situations come up often in → problems involving matrices; we call a nonzero vector − which satisfies u ...
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
Ask a homework question - tutors are online<|endoftext|>
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Square Root Calculator
Calculate Square Root
$$\begin{array}{l}\sqrt{a}\end{array}$$
Enter the ‘a’ value =
Square Root =
The square root calculator is a free online tool that displays the square root of the given number. BYJU’S online square root calculator tool makes the calculations faster and easier where it gives the square root of the given number in a fraction of seconds.
How to Use the Square Root Calculator?
The procedure to use the square root calculator is as follows:
Step 1: Enter the number in the respective input field
Step 2: Now click the button “Find Square Root” to get the output
Step 3: The square root of the given number will be displayed in the output field
What is Square Root?
The square root of a number is defined as the value, which gives the number when it is multiplied by itself. The radical symbol √ is used to indicate the square root. For example, √16 = 4. The radical symbol is also called a root symbol or surds. If a number is a perfect square, we can easily find the square root of the number. If the given number is not a perfect square number, the square root can be found using the long division method.
Standard Form
The standard form to represent the square root is given below:
The square root of a function is defined as: f(x) = √x
In other words, it is defined by √(x.x) = √(x)2 = x
Solved Examples on Square Root
Example 1:
Find the square root of 625
Solution:
Given: To find the square root of 625
√625 can be written as
√625 = √(25 × 25)
= √(25)2
Eliminate square and square root, we get
√625 = 25
Hence, the square root of 625 is 25.
Example 2:
Determine the value of 4√16
Solution:
Given: expression is 4√16
√16 = √(4×4)
√16 = √(4)2
Now, cancel out square and square root, we get
√16 = 4
Now substitute √16 = 4 in the given expression
Therefore, 4√16 = 4 × 4 = 16
Hence, 4√16 = 16
Frequently Asked Questions on Square Root Calculator
What is meant by a perfect square?
The perfect square of a number is defined as the squares of whole numbers. The square of the number is given as the number times itself. Some of the examples of the perfect squares are 1, 4, 9, 16, 25, 36, and so on.
What is the formula for square root?
The formula to find the square root of a number is given as: √(x^2) = x
How do you find the square root for non-perfect square numbers?
If the given number is not a perfect square number, the method called long division method is used.
What is the value of the square root of 2?
As the number 2 is not a perfect square, the value for the square root of 2 can be found using the long division method. The value of the square root of 2 is approximately equal to 1.414.
Write down the square and square root value of 4.
The square of 4 is 42, which is equal to 16
The square root of 4 is 2.<|endoftext|>
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The most destructive aspect of volcanoes are lahars and pyroclastic flows.
What are lahars?
Lahars are mudflows created when water (from rain or meltwater from glaciers) and volcanic ash mix. This deadly combination can have devastating results on the surrounding area. When lahars settle they can be metres thick and as hard as cement. Lahars can occur long after a volcanic eruption.
What are pyroclastic flows?
Pyroclastic flows are avalanches containing hot volcanic gases, ash and rock. They are the most deadly event to happen at a volcano. Pyroclastic flows have killed over 90,000 people since 1600AD. 30,000 people were killed by a pyroclastic flow from Mount Pelee in 1902.
A pyroclastic flow
Pyroclastic flows are incredibly dangerous for several reasons. They are fast moving. On steep volcanoes, pyroclastic flows can reach speeds of 450 miles per hour. Pyroclastic flows are incredibly hot. They come from explosive eruptions or from the collapse of a lava dome. Temperatures can be up to 1000°C. The clouds mask dense avalanches carrying rock. The force of a pyroclastic flow can destroy buildings and flatten trees.
Possibly one of the most famous pyroclastic flows was the one caused by the eruption of Mount Vesuvius in 79 AD which covered the Roman city of Pompeii with ash and volcanic debris. 1.5 million tons of rock, pumice and ash we released every second.
The video below shows a more recent pyroclastic flow at Mount Ontake, Japan.
Please Support Internet Geography
If you've found the resources on this page useful please consider making a secure donation via PayPal to support the development of the site. The site is self-funded and your support is really appreciated.
Use the images below to explore related GeoTopics.
If you've found the resources on this site useful please consider making a secure donation via PayPal to support the development of the site. The site is self-funded and your support is really appreciated.<|endoftext|>
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This project is solving the Asteroids 2025-2100 - Future History challenge. Description
This project gives educators a way to explore space science with their students and then to allow the students to engage their imagination about future space exploration. There is a collection of descriptions about various bodies in space, as well as examples of "what could happen" in the future. Students are able to create their own art to represent components of the visual representation, which are uploaded into the software and displayed when the planet is selected.
For pre-k through 2nd grade (beginner level), there are reading level appropriate descriptions of planets, the sun, asteroids belt, the moon and the ISS. Lessons can include imaging what the planet looks like (including geographical features like volcanoes and mountains), the atmosphere, and the path a rocket ship might take to get there. Young student can think about what space ships might need to look like to land on these place and work together to think about the sorts of thing people and robots can do on these space bodies.
For 3-6 graders (advanced), there is an increase in the details in the description of the space bodies. Additionally, the advance levels include Ceres in the Asteroid belt, the Kuiper Belt and Eris and Pluto as representations of dwarf planets. There are links for students to read about planned and past missions. Teachers can have students create space missions, including thinking about what types of resources they might need, how they might engineer a mission, and what that would look like. This can include using the asteroid belt as a jumping off point for exploring the outer planets and using rouge asteroids and comets as potential things to hitch a ride on to save on the need for fuel to travel around the solar system. Additionally, social issues about long term space adventures are included. This includes how you would pick people for the mission, how you would create a new society on a planet, and how you would create rule, to inte4rgreat the social sciences into the mission.
Teachers are encouraged to bring in literature as a resource for students to help think creative about potential space missions. A short list is included, but teachers will be encouraged to give fee back on resources they are using to expand this in the future.
License: GNU General Public License version 2.0 (GPL-2.0)
Source Code/Project URL: https://github.com/gwob/stellar-visage<|endoftext|>
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Similarity, Ratios, and Proportions
SIMILARITY, RATIOS, and PROPORTIONS
• PRACTICE (online exercises and printable worksheets)
Suppose you have a triangle that you'd like to enlarge.
That is, you want to keep the shape exactly the same, but you want it to be bigger.
Perhaps you want an enlarged copy where the length of each side is three times the size of the original,
as shown below:
Or, perhaps you have a quadrilateral that you'd like to reduce.
That is, you want to keep the shape exactly the same, but you want it to be smaller.
Perhaps you want a reduced copy where the length of each side is half the size of the original, as shown below:
In both cases, you are keeping the angles exactly the same,
and you are multiplying all the sides by an appropriate scaling factor.
When you made it three times as big, the scaling factor was $\,3\,$.
When you made it half as big, the scaling factor was $\,0.5\,$.
This idea of keeping the shape the same, but changing the size is made precise by the concept of similarity.
Very roughly, two geometric figures are said to be similar when they have the same shape, but not necessarily the same size.
To make the concept of similarity precise, we first need to review ratios and proportions.
DEFINITION ratio
Let $\,a\,$ and $\,b\,$ be real numbers, with $\,b\neq 0\,$.
The ‘ratio of $\,a\,$ to $\,b\,$’ is the quotient $\displaystyle\,\frac{a}{b}\,$.
A ratio automatically brings a scaling factor into the picture, as follows.
Start by giving the ‘ratio of $\,a\,$ to $\,b\,$’ a simpler name, $\,s\,$.
(We're calling it $\,s\,$ because it will be our scaling factor.)
$\displaystyle s := \frac{a}{b}$
(The notation $\,:=\,$ is frequently used in mathematics to mean equals, by definition.)
Solving for $\,a\,$ gives $\,a=sb\,$.
So, $\,b\,$ has been scaled by $\,s\,$ to give $\,a\,$.
If $\,s=3\,$, then $\,a=3b\,$, so that $\,a\,$ is three times bigger than $\,b\,$.
If $\,s=0.5\,$, then $\,a=0.5b\,$, so that $\,a\,$ is half as big as $\,b\,$.
When we scale geometric figures, equal ratios automatically enter the picture.
Consider the picture below, where a quadrilateral has been scaled by a factor of $\,s\,$ (here, $\,s\,$ is greater than one) to get a new quadrilateral:
Notice that $\,A=sa\,$ and $\,B=sb\,$ and $\,C=sc\,$ and $\,D=sd\,$.
Solving for $\,s\,$ in each of these four equations gives an equality of four ratios: $$s = \frac{A}{a} = \frac{B}{b} = \frac{C}{c} = \frac{D}{d}$$ Thus, equality of ratios arises very naturally in any scaling situation.
An equality of ratios is called a proportion:
DEFINITION proportion
A proportion is an equality of ratios.
Therefore, a proportion takes the form: $$\frac{A}{a} = \frac{B}{b}$$
Using just a little bit of algebra, many useful equalities arise from proportions:
THEOREM equivalent proportions
Let $\,a\,$, $\,b\,$, $\,A\,$, and $\,B\,$ be nonzero real numbers.
Then, the following are equivalent:
• $\displaystyle\frac{A}{a} = \frac{B}{b}$
• $\displaystyle\frac{a}{A} = \frac{b}{B}$
• $\displaystyle\frac{a}{b} = \frac{A}{B}$
• $\displaystyle\frac{b}{a} = \frac{B}{A}$
Think about these proportions in the context of a scaled quadrilateral: Some of the proportions, like $$\frac{A}{a} = \frac{B}{b}\ \text{ and }\ \frac{a}{A} = \frac{b}{B}$$ compare ratios between the two quadrilaterals. Some of the proportions, like $$\frac{a}{b} = \frac{A}{B}\ \text{ and }\ \frac{b}{a} = \frac{B}{A}$$ compare ratios within the two quadrilaterals. In both cases, attaching words to the proportions may be useful. For example, you can read $\ \displaystyle\frac{A}{a} = \frac{B}{b}\$ as ‘$\,A\,$ is to $\,a\,$ as $\,B\,$ is to $\,b\,$’.
Cross-multiplying is a useful technique whenever you find yourself working with proportions:
DEFINITION cross-multiplying
The process of going from the proportion $\displaystyle\,\frac{A}{a} = \frac{B}{b}\,$ to the equivalent statement $\,Ab = aB\,$ is called cross-multiplying.
Cross-multiplying is nothing other than an application of the Multiplication Property of Equality,
where both sides of the equation are multiplied by the product of the original denominators: $$\begin{gather} \frac{A}{a} = \frac{B}{b}\cr\cr \frac{A}{a}\cdot (ab) = \frac{B}{b}\cdot (ab)\cr\cr Ab = Ba \end{gather}$$
DEFINITION the phrase: ‘corresponding sides are proportional’
Suppose a triangle with sides $\,a\,$, $\,b\,$, and $\,c\,$ is scaled to get a new triangle
with corresponding sides $\,A\,$, $\,B\,$, and $\,C\,$.
In this situation, the phrase corresponding sides are proportional is used to describe the equality of ratios: $$\frac{A}{a} = \frac{B}{b} = \frac{C}{c}$$ This same terminology is used when scaling any polygon.
Finally, we are ready for the precise definition of similar polygons:
DEFINITION similar polygons
Two polygons are similar if and only if there exists a correspondence between their vertices such that corresponding sides are proportional and corresponding angles are equal.
In general, to prove that two polygons are similar, you must check two different things:
• that corresponding angles are equal
• that corresponding sides are proportional
It is an extremely useful result that for triangles (and triangles alone!) it suffices to check only that angles are equal;
Of course, if two angles are equal, then the third angles must be equal, and so we have the (unproved) theorem:
AA SIMILARITY THEOREM
Two triangles are similar if and only if
two angles of one triangle are equal to two angles of the other triangle.
This is NOT true for polygons other than triangles.
For example, look at the pictures below.
All the angles are the same, but the polygons are not similar.
Master the ideas from this section<|endoftext|>
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What is the atmosphere?
The atmosphere is a thin layer of gases that surrounds the Earth. It seals the planet and protects us from the vacuum of space. It protects us from electromagnetic radiation given off by the Sun and small objects flying through space such as meteoroids. Of course, it also holds the oxygen (O2) we all breathe to survive. In the same way that there are layers inside of the Earth, there are also layers in the atmosphere. All of the layers interact with each other as the gases circulate around the planet. The lowest layers interact with the surface of the Earth while the highest layers interact with space. On your level, you may feel the atmosphere as a cool breeze. Other times you will feel it as a hot or humid day that seems to push on you from all sides.
An Envelope of Gases
When compared to the diameter of the Earth, the atmosphere is very thin. The thickness of the atmosphere is a balance between the gravity of the Earth and energetic molecules that want to rise and move towards space. The molecules in the upper layers of the atmosphere become excited as energy from the Sun hits the Earth. The molecules in the lower layers are cooler and under greater pressure. If the Earth were larger, the atmosphere would be denser. The increased mass and related gravity of a larger planet would pull those gas molecules closer to the surface and pressure would increase. The atmosphere is more than just layers of gases surrounding the planet. It is also a moving source of life for every creature of the planet. While the majority of the atmosphere is composed of nitrogen (N2) molecules, there are also oxygen and carbon dioxide (CO2) which plants and animals need to survive. You will also find ozone (O3) higher in the atmosphere which helps filter harmful ultraviolet radiation from the Sun. The atmosphere also protects us from the vacuum and cold of space. Without our atmosphere, the Earth would be as barren and dead as the Moon or Mercury.
There is no single climate of the planet. Specialized climates are found in areas all over the planet and might include deserts, rainforests, or polar regions. The common trait of all of these climates is the atmosphere. The atmosphere circulates gases and particles between all of these regions. The hot air from the equator eventually moves north or south to other climate regions. That warmer air combines with cooler air, mixing begins, and storms form. The constant mixing of the atmosphere maintains a stable system that helps organisms survive. Oxygen will never run out in one area of the planet and temperatures will not skyrocket in another. The atmosphere balances the possible extremes of the Earth and creates an overall stability. A great example is seen in the way tropical cyclones (hurricanes) form over the Atlantic Ocean. Because of global atmospheric circulation, systems start over the Sahara Desert in Africa, move across the west coast of northern Africa, pick up large amounts of water as they pass over the warm Atlantic Ocean and Caribbean Sea, and finally dump all of the rain on the Caribbean or south eastern coast of the United States. In addition to the stormy weather, the atmosphere can also carry dust and particles from the Sahara to North America.<|endoftext|>
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In the early days of the Texas Republic (1835-1845), Texas was a vast, untamed wilderness. Abundant wild game and fertile soil created a land of opportunity. However, it also challenged settlers with difficulties such as changeable weather and unfriendly Indians. The infant Texas government was eager to attract homesteaders and build a civilization. In order to hasten settlement, the government granted large tracts of land, or headrights, to immigrants. This not only encouraged settlement of the empty land, but also established a tax base to provide revenue for the fledgling government. In order to facilitate the transfer of property into private hands, the government recognized the need to create a system to record and verify private land titles. Therefore, in 1837 The Republic of Texas created the General Land Office (GLO) to administer and record the land grant process. The government of Texas granted headrights to eligible heads of families and to single men who could prove residency and show themselves to be good citizens.
Basic Land Provisions
Before a land grant, or patent, was issued to a settler, surveyors measured and recorded the borders of the grant. The basic unit of measurement for land was the vara, a Spanish term of measurement which equaled thirty-three and one-third inches. Thirty-six varas equaled one-hundred feet. Land surveyors frequently used this term in their descriptions of distances and borders. Other units of measurement for land included leagues and labors. A league was 4,428.4 acres, and a labor was 177.1 acres. A usual headright for the earliest settlers in Texas was a league and a labor or about 4,600 acres.
Early surveyors did a remarkably accurate job of surveying and defining these parcels of land, especially taking into consideration that they were often surveying uncharted streams and unknown forests. A survey had to be made and approved before the grant could be issued. It was not unusual for a survey to use landmarks such as “a stake in the prairie” or descriptions of trees of a particular kind or diameter. The early price for this land was quite cheap. An eight-league grant could be bought for as little as $1000.i
The Republic of Texas set some strict guidelines to determine who was eligible to receive land grants. This list of requirements gives historians a framework for understanding the history of each grant and a profile of the person who received it. Researchers can glean information about a grantee by examining the type of grant received. Land grant records provide information such as the settler’s approximate time of arrival in Texas, their place of residence, and marital status.
Four Classes of Grants
The Texas GLO grouped land grants or headrights into four main classes. First class grants were given to settlers who had arrived in Texas prior to the signing of the Texas Declaration of Independence on March 2, 1836, but who had not yet received a land grant. Heads of families received one league and one labor of land while single men could receive one-third of a league or 1,476.1 acres. These grants were unconditional, allowing the grant to be transferred or sold even before the Certificate was issued.
Second class land grants were given to settlers who had arrived in Texas after March 2, 1836, but before October 1, 1837. Heads of families could receive 1,280 acres while single men were eligible for six-hundred forty acres. Initially, the government issued a Conditional Certificate that required three years of responsible citizenship. During the period of the Conditional Certificate, the grantee could not sell the property. After at least three years of residency in Texas, the grantee received an Unconditional Certificate which could lead to a Patent. The grantee was not required to live on his land grant; he was only required to live in Texas. The issuance of Conditional Certificates and qualifying citizenship requirements discouraged speculators from profiting by buying and selling cheap land. It also curtailed tendencies toward forging documents.
Third class land grants were given to settlers who arrived in Texas between October 1, 1837 and January 1, 1840. Heads of families were eligible for six-hundred forty acres while single men received three-hundred twenty acres. Third class grants required similar conditions to second class grants, including a Conditional Certificate followed by three years of responsible citizenship before the Unconditional Certificate was issued.
Fourth class grants were issued to immigrants who arrived in Texas between January 1, 1840 and January 1, 1842. The amount of land granted was the same as the third class grant. The conditions for ownership were also similar to the second and third class grants, with the additional requirement that at least ten acres be cultivated.
The type of land grant received indicates a settler’s approximate date of arrival. Earlier arrivals received more land than later arrivals, and heads of households received more land than single men. A man who had a wife or dependents qualified as a head of the household and could receive the full grant of land, but only if his dependents were living in Texas with him. If he came alone, then he only qualified for the land grant for single men. If his family arrived later, he could apply for an augmentation.ii
In all, Texas issued 17,382 First Class land grants, 6,056 Second Class land grants, and 37,670 Third Class grants. Though this seems like an enormous number of grants issued in the early years, it makes up only twenty-one percent of the 290,597 patents given by the state of Texas as of 1986.iii
The Grants for Pecan Springs
In the case of the Pecan Springs Ranch, two primary grants comprise the land holding. A section on the northwestern edge of the property was originally deeded to Lewis B. De Spain or D’Spain. He received a second class grant, so it can be deduced that he arrived in Texas before October 1, 1837. He applied for the grant on January 18, 1839 and received a Conditional Certificate for 640 acres. This indicates that he was either single or his family did not reside in Texas since 640 acres was the amount given to a single man for a second class grant. His application was submitted in San Augustine County. On September 21, 1841, Texas issued him an Unconditional Certificate after he met his residency requirements.iv De Spain continued to live in San Augustine County, and there are no records of him residing on or improving the property. After his death, his estate sold the land to John Dabney Sims on January 17, 1857.v The property remained in the Sims family for over 150 years.
A land grant given to Charles Merlin forms the larger portion of the Pecan Springs land. Merlin was born circa 1800 in France. He immigrated to Texas sometime before 1839 because he received a third class Conditional land grant from the state of Texas on March 11, 1839.vi The Texas government issued the Unconditional Certificate on October 7, 1844. The Merlin headright is unique and of historical significance because it was signed by Sam Houston, then President of the Republic of Texas. While it is not unusual to see a land grant signed by Houston, the majority of land grants were signed by Anson Jones, the last President of Texas.vii Mr. Jones was president during a majority of the years when the early grants were issued.
The surveyors described the Merlin grant in the fashion of most land grants of that time, delineating the borders by naming landmarks. The Merlin survey was described as:
Six hundred and forty acres (being his headright) of land situated and described as follows in Robertson County – on the waters of Chambers Creek a branch of the Trinity River. Beginning at the West corner of a survey of 640 acres made for W R Horne a stake in prarie (sic). Thence North 30 degrees West nineteen hundred varas with the South
West line of a survey of 640 acres made for John Levi to his West corner a stake in the prarie (sic). Thence South 60 degrees West nineteen hundred varas to a stake from which a Spanish Oak 14 inches in diameter bears North 53 degrees West 3 varas. And an Over Cup Oak 24 inches in diameter bears North 75 degrees east 24 varas. Thence South 30 degrees East nineteen hundred varas to the West corner of a survey of 640 acres made for Ann H. Stokes, a stake in the prarie (sic). Thence North 60 degrees East nineteen hundred varas with the North West line of said Ann H. Stokes survey to the place of Beginning.viii
Both the De Spain and Merlin grants are recorded as being located in Robertson County since Ellis County had not yet been formed at that time.
Good Land Awaiting Use
In spite of having prime black soil prairie land, neither De Spain nor Merlin seemed to take a personal interest in the land. They neither performed any improvements on the land nor planted any crops. It is unclear whether they even personally saw the land. Merlin filed for the land grant from the city of Houston in Harris County, and he resided there the rest of his life. According to 1840 tax records, Merlin owned a town lot in Houston.ix Merlin’s family consisted of his French-born wife Eliza Baiz (1815-1850) and their two daughters. Merlin died on October 13, 1854 from an apparent overdose. A report issued by the Houston Telegraph reads:
“Found Dead: Mr. Charles Merlin, an old resident of this city, and formerly keeper of the Alabama House, was found dead in his bed on yesterday morning. He had been on a spree several days before, and had a phial of black drops, from which he had taken a few drops to quiet his nerves on Thursday night. It was found that he had taken during the night the entire contents, which occasioned his death.”x
Merlin’s death left behind two young daughters: Celestine, age thirteen, and Julia, age eleven. Merlin’s rather sizeable holdings went to his daughters. In addition to the land that he owned in Ellis County, he also owned three hundred twenty acres in Erath County, thirteen and a half lots and a house in Houston, and thirteen hundred and ninety-eight acres of land in Harris County.xi The Merlin estate sold the land in Ellis County to Alfred Whitaker in 1863 for $1,280.xii By this time Celestine and Julia were both married and still residing in Houston. The terms of Merlin’s will divided the Ellis County land and the profits from its sale between the two sisters.xiii
Alfred Whitaker experienced some financial difficulties shortly after purchasing the Merlin land, and as part of bankruptcy proceedings sold several properties to Alfred Gee in 1867.xiv These properties included the Merlin grant. Fortunately for Whitaker, he was able to recoup some of his expenditures since Gee purchased the Merlin land for approximately $1,485, about $200 more than Whitaker paid for it. Gee then sold the 640 acre Merlin grant to Nicholas P. Sims in 1869.xv Nicholas demonstrated shrewd business sense because he was able to buy the land for $1,000 in gold. In 1878, he then sold the land to his nephew, Wilson Dabney Sims, for $4,500, earning a profit of $3,500.
i1 A.R. Stout, “Pioneer Country: Its Origin, Early Land Grants,” History of Ellis County, Texas (Waco: Texian Press, 1972), 13.
ii 2Gifford White, 1840 Citizens of Texas, Volume 1: Land Grants, Austin, Texas (St. Louis: Ingmire Publications, 1983), vii-xiii.
iii3 Gary Mauro, The Land Commissioners of Texas, 150 Years of the General Land Office, Austin, General Land Office of Texas, 1986 quoted in Charles E. Gilliland, David Carciere, and Zachry Davis, “Texas Title Trail.” Land Markets. Publication 1760 ( January 2006).
iv4 Ibid., 66.
v5 Ellis County Clerk, Land Grant Records, Vol. C, page 86.
vi6 White, Land Grants, Austin, Texas, 173.
vii7 A.R. Stout, “Pioneer Country: Its Origin, Early Land Grants,” History of Ellis County, Texas (Waco: Texian Press, 1972), 11.
viii8 Ellis County Clerk, Land Grant Records, Vol. E, page 163.
ix9 Gifford White, 1840 Citizens of Texas, Volume 2: Tax Rolls, Austin, Texas (St. Louis, Ingmire Publications, 1984). 67.
x10 John S. Ford, The Texas State Times (Austin, Tex,), Vol. 1, No. 46, Ed.1 Saturday, October 14, 1854. Austin, Texas: Ford, Walker & Davidson. The Portal to Texas History. http://texashistory.unt.edu/ark:/67531/metapth235731/m1/2/zoom/?q=charl… (accessed February 28, 2015).
xi11 Ancestry.com. Texas, Land Title Abstracts, 1700-2008 [database on-line]. (Provo, UT, USA: Ancestry.com Operations inc., 2000).
xii12 Ellis County Clerk, Land Grant Records, Vol. F, page 192.
xiii13 Ellis County Clerk, Land Grant Records, Vo. E, page 163.
xiv14 Ellis County Clerk, Land Grant Records, Vol. F, page 663.
xv15 Ellis County Clerk, Land Grant Records, Vol. I, page 414.<|endoftext|>
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Back before pumpkin pies were even a glimmer in the eyes of bakers, Pleistocene-era mastodons, mammoths and giant sloths were spreading the seeds of these fruit far and wide. Anywhere the huge animals collectively known as megafauna roamed became a dumping (pardon the pun) ground for the seeds of pumpkins, squash and other members of the genus Cucurbita, which would then spring up like weeds.
But while these wild fruits were nourishing giant animals, these ancestral pumpkins weren't yet part of the diet of humans or smaller animals due to the plants' toxicity and bitter taste.
A new study by an international group of researchers – who looked at, of all things, gourd/squash seeds in mastodon dung – has shown that the extinction of megafauna about 12,000 years ago led in a rather roundabout way to the evolution of Cucurbita from the toxic and bitter into the tasty pumpkins, squash and other fruits that we enjoy for Thanksgiving and, which in turn, evolved into the now-ubiquitous pumpkin-spice lattes, beers and ice cream.
But how could the extinction of megafauna back then lead us to autumn's most overused fall flavor now? Think coevolution – when two or more species mutually affect each other's evolution.
"There's a whole suite of plants that have coevolved – it's called dispersal mutualism – with animals," says Lee Newsome, co-author of the study and an associate professor of anthropology at Penn State. "There's a large number that are coevolved with mammals. Some just hitch a ride on mammal fur then ultimately fall off somewhere."
The fruit of other plants, such as the wild gourds giant sloths and woolly rhinos chowed upon, are eaten and their seeds expelled, maybe miles from where the original plant grew.
Imagine automobile-sized mastodons running rampant across the environment of what is now North Florida and into Georgia, Newsome says, eating wild gourds, then expelling the seeds still lodged in the dung that she and her team found and studied. As the environment warmed following the most recent ice age, and the large mammals became extinct, "the plants were left without their primary partner [and] disperser," she says.
No more megafauna? Enter a new partner: us.
"By then humans were present and were starting to make use of [wild gourds and squash] for containers," Newsome says, though our ancestors weren't eating them – yet. "And ultimately humans are planting [the seeds] and changing them."
As the plants evolved and adapted to the new environment, smaller animals found that some of the Cucurbita didn't taste as bitter anymore. Over the dozen millennia that have passed since the end of the Pleistocene Ice Age, wild gourds and squash evolved into the tasty foods we eat today.
Next time you're chowing down on a pumpkin pie, squash casserole or pumpkin spice latte, remember the mastodons and their dung. Without the demise of megafauna, pumpkins would've remained bitter and unpleasant to our palate, and we'd have a much less flavorful diet.<|endoftext|>
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The Power of Non-Violence
The concept of nonviolence has been used to promote change in the world. At first, this way of thinking was seen as unpractical. It was not until a man named Mahatma Gandhi began to speak the idea known as Ahimsa throughout India. More ideas similar to this were incorporated in Gandhi’s philosophy known as “Satyagraha”. Gandhi witnessed his beloved country of India be controlled by the British Empire. The ways that he acted against the violation of human rights were through nonviolent protest and fasting. These actions lead to Gandhi’s imprisonment from 1922 to 1924. Because of his revolutionary ideologies, India was liberated from British rule. On January 30, 1948 Gandhi was assassinated. Gandhi’s philosophy of ahimsa is useful because it shed insight on the letter written by Dr. Martin Luther King years later. Dr. King was imprisoned as a participant in non-violent demonstrations against segregation. From the Birmingham jail, Dr. king wrote a letter in response to a public statement of concern by white religious leader of the South. In this letter, he urges his fellow black brothers to participate in non-violent actions in order to be successful. There are various similarities between Gandhi and Dr. King. Some examples include the importance of acting without violence, the necessity of breaking laws if they are deemed unjust, and the dedication of their lives to helping people reach equality. Both men were strong leaders and their legacy will live on forever. It is so important to realize that the ideas that both Gandhi and Dr. King echo one another through their philosophies. The following of laws by the people is important for any government to function properly. Law is defined as the system of rules that a particular country recognized as regulating the actions of its members and may enforce by the imposition of penalty. Throughout the history of the world laws have either been beneficial or devastating to it....
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# The first 3 terms of the expansion of $\left(1+\frac{x}{2}\right)\left(2-3x\right)^6$
According to the ascending powers of $$x$$, find the first $$3$$ terms if the expansion of $$\left(1+\frac{x}{2}\right)\left(2-3x\right)^6$$ For the expansion of $$(2-3x)^6$$ The first 3 terms are $$64 -576x + 2160x^2$$ Now, are the required 3 terms are $$64-576x + 2160x^2$$ Or $$32x -288 x^2 +1080 x^3$$ Or otherwise ?
You got the first $$3$$ terms of $$(2-3x)^6$$ correct: $$64-576x+2160x^2$$.
$$\left(1+\dfrac x2\right)(64-576x+2160x^2...)=(64-576x+2160x^2...)+(32x-288x^2...)$$
$$=64-544x+1872x^2...$$
Add the two together, and keep the terms with $$x^0$$, $$x^1$$, and $$x^2$$. So the answer is $$64+(32-576)x+(2160-288)x^2$$
The Binomial Theorem gives: $$(a+b)^n= \sum_{i=0}^n \binom{n}{i} a^i b^{n-i}$$ Using this you can find each power you need from the $$(2-3x)^6$$ term. For instance, the $$x$$ term from this is $$\binom{6}{1} (-3x)^1 2^5= 6 \cdot 2^5 \cdot (-3) \cdot x= -576 x$$. You can then multiply this times the constant term from $$(1+x/2)$$. Then you just need to add that to the $$x/2$$ times the constant term of $$(2-3x)^6$$. Proceeding this way gives you all the terms. Once you have the idea, the rest is work, so you can also shortcut/check (more the latter) using this link here will give you the complete expansion near the bottom of the loaded page. $$(1+x/2)(2-3x)^6= 64 - 544 x + 1872 x^2 - 3240 x^3 + 2700 x^4 - 486 x^5 - 729 x^6 + (729 x^7)/2$$<|endoftext|>
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Another article under our guest contributor program; this time covering the basics of nuclear chemistry. Those who have missed the last article covering basics of chemistry, you may read it here. Now let’s us focus on Nuclear Chemistry which is an area given stress in most of the UPSC Preliminary question papers. Let’s start from a few basic concepts first.
Atomic number (Z)
Atomic number is the number of protons or electrons present in an atom (for every atom, the number of proton and electron are same).
Eg: Nitrogen (N) = 7, Calcium (Ca) = 20, Oxygen (O) = 8.
Mass number (A)
Mass number is the sum of protons and neutrons present in an atom (or it is the sum of electron and neutron present in an atom.)
Eg: Nitrogen (N) = 14, Calcium (Ca) = 40, Oxygen (O) = 16
Elements having same atomic number but different mass number are called isotopes.
Eg: Protium, Deuterium, Tritium.
Elements having same mass number but different atomic numbers are called isobars.
Eg: 40S, 40Cl, 40Ar, 40K, and 40Ca.
Different forms of a single element are called allotropes.
Eg: Diamond and graphite are two allotropes of carbon; ie. pure forms of the same element that differ in crystalline structure.
Unstable atomic nuclei will spontaneously decompose to form nuclei with a higher stability. The decomposition process is called radioactivity. Energy and particles released during the decomposition process are called radiation.
There are three major types of natural radioactivity : alpha, beta and gamma radiation.
23892U → 42He + 23490Th.
The helium nucleus is the alpha particle.
23490 → 0-1e + 23491Pa.
The electron is the Beta particle.
Gamma rays are high-energy photons with a very short wavelength. Gamma emission changes neither the atomic number nor the atomic mass.
Nuclear reactions are mainly two types :
- Nuclear fission.
- Nuclear fusion.
Nuclear fission takes place when an atom’s nucleus splits into two or more smaller nuclei. These smaller nuclei are called fission products. Particles (e.g., neutrons, photons, alpha particles) may also be released along fission.
23592U + 10n → 9038Sr + 14354Xe + 310n.
Nuclear fusion is a process in which atomic nuclei are fused together to form heavier nuclei. Large amounts of energy are released when fusion occurs.The reactions which take place inside the sun is an example of nuclear fusion.
11H + 21H → 32He.
32He + 32He → 42He + 211H.
11H + 11H → 21H + 0+1β.
– Article contributed by Deepesh S Rajan.<|endoftext|>
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# Confidence Interval Calculator
### Confidence Interval Calculator: An Insightful Guide to Understanding and Application
#### Abstract:
Understanding statistical inferences is pivotal in various disciplines, from health sciences to social research. One of the main components of statistical inference is the confidence interval (CI). The Confidence Interval Calculator aids in determining the range in which a population parameter is likely to fall. This comprehensive guide, provides a deep dive into the concept, workings, and significance of the Confidence Interval Calculator.
#### 1. Introduction:
Confidence intervals offer a range of values which is likely to contain an unknown population parameter. This mathematical concept is of paramount importance when attempting to make inferences about a population from a sample. The Confidence Interval Calculator emerges as an essential tool in determining these intervals without tedious manual computation.
#### 2. Basics of Confidence Intervals:
Before diving into the calculator itself, it’s crucial to understand the underlying concept:
• Point Estimate: The single best estimate of the population parameter.
• Margin of Error: The amount the sample mean will be expected to vary from the population mean.
By combining these two aspects, we obtain a confidence interval.
#### 3. Role of Confidence Level:
The confidence level (commonly set at 95%) is a measure of how certain we are about our computed interval. It represents the probability that the interval will capture the population parameter in repeated sampling.
#### 4. The Mathematics Behind the CI Calculator:
When using the calculator, it employs a specific formula to derive the confidence interval. For a population mean, using a normal distribution, it's calculated as:
�ˉ±�×(��)
Here:
• �ˉ = Sample mean
• = Z-value (from Z-table based on confidence level)
• = Population standard deviation
• = Sample size
#### 5. The Confidence Interval Calculator: A Closer Look:
Using the calculator generally requires the input of:
• Sample mean
• Standard deviation (or standard error)
• Sample size
• Desired confidence level
The calculator then computes the confidence interval based on these inputs.
#### 6. The Utility of the CI Calculator:
The significance of this calculator lies in:
• Time-efficiency: Quick computations without manual efforts.
• Accuracy: Eliminating human errors that might occur during manual calculations.
• Convenience: Easily re-calculate with different inputs or confidence levels.
#### 7. Interpreting Results:
Understanding the output is as crucial as the computation itself. A 95% confidence interval suggests that if we took multiple samples and computed an interval estimate for each of them, we expect about 95% of the intervals to contain the true parameter.
#### 8. Real-world Applications of Confidence Intervals:
From clinical trials in pharmaceuticals to market research in business, confidence intervals offer a nuanced understanding beyond just point estimates. They provide a range, acknowledging the potential variation and uncertainty in estimates.
#### 9. Limitations of the Confidence Interval Calculator:
While the CI calculator is undoubtedly a boon, it's also important to acknowledge its limitations:
• Assumes that the sample provides a good representation of the population.
• The chosen confidence level can significantly affect the width of the confidence interval.
• The accuracy is only as good as the data input into the calculator.
#### 10. Common Mistakes and Misunderstandings:
• Misinterpreting CI: A 95% CI doesn't mean there's a 95% probability the population parameter falls within the interval.
• Assuming Symmetry: Not all confidence intervals are symmetrical about the sample statistic.
#### 11. The Future of Confidence Intervals and Advanced Tools:
With the advancements in statistical tools, we are moving towards more user-friendly interfaces, integration with other analytical tools, and even predictive modeling that might use confidence intervals.
#### 12. Conclusion:
The Confidence Interval Calculator stands as an exemplar of the seamless blend of statistical theory with practical application. As we continue to venture deeper into an era driven by data, tools like these not only simplify computations but also aid in informed decision-making, enhancing the overall research quality.
To wrap up, understanding the Confidence Interval Calculator is more than just about using a tool. It’s about comprehending the statistical nuances, making informed judgments, and responsibly applying findings in diverse fields.<|endoftext|>
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Using a Function Box Backwards
# USING A FUNCTION BOX ‘BACKWARDS’
• PRACTICE (online exercises and printable worksheets)
Recall that a function can be viewed as a ‘box’:
you drop an input in the top, the function does something to the input, and a unique output drops out the bottom.
In this lesson, we're going to try and use a function box ‘backwards’.
That is, we'll pick up a number from the output pile, put it in the box ‘backwards’,
and try to see what input it came from.
Sometimes this works out nicely, and sometimes it doesn't!
## the Squaring Function: Using the Function Box ‘Backwards’ Fails
Consider the squaring function, $\,f(x) = x^2\,$. Note: $\,f(2) = 2^2 = 4\,$ $\,f(-2) = (-2)^2 = 4\,$ In other words: drop a $\,2\,$ in the top, and $\,4\,$ comes out the bottom; drop a $\,-2\,$ in the top, and again $\,4\,$ comes out the bottom. Now, pick up the number $\,4\,$ from the output pile. What input did it come from? Did it come from the input $\,2\,$? Did it come from the input $\,-2\,$? There's no way to know! The output $\,4\,$ does not have a unique corresponding input. Indeed, in this case, the output $\,4\,$ has two corresponding inputs, as the graph of the squaring function clearly shows. For the squaring function, trying to use the function box ‘backwards’ fails.
## the Cubing Function: Using the Function Box ‘Backwards’ Succeeds
Consider the cubing function, $\,f(x) = x^3\,$. Note: $\,f(2) = 2^3 = 8\,$ There is no other input whose output is $\,8\,$. Pick up the number $\,8\,$ from the output pile. What input did it come from? It came from the input $\,2\,$. The output $\,8\,$ has a unique corresponding input. Since every real number has a unique cube root, every output from the cubing function has a unique corresponding input. For the cubing function, using the function box ‘backwards’ succeeds.
## So—When Can a Function Box Be Used ‘Backwards’?
A function box can only be used ‘backwards’ when every output has exactly one corresponding input!
## the Horizontal Line Test: a Graphical Test to See if Each Output has Exactly One Corresponding Input
Imagine sweeping a horizontal line from top to bottom (or bottom to top) through the graph of a function.
At each location, the horizontal line hits the $\,y\,$-axis (but not necessarily the graph of the function!) in a unique point;
you're ‘testing’ each output to see how many corresponding input(s) it has.
If a horizontal line ever hits the graph at more than one point (as in the squaring function),
then there exists an output with more than one input.
For the squaring function, the horizontal line at height $\,4\,$ hits the graph at two points: $\ x=2\$ and $\ x=-2\$.
In this case, we say that the graph ‘fails the horizontal line test’.
If a horizontal line always intersects the graph at only one point,
then every output has only one input.
In this case, we say that the graph ‘passes the horizontal line test’.
For example, the graph of the cubing function passes the horizontal line test.
## Summary: One-to-one Functions
So—some functions are ‘nicer’ than others, with respect to using the function box ‘backwards’!
Which are the ‘nice’ ones?
The ones with graphs that pass a horizontal line test!
(Note: Every function already passes a vertical line test.)
For these ‘nice’ functions, there is a beautiful relationship between the inputs and outputs:
since it's a function, each input has exactly one output (that is, the graph passes a vertical line test);
since it additionally passes a horizontal line test, each output has exactly one input.
Put together, it is as if the inputs and outputs are connected with strings!
Pick up any input, follow the string to its unique corresponding output.
Pick up any output, follow the string back to its unique corresponding input.
There is a one-to-one correspondence between the inputs and outputs.
An input uniquely determines an output, and an output uniquely determines an input.
In the next lesson, we'll give these ‘nice’ functions a special name: one-to-one functions.
Soon after, we'll see that one-to-one functions have inverses that ‘undo’ what the function does,
and we'll study techiques for finding inverses.
Master the ideas from this section<|endoftext|>
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Balance is an important athletic skill. From Lionel Messi to Todd Gurley, many of the world's greatest athletes demonstrate exceptional balance on a regular basis.
Since it cannot be easily measured like strength or speed, we often have a hard time quantifying balance and figuring out the best ways to improve it. Balance refers to the ability to maintain equilibrium in relation to the force of gravity. When most people think of balance, they think of standing on one leg for a long period of time without falling over, which is a form of static balance. Static balance is the ability to maintain equilibrium while stationary.
The other form of balance is called dynamic balance, which is the ability to maintain equilibrium while moving. A great example of dynamic balance is riding a bicycle. To go a step further, a great example of both static and dynamic balance in action together would be a gymnastics routine on a balance beam.
The reason balance is such an important athletic skill is that it is intertwined with so many other basic movement skills like running, jumping, landing, twisting and rotating, which in turn make up, in some shape or form, many sports skills. One could say that the development of balance is the foundation off which all other movements are developed.
Since balance is seen in so many different athletic skills, then it only makes sense to train it early and often so children develop a good all around balance skill-set. According to youth training expert Józef Drabik, the most sensitive periods to train balance for boys is around 10-11 years of age, and ages 9-10 for girls. Balance reaches maturity around the ages of 12-14. The reality is that most active children are continually developing a sense of balance throughout pre-adolescence into early adolescence.
Before diving into ideas for training balance, it's important to encourage adults in charge of children to make sure children lead active lives which will encourage developing a good sense of balance. Things like playing at the playground, learning to ride a bike or roller skating, learning to swim, learning to ski or ice skate are all great examples of activities that offer a wealth balance development opportunities.
As already mentioned, balance is intertwined with many other basic movement skills. The following list includes some of the different types of balance and some examples of how to train each.
Supporting balance is supporting or maintaining equilibrium using different parts of the body. In its simplest form, standing on one leg is a form of supporting balance. But we can greatly expand that idea to include balance from multiple stances, on the hands, feet, knees, elbows or any combination of them, as well as transitioning to and from those positions. As a youth coach, I especially like supporting balance, because it helps develop all-around strength. In sports, some clear examples of supporting balance are a lineman blocking in football, a player battling in the low post in basketball, and a floor routine in gymnastics. The above video shows different forms of supporting balance, both static and dynamic, in the training environment.
In the competitive athletic arena, falling on the ground is bound to happen. Whether it's being tackled in football or rugby, diving for a loose ball in basketball, executing a slide tackle in soccer, or simply being tripped up or shoved down by an aggressive opponent, this type of balance will be tested.
Rolling/rotational/falling balance depends on knowing where your body is in space and in relation to the ground (i.e., spatial awareness) and second, knowing how to fall to best make a play or protect oneself. Training young athletes, we spend a significant amount of time practicing losing balance only to regain it by exploring different rolling, twisting, rotational activities. The above video shows some of the different ways to explore rolling and falling balance.
Sliding/gliding movements, seen in skiing and ice skating, as well as rapid change of direction/cutting movements seen in field and court sports, rely heavily on weight shifts laterally from side to side. The athlete needs to maintain balance as he or she attempts another turn or stride. Although we don't have ice or snow in the gym, we can easily train elements of gliding/sliding/cutting balance on other surfaces. One that comes to mind is a simple slide board where the athlete can work on developing strength and balance by pushing side to side on a dry slick surface. But the purpose on this article will focus on dryland activities to develop these unique balance skills. One of my favorite activities for gliding/sliding and cutting is the simple Skater Jump or Heiden Jump. A simple lateral jump from one foot to another helps develop this lateral balance. It's an easy exercise to learn for young children and a lot of fun to progress for more advanced athletes. The above video shows some examples of gliding/sliding/cutting activities.
In many sports, the athlete will find themselves airborne at some point. Actions involving jumping, throwing, sliding, diving and catching in sports like football, basketball, baseball, soccer, track and field, volleyball and gymnastics, for example, will present moments where the athletes needs to maintain balance and control in the air. Training air balance is fun for the both the coach and athlete. There are many combinations of movements to explore and one is only limited their creativity and imagination. The above video shows some of the various ways to train air balance.
The best time to train balance activities is to begin in early pre-adolescence into early adolescence, typically between the ages of 6-12. The goal is to continually and consistently expose the young athlete to years of versatile and diverse movement challenges. Balance is much more that just standing on one leg. Balance encompasses just about everything we do as human beings, and the better we develop our all-around sense of balance, the better movers we will be.
If you enjoyed this article, be sure to follow Jeremy on Twitter!
Wormhoudt René. The Athletic Skills Model: Optimizing Talent Development through Movement Education. Routledge, 2018.
Drabik Józef. Children and Sports Training: How Your Future Champions Should Exercise to Be Healthy, Fit, and Happy. Stadion, 1996.
Gabbard, Carl, et al. Physical Education for Children: Building the Foundation. Prentice-Hall, 1994.
Grasso, Brian. "Coordination & Movement Skill Development – The Key to Long-Term Athletic Success." MFA - Perform Better.
- Kids Play Fewer Pick-Up Games, And It's Hurting Youth Sports
- Why Athletes Need 'Movement Variability', And How Coaches Can Deliver It
- Study Finds Just 12 Percent of NHL and NCAA Hockey Players Specialized Before Age 12<|endoftext|>
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National Register of Historic Places in Clark County
Hoover Dam is located in the Black Canyon on the Colorado River twenty-eight miles southeast of Las Vegas. The west wall of Black Canyon is in Clark County, Nevada, and the east wall is in Mohave County, Arizona.
The drainage area "above Hoover Dam comprises 167,800 square miles including parts of the states of Wyoming, Utah, Colorado, Nevada, Arizona, and New Mexico.
The Colorado River above Hoover Dam rises in the Rocky Mountains of Wyoming and Colorado and flows southwestward to Lake Mead for a distance of about nine hundred miles. Principal tributaries of the Colorado River that feed into Lake Mead are the Green, Yampa, White, Uinta, Duchesne, Price, San Rafael, Muddy, Fremont, Escalante, Gunnison, Dolores, San Juan, Little Colorado, and Virgin Rivers.
The construction of Hoover Dam began June 6, 1933 and was completed September 30, 1935, two years ahead of schedule. Hoover Dam is a concrete, arch-gravity storage dam. The water load in this type of dam is carried by both gravity action and horizontal action.
Curbing the Colorado River by means of building Hoover Dam was the greatest task in hydraulic engineering that had been attempted since digging the Panama Canal.
The proposed dam to be built in the Black Canyon had to achieve several purposes:
To achieve the goals stated above, a huge dam would have to be built. Hoover Dam was the highest dam in the world - 726.4 feet from bedrock to the crest. A structure of this height would create a reservoir large enough to store safely the normal flow of the river for two years and create the largest artificial lake in the world measured by volume.
The proposed dam was of such magnitude that there was serious opposition and concern about the engineering expertise needed. Questions were raised concerning the possibility of the collapsing of the dam.
Along with the engineering problems were other factors. The extreme remoteness of the damsite, the ruggedness of the terrain surrounding the site, and the extreme climatic conditions; summer temperatures of 125º in the canyon, cloud bursts, high winds and sudden floods all made the work difficult.
Source: NRHP Nomination Form
National Industrial Recovery Act of 1933
President Franklin Roosevelt came into office during the worst depression the nation had ever known. Fulfilling a campaign promise to put people to work, he instituted the New Deal to bring economic recovery to the depression-wrought country.
The National Industrial Recovery Act (NIRA) of 1933 authorized the Public Works Administration (PWA) to provide jobs, stimulate business activity, and increase purchasing power through the construction of permanent and socially useful public works. The Federal Government and local city, county and state governments formed a working partnership resulting in the greatest single construction program in history.
PWA construction projects, in addition to providing employment for the skilled, generated a volume of jobs for the unskilled. The PWA provided loans and grants up to forty percent of the total cost of the project to states, and many other public bodies, including schools.
From 1933-1935, the PWA underwrote projects in 3,040 of the 3,073 counties in all forty-eight states. Of the 3.76 billion dollars of
the NIRA fund, 2.56 billion dollars was spent on 19,004 construction projects.
Narrative adapted in part from the NRHP nomination for Tulare Union High School Auditorium and Administration Building
dated 16 November 1999.
Narrative adapted in part from the NRHP nomination for Tulare Union High School Auditorium and Administration Building dated 16 November 1999.
Many buildings funded by the PWA have been recognized for their historic significance and architectural excellence. Among them are:
Adobe Chapel of the Immaculate Conception in San Diego
Aquatic Park in San Francisco
Beach Chalet Murals in San Francisco
Big Basin Redwood State Park Headquarters Building
Big Creek Bridge in Big Sur
Feather River Scenic Byway Tunnels
Federal Building in Merced
Federal Writers and Artists Projects in San Francisco
Gasquet Ranger Station
Mariposa County High School Auditorium
McClatchy Senior High School in Sacramento
Monterey County Courthouse in Salinas
Mountain View Adobe
Police Headquarters, Jail & Courts in San Diego
Rincon Annex Post Office in San Francisco
Sacramento Junior College
San Francisco State Teachers College
Sitka Main Post Office and Court House in Alaska
National Park Service Southwest Regional Office in Santa Fe
New Mexico School for the Deaf Building 2 in Santa Fe
New Mexico School for the Deaf Hospital in Santa Fe
New Mexico Supreme Court in Santa Fe
Butte Falls Ranger Station in Butte Falls
Dead Indian Soda Springs Shelter in Rogue River National Forest
Fish Lake Shelter in Rogue River National Forest
Gold Beach Ranger Station
Lake of the Woods Ranger Station in Fremont-Winema National Forest
Lithia Park in Ashland<|endoftext|>
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A Columbia-led team has discovered a new method to manipulate the electrical conductivity of this game-changing material, the strongest known to man with applications ranging from nano-electronic devices to clean energy.
Graphene has been heralded as a wonder material. Not only is it the strongest, thinnest material ever discovered, its exceptional ability to conduct heat and electricity paves the way for innovation in areas ranging from electronics to energy to medicine.
Now, a Columbia University-led team has developed a new method to finely tune adjacent layers of graphene—lacy, honeycomb-like sheets of carbon atoms—to induce superconductivity. Their research provides new insights into the physics underlying this two-dimensional material’s intriguing characteristics.
The team’s paper is published in the Jan. 24 issue of Science.
“Our work demonstrates new ways to induce superconductivity in twisted bilayer graphene, in particular, achieved by applying pressure,” said Cory Dean, assistant professor of physics at Columbia and the study’s principal investigator. “It also provides critical first confirmation of last year’s MIT results—that bilayer graphene can exhibit electronic properties when twisted at an angle—and furthers our understanding of the system, which is extremely important for this new field of research.”
In March 2018 researchers at the Massachusetts Institute of Technology reported a groundbreaking discovery that two graphene layers can conduct electricity without resistance when the twist angle between them is 1.1 degrees, referred to as the “magic angle.”
But hitting that magic angle has proven difficult. “The layers must be twisted to within roughly a tenth of a degree around 1.1, which is experimentally challenging,” Dean said. “We found that very small errors in alignment could give entirely different results.”
So Dean and his colleagues, who include scientists from the National Institute for Materials Science and the University of California, Santa Barbara, set out to test whether magic-angle conditions could be achieved at bigger rotations.
“Rather than trying to precisely control the angle, we asked whether we could instead vary the spacing between the layers,” said Matthew Yankowitz, a postdoctoral research scientist in Columbia’s physics department and first author on the study. “In this way any twist angle could, in principle, be turned into a magic angle.”
Applying pressure transformed the material from a metal into either an insulator—in which electricity cannot flow—or a superconductor—where electrical current can pass without resistance—depending on the number of electrons in the material.
“Remarkably, by applying pressure of over 10,000 atmospheres we observe the emergence of the insulating and superconducting phases,” Dean said. Additionally, the superconductivity develops at the highest temperature observed in graphene so far, just over 3 degrees above absolute zero.”
To reach the high pressures needed to induce superconductivity the team worked closely with the National High Magnetic Field user facility, known as the Maglab, in Tallahassee, Florida.
“This effort was a huge technical challenge,” said Dean. “After fabricating one of most unique devices we’ve ever worked with, we then had to combine cryogenic temperatures, high magnetic fields, and high pressure—all while measuring electrical response. Putting this all together was a daunting task and our ability to make it work is really a tribute to the fantastic expertise at the Maglab.”
The researchers believe it may be possible to enhance the critical temperature of the superconductivity further at even higher pressures. The ultimate goal is to one day develop a superconductor which can perform under room temperature conditions, and although this may prove challenging in graphene, it could serve as a roadmap for achieving this goal in other materials.
Andrea Young, assistant professor of physics at UC Santa Barbara, a collaborator on the study, said the work clearly demonstrates that squeezing the layers has same effect as twisting them and offers an alternative paradigm for manipulating the electronic properties in graphene.
“Our findings significantly relax the constraints that make it challenging to study the system and gives us new knobs to control it,” Young said.
Dean and Young are now twisting and squeezing a variety of atomically-thin materials in the hopes of finding superconductivity emerging in other two-dimensional systems.
“Understanding ‘why’ any of this is happening is a formidable challenge but critical for eventually harnessing the power of this material—and our work starts unraveling the mystery,'” Dean said.
Explore further: Twisted electronics open the door to tunable 2-D materials
More information: “Tuning superconductivity in twisted bilayer graphene” Science (2019). science.sciencemag.org/lookup/ … 1126/science.aav1910<|endoftext|>
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# Video: Calculating the Amount of Work Done by a Force given the Force Expression
A body moves along the π₯-axis under the action of a force, πΉ. Given that πΉ = (8π + 12) N, where π m is the displacement from the origin, determine the work done on the body by πΉ when the body moves from π = 7 m to π = 8 m.
02:29
### Video Transcript
A body moves along the π₯-axis under the action of a force, πΉ. Given that πΉ is equal to eight π plus 12 newtons, where π metres is the displacement from the origin, determine the work done on the body by πΉ when the body moves from π equal seven metres to π equals eight metres.
We know that when applying a constant force, work done is equal to the force multiplied by the displacement. The work done will be measured in joules, the force will be measured in newtons, and the displacement in metres. In this question, however, the force is not constant. It is a function in terms of π , the displacement. We will, therefore, calculate the work done using integration. The work done is equal to the definite integral of π of π between the two limits, π and π. Our function πΉ of π is equal to eight π plus 12. We need to integrate this with respect to π . We need to calculate the work done between π equals seven metres and π equals eight metres.
Therefore, our limits are seven and eight. Integrating eight π gives us eight π squared over two. We increase the power by one and divide by the new power. This can be simplified to four π squared. Integrating the constant 12 with respect to π gives us 12π . We need to evaluate this between the limits, eight and seven. We, firstly, substitute eight into the expression. This gives us four multiplied by eight squared plus 12 multiplied by eight. This is equal to 352. Our next step is to substitute in π equals seven. This gives us four multiplied by seven squared plus 12 multiplied by seven. This is equal to 280. 352 minus 280 is equal to 72. The work done under the action of the force πΉ is 72 joules.<|endoftext|>
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Home > Grade 5 > Sum of Fractions Closest to 10
Sum of Fractions Closest to 10
Directions: Using the digits 1 through 9, at most one time each, fill in the boxes to make the statement true.
Hint
How do we add fractions? If we want to get the closest to 10, what does that mean for each of the fractions?
The closest to 10 so far is 9.8. There are few ways to get this, for example: 34/5 + 6/2 = 9.8
Source: Nanette Johnson, based on Giselle Garcia’s problem
Balanced Equations 2
Directions: Use the operation symbols (+, -, x, ÷) and equal sign (=) to make …
1. I had a student come up with 9.8 as a solution:
34/5 + 6/2 = 9.8
• How??? Where should a kid start? My child needs to show the steps to solve this problem. Where do we start?
• Since the answer contains a whole number and partial number (__.__), you can plug digits into the improper fraction that would get you a whole number 9, which is as close to 10 as you can get, and then try to plug in other digits that would give you the greatest value of partial number. You have to play with it more than follow steps because of the instruction that digits can only be used once.
• Thank you for finding a 9.8 answer. The solution has been updated.
2. 63/7 + 4/5=9.8
27/3+4/5=9.8
3. My student did 56/7 + 2/1 which equals 10. Does that work?
• The rule of the problems is that digits cannot be repeated. In your answer, the digit ‘1’ is used twice. Also, the answer must be a 2 digit number that has a digit in the ones place and the tenths place. 10 has a digit in the tens place and a digit in the ones place. So no, that doesn’t work.
4. I am a student in eighth grade and a solution I got was
76/8+5/9=9.944444444444444444444444444444444<|endoftext|>
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# Validating a solution to a combinatorics problem
I just started studying basic combinatorics (as part of discrete mathematics for CS) and was given the following problem in a homework assignment. I believe I've solved it but I want to check my solutions:
$$\sim$$
Problem: Given a group of 10 adults, 40 girls, and 20 boys:
a) How many ways are there to arrange them in a line with the adults first, then the girls, then the boys?
b) How many ways are there to arrange them in a line with exactly six children between every two adults? (There may be children at either end of the line)
c) How many ways are there to arrange them in a line with exactly 4 girls and 2 boys between every two adults? (There may be children at either end of the line)
$$\sim$$
My solutions are as follows:
a) We have 10 spaces for the adults, then 40 spaces for the girls, then 20 spaces for the boys. So the answer is $10!\cdot40!\cdot20!$
b) For the "center" portion of the line (enclosed between the first and last adult): we have $10!$ ways to arrange the adults with 9 spaces in between then. The combined number of children is 60 and we want to get all the 54-tuples of them, so that's $\dfrac{60!}{6!}$, multiplied by all the ways to order 9 6-tuples of them: $9!$
For the ends we have 6 children left, so we want all the $6!$ possible ways to order them multiplied by the 7 ways the center portion of the line can be pushed between them.
$$10! \cdot \dfrac{60!}{6!} \cdot 9! \cdot 6! \cdot 7 = 70 \cdot (9!)^2 \cdot 60!$$
c) In a similar fashion to the previous question, we'll address the center of the line first: $10!$ possible ways to arrange the adults. For each of these we'll take one of the $\dfrac{40!}{4!}$ 36-tuples of the girls and one of the $9!$ ways to arrange 9 4-tuples of them. For each of these we'll take one of the $\dfrac{20!}{2!}$ 18-tuples of the boys and one of the $9!$ ways to arrange 9 2-tuples of them. For each of these we'll pick 2 of the 5 spaces in the 4-tuple of girls to which they were assigned to "push" the two boys between the girls in the 4-tuple - there are $\dfrac{5!}{3!}$ ways to do this.
For the ends we have 6 children, so we want all the $6!$ ways to arrange them multiplied by all the 7 ways the center portion of the line may be pushed between them.
$$10! \cdot \dfrac{40!}{4!} \cdot 9! \cdot \dfrac{20!}{2!} \cdot 9! \cdot \dfrac{5!}{3!} \cdot 6! \cdot 7$$
$$=70 \cdot (9!)^3 \cdot 6! \cdot \dfrac{40!\cdot20!\cdot5!}{4!\cdot3!\cdot2!}$$
$$\sim$$
This feels correct to me but since the numbers involved are so large, I don't really have any way to validate my solution. Even if I were to cut it down to 2 adults, 8 girls and 4 boys, the number of possibilities is much, much too great to check manually.
So I have two questions:
1) Are my answers correct and if not, why?
2) Generally speaking, is there a "simpler" method to approaching problems like this? Is/are there any way/s to validate the correctness of a solution without manually checking all the cases?
For the second part, there are $7$ possible patterns (in terms of which spaces have adults and which have children) for the line: there are exactly $54$ children between the first and last adult, so there are $6$ left over to go at the ends, any number form $0$ to $6$ of which can be at the left.
Once you've chosen a pattern, the adults can go in the $10$ adult spaces in any order, and the children can go in the $60$ child spaces in any order. So the answer is $7\times 60!\times 10!$.<|endoftext|>
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## Profit & Loss
The chapter of profit and loss is a part & parcel of every commercial activity.
This chapter has a wide-ranging importance in exams. The concepts of application percentage rule, fraction to percentage change, are applicable to profit &loss.
Cost Price : The price at which an article is brought is called its cost price, abbreviated as C.P.
Selling Price : The price at which an article is sold is called its selling price, abbreviated as S.P.
Profit or Gain = (S.P.) - (C.P.)
Loss = (C.P.) - (S.P.)
Loss or Gain is always reckoned on C.P.
Gain % = Gain x
Loss % = Loss x
If Profit is P% If Loss is L%
SP = x CP SP = x CP
CP = x SP CP = x SP
Value of Profit = x CPValue of Loss = x CP
Example 1: CP is 40% of SP. What is the % gain on CP?
Solution: Let the SP = Rs. 100
CP = Rs. 40
Gain = (60/40) x 100% = 150%
Example 2:The loss when SP is Rs 27.8 is equal to the gain when SP is Rs 132.2. Find CP.
Solution: CP = = Rs. 80
Example 3:By selling 24 lemons the loss is equal to selling price of 8 lemons. Find the % Loss.
Solution:Let SP of 1 lemon = Rs. 1
Loss = x 100% = 25%
Example 4:By selling 24 lemons the gain is equal to the selling price of 8 lemons.
Find the gain %
Solution:Let SP of 1 lemon = Re. 1
Loss = x 100% = 50%
Example 5:A pretends to sell the goods at cost price but uses a weight of 960 g for 1 kg. Find his gain%.
Solution:Gain % = x 100 = 4.17 %
(Note: we use 960 because his cost is on Rs 960 and not on Rs 1000)
Example 6:A pretends to gain 20% but also uses a weight of 800 g for 1 kg. His actual gain is.
Solution:Let CP of 1 kg = Rs. 100
therefore, SP of 800 g = Rs. 120 (Sells it at 20% gain)
Gain = x 100%Therefore, Gain is 50%
Example 7:Selling price of two articles is Rs. 910 each. Gain on one is 30%. Loss on other is 30%. Find the final position.
Solution:CP of both = Rs. 910 x + Rs. 910 x = Rs. 2000
Therefore Loss = Rs. 2000 - 2. (Rs. 910) = Rs. 180
Example 8:Selling price of two articles is same. Gain on one is 20%. Loss on other 20%. On the whole the loss is ? %.
Solution:Loss = 20 x % = 4%, (If SP is same.)
Example 9:If SP is Rs. 48, loss is 37%. Find the SP to gain 26%.
Solution:New S. P = 126 % of CP
x = Rs. 96
Example 10:CP of 11 books = Rs. 5 SP of 10 books = Rs. 6. Find the gain percent.
Solution:Let the number of books = 110 (the LCM of 11 & 10)
CP = Rs. 50, SP = Rs. 66
Gain = x 100 = 32%
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Students learning in green buildings have higher levels of environmental knowledge and behavior
COLUMBIA, Mo. – The energy used to create electricity to power lights, heating and air conditioning, and appliances within buildings causes nearly 50 percent of all fossil fuel emissions in the United States. Educating children about the importance of having environmentally friendly, or “green,” buildings could be a key factor in whether they grow to up to own and operate buildings that are green. Now, a researcher at the University of Missouri has found that students who attend school in buildings specifically designed to be “green” exhibit higher levels of knowledge about energy efficiency and environmentally friendly building practices.
For her study, Laura Cole, an assistant professor of architectural studies in the MU College of Human Environmental Sciences, examined five middle schools from across the country. The schools were housed in buildings ranging from older, energy inefficient designs to new buildings architecturally designed as “teaching green” buildings.
“These ‘teaching green’ buildings are specifically designed as a kind of museum for environmentally friendly building designs,” Cole said. “The idea is that by being exposed to this innovative design every day at school, along with a sustainable school culture fostered by eductors, students will inherently learn and appreciate the importance of green buildings. This study found this idea to be true in that the students from the ‘teaching green’ schools had much higher levels of knowledge about environmentally friendly practices than students who attended school in more inefficient buildings. These students also had much higher levels of environmentally friendly behaviors while at school, such as recycling and turning off lights.”
“Teaching green” schools include a variety of design features to immerse students in an environmentally friendly atmosphere. These features can include open-air hallways, which greatly reduce heating and cooling costs; exposed beams and girders where students can see the materials required to erect such large structures; dedicated waste and recycling spaces that are easily accessible; and the use of recycled and repurposed construction materials. Cole says even if schools cannot afford to build expensive new “teaching green” buildings, other options are available to help teach their students by creating smaller interventions in the building or school yard.
“The study also showed that even a school with a relatively inefficient building design had students with a high level of green building literacy because the school had a very nice outdoor landscaped teaching space, including an outdoor classroom and a learning garden,” Cole said. “Anything educators can do to utilize existing space can help their students’ green building literacy. We all use buildings every day. Our children will soon be the people buying and constructing homes, offices and other buildings. Learning and translating that knowledge into future green building design will play a huge part in solving our environmental problems.”
This study was published in Children, Youth and Environments.<|endoftext|>
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Areas & Perimeters Part 2
Welcome to the second part of our Topic:-
# Area of rectangle:
Consider a rectangle of length 5 cm and width 3 cm.
Using the method of counting squares, we find that the area of the rectangle is 15 cm2.
Clearly, the rectangle contains 3 rows of 5 squares. Therefore:
Area = 5cm x 3cm
Area = 15cm2
This suggests that:
The area of a rectangle is equal to its length multiplied by its width. That is:
Area = Length x Width
Using the pro numerals A for area, l for length and w for width, we can write it simply as:
A = l x w
This is the formula for the area of a rectangle.
#### Example 1
Find the area of a rectangular field 20 m long and 10 m wide.
##### Solution:
A = lw
A=20m x 10m
Area of the field = 200m2
###### Note:
To find the area of a region enclosed within a plane figure, draw a diagram and write an appropriate formula. Then substitute the given values and use a calculator, if necessary, to obtain the required area.
SQUARE:
A rectangle is a quadrilateral with four right angles and four equal sides, this defer from rectangle because only two opposite sides are equal.
# Perimeter of a square
Finding the perimeter of a square when a side is given and finding the length of a side when the perimeter is given are the goals of this lesson.
Given a square with side s, the perimeter (P) or distance around the outside of the square can be found by doing
P = s + s + s + s = 4 × s
Example 1:
Find the perimeter of a square when s = 3 cm
Solution:
P = 4 × s = 4 × 3 = 12 cm
Notice that it is perfectly ok to do P = 3 + 3 + 3 + 3 = 12
However, it is usually easier and quicker to do 4 times 3 than adding 3 four times
Example 2:
Find P when s = 5 cm
Solution:
P = 4 × s = 4 × 5 = 20 cm
Example 3:
Find perimeter of square with side = 2/8 cm.
Solution:
P = 4 × s = 4 × 1/8 = (4/1) × (2/8) = (4 × 2)/ ( 1 × 8 )= 8/8 = 1 cm
Example 4:
A square has a perimeter of 12 inches. Find its side
Solution:
Here, given the perimeter, you are asked to find the length of a side of the square.
We know that P = 4 × s
You should replace P by 12 because that is what they gave you.
So, 12 = 4 × s
The problem becomes a multiplication equation that you need to solve
However, you can solve this equation with mental math. Replace s by a question mark(?) and ask yourself the following:
4 times ? = 12 or 4 times what will give me 12? The answer is 3, so s = 3
The perimeter of a square is 64 cm. What is the length of one side?
Again, since P = 4 × s, we get 64 = 4 × s after replacing P by 64
Ask yourself 4 times what will give me 64? Since 4 times 16 is 64, s = 16
You don't have to guess 4 times what will give 64. You can also divide 64 by 4 to get 16
In fact, whenever you are looking for s and P is a big number, you should always divide p by 4 to get s
Notice that to get the perimeter of a rhombus, you can do the exact same thing we did above for the square.
Since just like the square, all sides of a rhombus are equal, there is absolutely no difference as far as getting the perimeter!
Therefore, P is also equal to s + s + s + s = 4 × s
Continue to the next page for the area of Square:-<|endoftext|>
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# How to Determine the Number of Divisors of an Integer
Co-authored by wikiHow Staff
Updated: March 29, 2019
A divisor, or factor, is a number that divides evenly into a larger integer.[1] It is easy to determine how many divisors a small integer (such as 6) has by simply listing out all the different ways you can multiply two numbers together to get to that integer. When working with larger integers, finding the number of divisors is more difficult. However, once you have factored the integer into prime factors, you can use a simple formula to reach your answer.
### Part 1 of 2: Factoring the Integer
1. 1
Write the integer at the top of the page. You need to leave enough room so that you can set up a factor tree below it. You can use other methods to factor a number. Read Factor a Number for more instructions.
• For example, if you want to know many divisors, or factors, the number 24 has, write ${\displaystyle 24}$ at the top of the page.
2. 2
Find two numbers you can multiply together to get the number, not including 1. These are two divisors, or factors, of the number. Draw a split branch coming down from the original number, and write the two factors below it.
• For example, 12 and 2 are factors of 24, so draw a split branch coming down from ${\displaystyle 24}$, and write the numbers ${\displaystyle 12}$ and ${\displaystyle 2}$ below it.
3. 3
Look for prime factors. A prime factor is a number that is only evenly divisible by 1 and itself.[2] For example, 7 is a prime number, because the only numbers that evenly divide into 7 are 1 and 7. Circle any prime factors so that you can keep track of them.
• For example, 2 is a prime number, so you would circle the ${\displaystyle 2}$ on your factor tree.
4. 4
Continue to factor non-prime numbers. Keep drawing branches down from the non-prime factors until all of your factors are prime. Circle the prime numbers to keep track of them.
• For example, 12 can be factored into ${\displaystyle 6}$ and ${\displaystyle 2}$. Since ${\displaystyle 2}$ is a prime number, you would circle it. Next, ${\displaystyle 6}$ can be factored into ${\displaystyle 3}$ and ${\displaystyle 2}$. Since ${\displaystyle 3}$ and ${\displaystyle 2}$ are prime numbers, you would circle them.
5. 5
Write an exponential expression for each prime factor. To do this, look for multiples of each prime factor in your factor tree. The number of times the factor appears equals the exponent of the factor in your exponential expression.[3]
• For example, the prime factor ${\displaystyle 2}$ appears three times in your factor tree, so the exponential expression is ${\displaystyle 2^{3}}$. The prime factor ${\displaystyle 3}$ appears 1 time in your factor tree, so the exponential expression is ${\displaystyle 3^{1}}$.
6. 6
Write the equation for the prime factorization of the number. The original number you are working with is equal to the product of the exponential expressions.
• For example ${\displaystyle 24=2^{3}\times 3^{1}}$.
### Part 2 of 2: Determining the Number of Factors
1. 1
Set up the equation for determining the number of divisors, or factors, in a number. The equation is ${\displaystyle d(n)=(a+1)(b+1)(c+1)}$, where ${\displaystyle d(n)}$ is equal to the number of divisors in the number ${\displaystyle n}$, and ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are the exponents in the prime factorization equation for the number.[4]
• You might have less than three or more than three exponents. The formula simply states to multiply together whatever number of exponents you are working with.
2. 2
Plug in the value of each exponent into the formula. Be careful to use the exponents, not the prime factors.
• For example, since ${\displaystyle 24=2^{3}\times 3^{1}}$, you would plug in the exponents ${\displaystyle 3}$ and ${\displaystyle 1}$ into the equation. Thus the equation will look like this: ${\displaystyle d(24)=(3+1)(1+1)}$.
3. 3
Add the values in parentheses. You are simply adding 1 to each exponent.
• For example:
${\displaystyle d(24)=(3+1)(1+1)}$
${\displaystyle d(24)=(4)(2)}$
4. 4
Multiply the values in parentheses. The product will equal the number of divisors, or factors, in the number ${\displaystyle n}$.
• For example:
${\displaystyle d(24)=(4)(2)}$
${\displaystyle d(24)=8}$
So, the number of divisors, or factors, in the number 24 is 8.
## Community Q&A
Search
• Question
Is 8 the number of divisors excluding the numbers 24 and 1? Would 10 be a more apt answer?
No. The 8 divisors include the factors 24 and 1. To see this, you can list out all the ways to multiply two numbers to get to 24, and count all the unique factors. 1 x 24 2 x 12 3 x 8 4 x 6 So, as shown above, there are 8 different divisors of 24, including 1 and 24.
• Question
How do you find the odd divisors of an integer?
One way to do this would be to make a factor tree, and then look for all of the odd divisors.
• Question
What is the sum of the divisors of 600?
Donagan
The sum of the divisors is 19. The number of divisors is 6.
• Question
How can I find more examples so it's easier for me to figure out how to do my homework?
If you navigate to the sources cited in the article, you can find more examples.
• Question
What is the smallest positive integer with 6 divisors?
Donagan
If you mean six different divisors (and assuming all divisors must be positive integers), the smallest possible integer would be 1 x 2 x 3 x 4 x 5 x 6 = 720.
• Question
Is there an equation to find the number of divisors for an integer number?
Donagan
Yes. It's given and explained in Step 2 above.
200 characters left
## Tips
• When the number is a perfect square (such as 36), the number of divisors will be odd. When it's not a square, the number of divisors will be even.
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## Video.By using this service, some information may be shared with YouTube.
Co-Authored By:
wikiHow Staff Editor
This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article meets our high standards.
Co-authors: 9
Updated: March 29, 2019
Views: 151,463
Article SummaryX
If you need to determine the number of divisors of an integer, factor that integer and write the equation for the prime factorization of the number. Plug in the value of each exponent into the formula for determining the number of divisors, or factors, in a number. Once you’ve put the values into the formula, add the values in parentheses, then multiply all of the values in the parentheses. The product will equal the number of divisors in the integer. To learn the formula for determining the number of divisors, keep reading!
Thanks to all authors for creating a page that has been read 151,463 times.
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Sabbir Hossain
Apr 24, 2017
"It helps me a lot by giving me a short technique. "
• SS
Sandeepa Singh
Jan 15, 2017<|endoftext|>
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MIT engineers have done it again – this time creating an autonomous glider drone that can survey hundreds of miles of ocean surface all while being powered naturally by the wind.
Created to mimic the albatross, the drone can dip a keel into the water in calm winds to travel like a sailboat – scampering across the sea 10 times faster than an average sailboat. When winds pick up, the winged UAV skims the surface of the ocean – all while using one-third as much wind as its avian counterpart.
Weighing in at a slim six pounds with an almost 10-foot wingspan, the as-yet unnamed drone design can range in speeds 5.75 to 23 mph.
The oceans remain vastly undermonitored,” says Gabriel Bousquet, a former postdoc in MIT’s Department of Aeronautics and Astronautics, who led the design of the robot as part of his graduate thesis. “In particular, it’s very important to understand the Southern Ocean and how it is interacting with climate change. But it’s very hard to get there. We can now use the energy from the environment in an efficient way to do this long-distance travel, with a system that remains small-scale.”
The journey from fine-feathered observations to a working drone design began as the team created an academic paper about albatross flight a few years ago.
Researchers discovered that an albatross can execute a maneuver called “a transfer of momentum, in which it takes momentum from higher, faster layers of air, and by diving down transfers that momentum to lower, slower layers, propelling itself without having to continuously flap its wings.”
Bousquet also noted that the bird’s flight propulsion resembles sailboat travel. “Both the albatross and the sailboat transfer momentum in order to keep moving. But in the case of the sailboat, that transfer occurs not between layers of air, but between the air and water.”
The team will soon present their findings at the IEEE’s International Conference on Robotics and Automation, in Brisbane, Australia.
MIT continues to produce game-changing drone research. Recently, researchers developed a VR training platform that “fools” drones into “seeing” and flying around virtual obstacles despite operating in an open space. The platform may prove useful in the growing trend of FPV drone flight.<|endoftext|>
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A 3-D view of remote galaxies
For decades, distant galaxies that emitted their light six billion years ago were no more than small specks of light on the sky. With the launch of the Hubble Space Telescope in the early 1990s, astronomers were able to scrutinise the structure of distant galaxies in some detail for the first time. Under the superb skies of Paranal, the VLT's FLAMES/GIRAFFE spectrograph — which obtains simultaneous spectra from small areas of extended objects — can now also resolve the motions of the gas in these distant galaxies.
"This unique combination of Hubble and the VLT allows us to model distant galaxies almost as nicely as we can close ones," says François Hammer, who led the team. "In effect, FLAMES/GIRAFFE now allows us to measure the velocity of the gas at various locations in these objects. This means that we can see how the gas is moving, which provides us with a three-dimensional view of galaxies halfway across the Universe."
The team has undertaken the Herculean task of reconstituting the history of about one hundred remote galaxies that have been observed with both Hubble and GIRAFFE on the VLT. The first results are coming in and have already provided useful insights for three galaxies.
In one galaxy, GIRAFFE revealed a region full of ionised gas, that is, hot gas composed of atoms that have been stripped of one or several electrons. This is normally due to the presence of very hot, young stars. However, even after staring at the region for more than 11 days, Hubble did not detect any stars! "Clearly this unusual galaxy has some hidden secrets," says Mathieu Puech, lead author of one of the papers reporting this study. Comparisons with computer simulations suggest that the explanation lies in the collision of two very gas-rich spiral galaxies. The heat produced by the collision would ionise the gas, making it too hot for stars to form.
Another galaxy that the astronomers studied showed the opposite effect. There they discovered a bluish central region enshrouded in a reddish disc, almost completely hidden by dust. "The models indicate that gas and stars could be spiralling inwards rapidly," says Hammer. This might be the first example of a disc rebuilt after a major merger.
Finally, in a third galaxy, the astronomers identified a very unusual, extremely blue, elongated structure — a bar — composed of young, massive stars, rarely observed in nearby galaxies. Comparisons with computer simulations showed the astronomers that the properties of this object are well reproduced by a collision between two galaxies of unequal mass.
"The unique combination of Hubble and FLAMES/GIRAFFE at the VLT makes it possible to model distant galaxies in great detail, and reach a consensus on the crucial role of galaxy collisions for the formation of stars in a remote past," says Puech. "It is because we can now see how the gas is moving that we can trace back the mass and the orbits of the ancestral galaxies relatively accurately. Hubble and the VLT are real 'time machines' for probing the Universe's history", adds Sébastien Peirani, lead author of another paper reporting on this study.
The astronomers are now extending their analysis to the whole sample of galaxies observed. "The next step will then be to compare this with closer galaxies, and so, piece together a picture of the evolution of galaxies over the past six to eight billion years, that is, over half the age of the Universe," concludes Hammer.
The results reported here are either in print or to be printed in Astronomy and Astrophysics:
- Puech et al. 2009, A&A, 493, 899, A forming disk at z~0.6: Collapse of a gaseous disk or major merger remnant?
- Peirani et al. 2009, A giant bar induced by a merger event at z=0.4?
- Hammer et al. 2009, A forming, dust enshrouded disk at z=0.43: the first example of a late type disk rebuilt after a major merger?<|endoftext|>
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This module is a resource for lecturers
Journalism has long been considered a pillar of democracy, given its function of communicating vital information to the public regarding institutions and individuals in positions of power. An informed citizenry is critical for good governance and essential for exposing and preventing corruption. This assumes that the information is accurate, truthful and non-biased. Indeed, these are some of the ethical responsibilities of media professionals that the Module explores. The discussions are relevant to all students who are media consumers and wish to understand what ethical obligations they can expect media professionals to uphold. In addition to consuming media, many students play an active role in the production of media, especially social media. Therefore, after discussing the ethical obligations of media professionals, the Module proceeds to address the responsibility of all individuals to practice ethical behaviour in the creation and dissemination of social media. The Module first examines key terms and concepts.
Terms and concepts
Two key concepts used in this Module are "media" and "ethics." The word ethics comes from the Greek ethos, which means character, or what a good person is or does to have good character. The concept of ethics is explored in detail in Integrity and Ethics Module 1 (Introduction and Conceptual Framework), which introduces students to Richard Norman's definition of ethics: "the attempt to arrive at an understanding of the nature of human values, of how we ought to live, and of what constitutes right conduct" (Norman, 1988, p. 1). Media is defined by the Merriam-Webster's Dictionary as "the system and organizations of communication through which information is spread to a large number of people". A more current and relatable definition for students is provided by Dictionary.com, which defines media as "the means of communication, [such] as radio and television, newspapers, magazines, and the Internet, that reach or influence people widely".
The concept of "media ethics" refers broadly to the proper standards of conduct that media providers and disseminators should attempt to follow. With modern technology and increased globalization in today's world, there are many more branches of media than there were in earlier times in history. These new media forms trigger new ethical issues. For example, today many ethical issues arise in relation to the Internet, which did not exist just 40 years ago. As a result of the wide range of media platforms and vast accessibility, different issues may surface depending upon the branch of media in question.
Technology has also led to the emergence of so-called to "citizen journalists" as people are recording, photographing and videotaping newsworthy events as they unfold (Bulkley, 2012). Citizen journalists further compound media ethics issues and will be discussed in one of the exercises. An underlying theme in the subject of media ethics across many different branches of media is the potential conflict between the standards for ethical behaviour and companies' desire for profit. This and similar issues are discussed in Module 11 (Business Integrity and Ethics), which delves deeper into ethical issues confronting private sector actors, and Module 14 (Professional Ethics), which discusses the issues of professional codes of ethics and role morality.
It is important for students to grasp that conflicts of interest exist across many domains within the media and assess what ethical standards are required for journalists, consumers, and companies or individuals who play a role in the provision and dissemination of information to the public. As all parties involved must adhere to ethical standards, this Module considers the ethical principles for both media professionals and non-professionals who engage in creation and dissemination of media. It then allows students to holistically engage in the material through exercises.
Ethical principles for journalists and other media providers
While the Module considers ethical obligations of both media professionals and non-professionals, it should be noted that media professionals are held to higher ethical standards compared to non-professionals. They have duties to provide society with accurate, truthful and non-biased information. Media professionals have ethical obligations towards society simply by virtue of their activities as journalists, reporters, anchors, or owners of media corporations. The role of the media in contemporary times is affected by the commercialization and diversity of media actors, which include grass roots and independent media, corporate media, advocacy groups, consolidated media companies, state-owned and privatized media. Media ethical obligations apply to all of these.
Many media houses, online platforms, professional associations and other organizations have developed ethical codes for journalists. Over 400 ethical codes for journalists have been adopted worldwide, many of which can be accessed at the database of the Accountable Journalism Site. The Code of Principles adopted by the International Federation of Journalists (IFJ) in 1954 has been regarded as a universal statement about ethics in journalism. According to the IFJ Code, the core values of journalism are truth, independence and the need to minimise harm. Another influential ethics code for journalists is the one adopted in 2014 by the U.S.-based Society of Professional Journalists (SPJ). The SPJ Code of Ethics is available in numerous languages including Arabic, Chinese, French, German, Persian, Portuguese, Russian, and Spanish. Its Preamble states that "an ethical journalist acts with integrity" and the code has four foundational principles that call on journalists to: (1) seek truth and report it, (2) minimize harm, (3) act independently, and (4) be accountable and transparent. Under each principle, the SPJ Code of Ethics contains further guidance and calls on journalists to approach their work with the highest standard of ethics in mind. These principles, which are discussed in further detail below, apply to traditional journalism as well as modern forms of social media such as Facebook, YouTube, Instagram, Twitter, and LinkedIn. While some of the examples below are from the SPJ Code, they apply universally since similar principles and values are embraced by journalism codes around the world.
(1) "Seek truth and report it"
With regard to the first principle (seek truth and report it), the SPJ Code calls on journalists to take responsibility for the accuracy of their work, confirm information before releasing it, and rely on original sources whenever possible. The Code promotes and encourages journalists to use their work to facilitate greater transparency of those in power. For example, the Code requires that journalists be persistent and brave in their constant effort to hold those in power accountable. Journalists, according to the SPJ Code, must provide a platform for those in society who may not have a voice. It also states that journalists should be supportive of open and civil dialogue in which different points of view are exchanged, even if the journalists themselves find those views objectionable. Journalists have a special responsibility to be watchdogs over the government and public affairs. Furthermore, journalists should endeavour to ensure the transparency of public records and public business. In this sense, the SPJ Code appears to promote the idea that journalists owe a duty to the public to provide accurate information, to facilitate open access and transparency of the government and other individuals in authoritative positions, and to provide those without a voice in society the opportunity to speak and share their beliefs, perspectives, and experiences.
Experts on media ethics echo the value and importance of truth-seeking by journalists. Journalists and news organizations should be truthful and their reporting should accurately represent the issues or stories being reported. However, with this in mind, it is also critically important that journalists maintain respect for individual privacy while seeking the truth. At times, the individual's right to privacy may clash with the public's need to know information. There are ethical obligations on both sides of every decision and therefore journalists face difficult choices.
(2) "Minimize harm"
The drafters of the SPJ Code emphasize under the second principle that journalists must also minimize harm that could be caused by their reporting and that ethical journalism demands that sources, subjects, colleagues and members of the public are treated as human beings deserving of respect. As such, journalists should consider individual privacy rights as well as the impact their reporting may have on individuals in general. The Code states that journalists must show compassion for individuals who may be affected by news coverage, which may include juveniles or victims of crimes. Journalists should also be mindful of cultural differences when reflecting on the ways in which news or information may be received. The Code advises journalists to show "heightened sensitivity" in these circumstances (Society of Professional Journalists, 2014).
The tension between the competing goals of publishing information for the greater good of the public and refraining from sharing such information in order to protect individual privacy rights raises ethical questions and requires journalists to consider and weigh various factors in these strategic decisions. Harm to the individual may take the form of invasion of privacy or the dissemination of information that offends or damages him or her in some way.
In these decisions, journalists may consider various schools of thought, including virtue ethics, utilitarianism and deontology (Ess, 2013, p. 262), which are discussed in Integrity and Ethics Module 1. The basic premise of utilitarianism is that the morality of an action depends on whether it maximizes overall social 'utility' (or happiness). More specifically, utilitarianism is the idea that the goal of an action should be the largest possible balance of pleasure over pain or the greatest happiness for the largest number of people (see Module 1 for further explanation and sources).
Utilitarianism can either justify the release of information to the public despite a slight violation of privacy rights, or it can justify the withholding of information in order to protect privacy rights in certain circumstances. Utilitarianism can justify individual privacy, and correlatively, property rights, insofar as these things lead to the greatest happiness for the largest number of people, as opposed to just that individual. Utilitarianism can be used to justify sacrificing the privacy of a few individuals if it would facilitate greater access to information for the general public.
Deontologists, on the other hand, present a competing perspective. They provide a more straightforward defence of individual privacy rights, because these rights are arguably necessary to our basic existence and practices as autonomous moral agents. Thus, deontologists would favor the protection of individual privacy over the release of information that would serve the greater good to the detriment of the individual. Deontology is also defined in Module 1. Its basic premise, according to that Module, is that morality depends on conformity to certain principles or duties irrespective of the consequences. Therefore, the deontologists' response to this question in media ethics would be that we should not violate individual privacy rights of others, as we would not want our own privacy rights violated.
These competing perspectives inform approaches to questions in media ethics and are particularly relevant when addressing questions of protecting individual privacy and minimizing harm to the individual, on the one hand, and serving the greater public good on the other.
(3) "Act independently"
Journalists are also called on to act independently, which is the third principle outlined in the SPJ Code. Under this principle, the drafters of the Code emphasize that the primary responsibility of ethical journalism is to serve the public (Society of Professional Journalists, 2014, footnote 6). As such, journalists must put the public first and reject any special treatment to advertisers, donors, or any other special interests, and resist internal and external pressure to influence coverage. This requires journalists to refuse gifts and to avoid any conflicts of interest.
An example of conduct that falls short of the principle to act independently occurred in Canada in 2015. Leslie Roberts, a news anchor for Global Toronto, a news agency in Canada, resigned from his position at the network due to serious allegations of conflict of interest (Global News, 2015). Roberts publicly admitted that he was secretly a part owner of a public relations firm whose clients appeared on Global News programmes. Mr. Roberts never informed Global News management of his connection to the public relations firm. Such a conflict of interest raises serious ethical concerns, as the media has a duty to provide unbiased and independent information. If an anchor and news agency are presenting information to the public that is skewed by preference in guests on the show who are perhaps incentivized to send a certain message to the public, the duty of presenting the truth has been violated. This conduct flies in the face of the principle of acting independently laid out in the Code.
Another example of conduct that falls short of the principle that journalists should act independently is the handing out of "brown envelopes" containing large amounts of cash to journalists at press briefings, in exchange for publishing their stories. This trend of "brown envelope" journalism is fundamentally opposed to the principle of independence in journalism and allows the media to present skewed, or biased, information to the public (Nwaubani, 2015).
(4) "Be accountable and transparent"
The SPJ Code advises journalists that they must be accountable and transparent, which is the fourth principle. Journalists should explain ethical choices and processes to audiences and should recognize and publicly acknowledge any mistakes. They should also correct these mistakes promptly and prominently. In the example referred to above, Leslie Roberts came to recognize that journalists have a duty to expose unethical conduct in journalism to the public, including any behaviour within their own organizations, as the Code also states that ethical journalism means "taking responsibility for one's work and explaining one's decision to the public." Global News recognized the unethical conduct of Mr. Roberts and made the following statement after Mr. Roberts resigned: "Global News remains committed to balanced and ethical journalism produced in the public's interest." Global News also made public Mr. Roberts' letter of resignation, in which he acknowledged that his own unethical conduct was the cause of his resignation from the network and apologized by stating: "I regret the circumstances, specifically a failure to disclose information, which led to this outcome" (Global News, 2015). The morally upstanding way in which Global News handled this conflict of interest is one that preserves the principles laid out in the Code and upholds the high standards of behaviour for journalists that are a fundamental part of any discussion on media ethics.
In summary, journalists have a duty to (1) seek the truth and report it, (2) disseminate information in a way that minimizes harm to the public, (3) act independently in providing such information, and (4) be accountable and transparent in the process. These ethical duties of journalists are fundamental concepts in media ethics.
A fifth ethical duty or principle that can be discussed in class is the concept of objectivity. Long considered a norm in journalism, objectivity is currently the subject of significant debate. That debate tends to recognize transparency to be a preferable principle. While human beings may never be truly objective, we can at least disclose our frames of reference. In her article Objectivity and Journalism: Should We Be Skeptical? , Alexandra Kitty elaborates further on this idea (2017).
Ethical principles for citizen journalists and media consumers
While the most widely known ethical obligations in the world of media are those owed by journalists to the public, individuals who are not media professionals still have a responsibility to act with integrity in their use and consumption of media.
To illustrate this obligation, this Module will first look at the case of people who are often referred to as "citizen journalists." These people are not media professionals. Often, they are simply bystanders with a smart phone. However, these people sometimes have access to unfolding events that traditional journalists do not always have. Because of the global nature of social media, they are able to share their recordings and photos with a nearly limitless number of people. Examples of this phenomenon were seen when thousands of people on the ground were posting their experiences online during Hurricane Sandy of 2012, the Fukushima earthquake of 2011, the Boston Marathon bombing of 2013, the Paris terror attacks of 2015, and various global conflicts. While this was helpful in many ways, it also led to the spread of dangerous rumours and untrue statements.
Ordinary citizens should never be discouraged from sharing what they see. However, the increased power of their position due to social media platforms creates an ethical duty to act with care. Citizen journalists should strive to possess the same integrity that is expected from the professional news media. This means asking questions such as the following before sharing material online:
- Is what I am posting accurate?
- Have my sources of information been verified?
- Will anyone be harmed by sharing this information?
After all, the goal of citizen journalists should be to contribute to a better societal understanding of whatever they are reporting. This standard does not only apply to people who are posting online about ongoing events, but also to those who are blogging or creating content in any way.
The ethical obligations of users of social media also deserve attention. While social media users may not always be creating new content, they still often make decisions about which content to share with others. Unlike in the print media of the past, much of the media now published online is no longer clearly demarcated as news or opinion. Ads often resemble statements of fact and articles frequently do not list a writer or source. This confusion can be seen in the discussions globally regarding the issue of "fake news."
In the United States, a poll published by the Pew Research Center on 16 December 16 2016, shortly after the U.S. elections, showed that 23 percent of respondents had shared a made-up news story on social media, either knowingly or not. According to the same poll, 64 percent of respondents said that the phenomenon of fake news had caused significant confusion regarding current events.
Sharing and promoting these stories on social media may not only cause confusion, but it may also be harmful. Rumours and mistruths can damage reputations and even put others in danger. While one person sharing a false story may go unnoticed, there is often a collective impact.
The ethical course of action for social media users is to refrain from contributing to the spread of misinformation. In order to avoid this, social media users must critically evaluate content before sharing.
To assess the credibility of an article or story, social media users should ask themselves questions similar to the following:
- Who is the source? The author?
- Is the writer asserting fact or opinion?
- Does the piece contain sources or quotes that can be verified?
- Does the piece use language intended to provoke emotional reactions?
Studies show that members of the public have significant difficulty in assessing the credibility of content on social media. Knowing this, it is even more important to think about the ethical implications of what we share online.
To conclude, this Module illustrates that media ethics applies to all of us, whether or not we intend to become media professionals. With this in mind, the following section suggests class activities through which students can engage with the issues discussed above.
- Barthel, Michael, Amy Mitchell and Jesse Holcomb (2016). Many Americans believe fake news is sowing confusion. Pew Research Center: Journalism and Media, 15 December.
- Bulkley, Kate (2012). The rise of citizen journalism. The Guardian, 10 June.
- Ess, Charles M. (2013). Global media ethics? Issues, requirements, challenges, resolutions. Stephen J.A. Ward, In Global Media Ethics: Problems and Perspectives, Stephen J.A.Ward, ed.Chichester, Sussex, United Kingdom: John Wiley & Sons.
- Global News (2015). Leslie Roberts resigns from Global News in wake of internal investigation, 15 January.
- Kitty, Alexandra (2017). Objectivity and Journalism: Should We Be Skeptical?
- Norman, Richard (1998). The Moral Philosophers. Oxford: Oxford University Press.
- Nwaubani, Adaobi Tricia (2015). Nigeria's 'brown envelope' journalism. BBC News, 5 March.
- Society of Professional Journalists (2014). SPJ code of ethics, 6 September.<|endoftext|>
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Radiotherapy is one method doctors can use to treat cancer. Radiotherapy can be performed without surgery and avoids the side effects that a patient may experience during chemotherapy.
In the most common form of radiotherapy, high energy x-rays are used to kill the cells in the tumour. The treatment must be planned carefully to make sure that healthy cells are not damaged by the x-rays. The dose required to kill the cancerous cells is normally delivered in smaller doses and at different angles (as shown in the diagram) to make sure that only the cells in the tumour are destroyed.
- Each beam delivers one third of the dose required.
- The beams overlap at the tumour, which receives the full dose.
Here are some slides about radiotheraphy. I have attached a copy of them (as a pdf file) to the bottom of this post.<|endoftext|>
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### Complex Conjugates
Every complex number has a complex conjugate. The complex conjugate of a + bi is a - bi. For example, the conjugate of 3 + 15i is 3 - 15i, and the conjugate of 5 - 6i is 5 + 6i.
When two complex conjugates a + bi and a - bi are added, the result is 2a. When two complex conjugates are subtracted, the result if 2bi. When two complex conjugates are multiplied, the result, as seen in Complex Numbers, is a2 + b2.
### Dividing Complex Numbers
To find the quotient of two complex numbers, write the quotient as a fraction. Then multiply the numerator and the denominator by the conjugate of the denominator. Finally, simplify the expression:
= = = .
Examples:
(3 + 2i)÷(4 + 6i) =?
= = = = = = .
(6 + 3i)÷(7 - 2i) =?
= = = = = .
(3 - i)÷(- 5 + i) =?
= = = = = = .<|endoftext|>
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Sustainability in Everyday Life
Daily headlines warn of new chemical dangers, species on the edge of extinction, global warming-- framing our planet as “at risk.”
Some people stop listening, others get alarmed, while some others want to learn more in hopes of making a difference in the world.
Do you want to live a more sustainable life? In this environmental studies course, you will learn how to make more informed choices about your ecological footprint and gain a better understanding of how your choices impact on our world.
This course is organized into five key themes: chemicals, globalization, climate change, food and energy. These five themes represent challenges that people face day-to-day managing choices relating to sustainability.
In the final task of this course, you will demonstrate how you have acquired the skills and knowledge to organize your everyday life sustainably.
What you'll learn
- Appreciate the complexity of sustainable development and understand how it relates to everyday life
- Critically evaluate and reflect on the information flow from the public media
- Develop cognitive and decision-making skills that can be applied to issues and problems in everyday life
- How to discuss these topics appropriately, and encourage others to make informed decisions regarding sustainable living
|University:||Chalmers University of Technology|
|Start:||currently not in session|
|Workload per week:||7.0 h|
|Tags||environment , environmental justice , environmental pollution , environmentalism , sustainability , sustainable development|<|endoftext|>
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Circuit Fundamentals
# Resistors in parallel. Current divider circuit and equation
Two or more resistors are said to be in parallel, if identical voltage is across all the resistors.
In accordance to Kirhhoff’s Law the current ${I}_{S}={I}_{1}+{I}_{2}+...+{I}_{N}$, and ${V}_{1}={V}_{2}=...={V}_{N}$$=V$.
Here . So ${I}_{S}=V\left(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+...\frac{1}{{R}_{N}}\right)=\frac{V}{{R}_{eq}}$.
Here parallel resistors can be replaced by the equivalent resistance $\frac{1}{{R}_{eq}}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+...\frac{1}{{R}_{N}}$, and ${R}_{eq}=\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+...\frac{1}{{R}_{N}}}$.
Then current through a resistor in parallel circuit is ${I}_{n}=\frac{\frac{1}{{R}_{n}}}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+...\frac{1}{{R}_{N}}}{I}_{S}$. This equation is called current divider. Current divider is a linear circuit, producing output current equal to a fraction of the input current.
For example, if ${R}_{1}=1$Ohm, ${R}_{2}=2$ Ohm, Ohm, ${R}_{4}=2$Ohm and ${I}_{s}=4$A, then ${I}_{3}=\frac{4}{9}$A.<|endoftext|>
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+0
# Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that sh
+1
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2
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Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that she has exactly one pair of socks with the same color?
Mar 1, 2018
#1
+22006
+2
Gretchen has eight socks, two of each color: magenta, cyan, black, and white.
She randomly draws four socks.
What is the probability that she has exactly one pair of socks with the same color?
$$\text{Let {\color{magenta}{m}} = {\color{magenta}{magenta}} } \\ \text{Let {\color{cyan}{c}} = {\color{cyan}{cyan}} } \\ \text{Let {\color{black}{b}} = {\color{black}{black}} } \\ \text{Let {\color{grey}{w}} = {\color{grey}{white}} }$$
The Set is$$\{ {\color{magenta}{m_1}},{\color{magenta}{m_2}}, {\color{cyan}{c_1}},{\color{cyan}{c_2}}, {\color{black}{b_1}},{\color{black}{b_2}}, {\color{grey}{w_1}},{\color{grey}{w_2}} \}$$
The number of all the possibilities is $$^8C_4=\dbinom{8}{4} = \mathbf{70 }$$
$$\begin{array}{|rcll|} \hline && \dfrac{ \dbinom{ {\color{magenta}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{cyan}{2} } }{ 2 }\times \left[ \dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{magenta}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{black}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{grey}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{magenta}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{1} \right] } {70} \\\\ &=& \dfrac{ 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] } {70} \\\\ &=& \dfrac{ ( 4+4+4 ) + (4+4+4) + (4+4+4) + (4+4+4) } {70} \\ &=& \dfrac{ 12 + 12 + 12 + 12 } {70} \\ &=& \dfrac{ 4\cdot 12 } {70} \\\\ &=& \dfrac{ 2\cdot 12 } {35} \\\\ &\mathbf{=}&\mathbf{ \dfrac{ 24 } {35} } \\ \hline \end{array}$$
The probability that she has exactly one pair of socks with the same color is $$\mathbf{ \dfrac{ 24 } {35} }$$
Mar 1, 2018
#1
+22006
+2
Gretchen has eight socks, two of each color: magenta, cyan, black, and white.
She randomly draws four socks.
What is the probability that she has exactly one pair of socks with the same color?
$$\text{Let {\color{magenta}{m}} = {\color{magenta}{magenta}} } \\ \text{Let {\color{cyan}{c}} = {\color{cyan}{cyan}} } \\ \text{Let {\color{black}{b}} = {\color{black}{black}} } \\ \text{Let {\color{grey}{w}} = {\color{grey}{white}} }$$
The Set is$$\{ {\color{magenta}{m_1}},{\color{magenta}{m_2}}, {\color{cyan}{c_1}},{\color{cyan}{c_2}}, {\color{black}{b_1}},{\color{black}{b_2}}, {\color{grey}{w_1}},{\color{grey}{w_2}} \}$$
The number of all the possibilities is $$^8C_4=\dbinom{8}{4} = \mathbf{70 }$$
$$\begin{array}{|rcll|} \hline && \dfrac{ \dbinom{ {\color{magenta}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{cyan}{2} } }{ 2 }\times \left[ \dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{magenta}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{magenta}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{black}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{magenta}{2}}}{0}\dbinom{\color{grey}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{magenta}{2}}}{1}\dbinom{\color{grey}{2}}{1} \right] \\ + \dbinom{ {\color{grey}{2} } }{ 2 }\times \left[ \dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{0} +\dbinom{{\color{cyan}{2}}}{1}\dbinom{{\color{black}{2}}}{0}\dbinom{\color{magenta}{2}}{1} +\dbinom{{\color{cyan}{2}}}{0}\dbinom{{\color{black}{2}}}{1}\dbinom{\color{magenta}{2}}{1} \right] } {70} \\\\ &=& \dfrac{ 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] \\ + 1\times \left[ 2\cdot 2\cdot 1 + 2\cdot 1\cdot 2 + 1\cdot 2\cdot 2 \right] } {70} \\\\ &=& \dfrac{ ( 4+4+4 ) + (4+4+4) + (4+4+4) + (4+4+4) } {70} \\ &=& \dfrac{ 12 + 12 + 12 + 12 } {70} \\ &=& \dfrac{ 4\cdot 12 } {70} \\\\ &=& \dfrac{ 2\cdot 12 } {35} \\\\ &\mathbf{=}&\mathbf{ \dfrac{ 24 } {35} } \\ \hline \end{array}$$
The probability that she has exactly one pair of socks with the same color is $$\mathbf{ \dfrac{ 24 } {35} }$$
heureka Mar 1, 2018
#2
+1
There are 4 possible pairs Gretchen can pick, and (6C2)−3=12 ways for her to pick socks of two other colors. There are (8C4)=70 total ways Gretchan can pick socks, and thus there is a 4⋅12/70 =24/35 probability Gretchen picks exactly 1 pair.
This number might seen large, but considering there is a 2^4/ 70 =8/35 chance of picking no pairs, and a (4C2)/70 =3/35 chance of two pairs, it adds up (to 1).
https://math.stackexchange.com/questions/2660495/probability-of-choosing-exactly-1-pair-out-of-4
Mar 1, 2018<|endoftext|>
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## Question
Gauthmathier3459
Grade 12 ·
YES! We solved the question!
Check the full answer on App Gauthmath
The following week they put socks right next to the shoes to see how it would affect Saturday sales. The results were as follows; a total of 163 people bought socks, shoes, or both. Of those 52 bought only socks, 76 bought only shoes and the remainder bought both. What is the probability that a person bought socks, if they purchased shoes?
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## Gauth Tutor Solution
Camila
Universidad Industrial de Santander
Physics teacher
Answer
\begin{align*}P(\text{socks} | \text{shoes}) = \frac{35}{111} = .315 \ \text{or} \ 31.5 \%\end{align*}
Explanation
Given that,
\begin{align*}P(\text{only socks}) = 52\end{align*}
\begin{align*}P(\text{only shoes}) = 76\end{align*}
\begin{align*}\text{Total number of people} = 163\end{align*}
\begin{align*}\eqalign{ P(\text{shoes and socks}) &=\text{Total number of people} - P(\text{only socks}) - P(\text{only shoes}) \\ P(\text{shoes and socks}) &=163 - 52 - 76 \\ P(\text{shoes and socks}) &=111 - 76 \\ P(\text{shoes and socks}) &=35 }\end{align*}
To find \begin{align*}P(\text{shoes})\end{align*}:
\begin{align*}\eqalign{ P(\text{shoes}) &=\text{Total number of people} - P(\text{only socks}) \\ P(\text{shoes}) &=163 - 52 \\ P(\text{shoes}) &=111 }\end{align*}
To find the probability of that a person bought shoes:
\begin{align*}\eqalign{ P(\text{socks} | \text{shoes}) &=\frac{P(\text{shoes and socks})}{P(\text{shoes})} \\ P(\text{socks} | \text{shoes}) &=\frac{35}{111} \\ P(\text{socks} | \text{shoes}) &=0.315 \\ P(\text{socks} | \text{shoes}) &=0.315 \times 100\% \\ P(\text{socks} | \text{shoes}) &=31.5\% }\end{align*}
Option A is the correct answer.
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# Associative property of multiplication
– [Instructor] So, what we’re gonna do is get a little bit of practicing
multiple numbers together and we’re gonna discover some things. So, first I want you to figure out what four times five times two is. Pause the video and try to
figure it out on your own. Alright, so whatever your answer is, some of you might have done it this way, some of you might have said hey, what is four times five and then you multiplied it by two, so what you would really have done is you would have done
four times five first, so that’s why I put
parentheses around that and then you would have multiplied by two and what would you have gotten? Weil, the four times five part, that is of course 20 and then you multiply that times two and you would get 40 which of course would be correct, four times five times two is indeed equal to 40. Now, what I want you to do now is as quickly as possible try to figure out what five times two times four is. Really quick, pause the
video, try to figure that out. Well, some of you might have tried and you might have done
it in a similar way where you tried to figure
out five times two is first and you said okay, five times two is equal to 10 and then I’d multiply that times four and then you would say well, gee, this is same thing as I got last time. Is there something interesting going on? And the interesting thing that you might realize is in both cases we’re multiplying the same three numbers. We are just doing it in a different order. Here we multiplied four times, we wrote it out in a different order, four times five times two. Here we wrote five times two times four. Here we did the four times five first, here we did the five times two first but notice we got the same result. Now, I’d encourage you, pause this video. Try to multiply these numbers in any order, maybe you do two times four first. In fact, let’s just do that. Let’s do two times four, two times four and then multiply that by five. What is this going to be equal to? Well, you might notice again this is two times four is eight, you multiply that times five. Well, once again, we got 40, so you might see a pattern here. It doesn’t matter which order
we multiply these things in. In fact, you could write
four times five times two. You could do the four times five first, four times fives times two or you could do four times five times two, so you could do four times five times two. So, it doesn’t matter which order you multiply these things in. In every case you are going to get 40. Now, there’s a very fancy term for this, the associative property of multiplication but the main realization
is and it’s not just true with the three numbers, in fact, you’ve seen something similar with two numbers where it doesn’t matter what order you multiply them in but what you see with three numbers and even if you tried it with four or five or really 1,000 numbers being multiplied together, as long as you’re just
multiplying them all, it doesn’t matter what
order you’re doing it with. It doesn’t matter in what order you associate them with. Here we did four times five first, four times five first, here we did five times two first but in either case we got the same result and I’d encourage you, after this video, try to draw it out. Try to think about why that
actually makes intuitive sense, why this is true in the world and it’s nice because
it simplifies our life when we’re doing mathematics and not only now but in our
future mathematical career.
## 6 thoughts on “Associative property of multiplication”
1. WOOHOO I'm first comment!! OH YEAH!
2. Hi ! I'm looking for the software used to make this video if someone know the name
3. Which playlist can I find the new uploaded videos in khan academy ?
4. Fourth
5. The numbers can be changed in any order in multiplication and also in addition.
6. hi<|endoftext|>
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If A = <2 ,-5 ,9 >, B = <-9 ,1 ,-5 > and C=A-B, what is the angle between A and C?
Dec 13, 2017
The angle is $= {25.4}^{\circ}$
Explanation:
Let's start by calculating
$\vec{C} = \vec{A} - \vec{B}$
vecC=〈2,-5,9〉-〈-9,1,-5> = <11,-6,14>
The angle between $\vec{A}$ and $\vec{C}$ is given by the dot product definition.
vecA.vecC=∥vecA∥*∥vecC∥costheta
Where $\theta$ is the angle between $\vec{A}$ and $\vec{C}$
The dot product is
vecA.vecC=〈2,-5,9〉.〈11,-6,14〉=22+30+126=178
The modulus of $\vec{A}$= ∥〈2,-5,9〉∥=sqrt(4+25+81)=sqrt110
The modulus of $\vec{C}$= ∥〈11,-6,14〉 ∥=sqrt(121+36+196)
$= \sqrt{353}$
So,
costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=178/(sqrt110*sqrt353)=0.903
$\theta = {25.4}^{\circ}$<|endoftext|>
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calculator.org
# property>Henry's law constant
## What is the Henry's Law Constant?
Henry's Law describes how an amount of a given gas dissolved in a given type and volume of liquid is proportional to the partial pressure of the gas in equilibrium with the liquid. Another way many people understand this would be to say that the solubility of a gas within a liquid will be proportional to the pressure of the gas above the liquid. This is actually a phenomenon that we see almost every day when we open up a can of soda or a bottle of champagne. When the bottle is closed, there is a very high pressure section of carbon dioxide present on top of the champagne. As soon as the bottle is opened, that pressure is lowered and the carbon dioxide dissolved in the champagne begins to come out of solution. Over time, if one were to leave a glass of champagne, the drink would lose all of its carbonation and go flat.
The Henry's law constant appears in the following mathematical formula that relates a partial pressure p to the concentration of a solute in solution.
p = kh.c
where c is the concentration and kh is the Henry's law constant. The "constant" is actually dependent on the solvent, the gas solute and the temperature. Henry's law constant for some common gases are listed:
• Hyrdogen (H2): 1228 l.atm/mol
• Carbon Dioxide (CO2): 30 l.atm/mol
• Oxygen (02): 757 l.atm/mol
• Nitrogen (N2): 1600 l.atm/mol
• Helium (He): 2865 l.atm/mol
To complicate matters further, Henry's law is often written in different ways and uses different units. Many of these use mass rather than number of moles, or they arrange them as different ratios. For example:
kh,pc = p/c
In the above equation, the units for kh are the same as the first equation, having a value of l.atm/mol. The next equation switches them:
kh,cp = c/p
In this case, the units on the bottom move to the top and the units on the top move to the bottom, so we get mol/(l.atm). There are others that have units of atmospheres or are dimensionless as well.
Some people prefer to call the Henry's law constant the Henry's law coefficient because of its temperature dependence. For a given temperature T, and a known temperature Tref, the Henry's law constant is given by:
kh = kh,ref Tref.exp(-C.((1/T) - (1/Tref)))
Bookmark this page in your browser using Ctrl and d or using one of these services: (opens in new window)<|endoftext|>
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Bearings are measured as the angle from due north and due south. They are stated as which quadrant they are in, as identified by the first and last letter of each bearing.
Survey distances are expressed in horizontal feet. Horizontal distances are the same as "map" or "flat" distances as would be viewed from a high elevation. The slope of the surface of the ground is taken into account when performing surveying measurements. Steep slopes and features such as streams can result in a slope distance, or distance along the surface, that is considerably longer than the horizontal distance, as illustrated above.
High precision angular measurements are stated in degrees, minutes, and seconds. There are 360 degrees in a full circle, 60 minutes in a degree, and 60 seconds in a minute. This is often but incorrectly confused with GPS coordinates.
Acreage is a measurement of surface area. It is not fixed as a square or other shape. Any shape has area, even if challenging to calculate. Each of the shapes above contain one acre of surface area. As can be seen, the shape and distance around the outside of the shape can vary greatly yet still contain the same surface area.
GPS coordinate, also known as geodetic coordinates or "lat/long", consists of two components. The first is the latitude. This is the angular measure from the center of the earth as referenced to the equator as being at zero degrees and the north pole as ninety degrees. The closer to the equator a location is, the lower the value of the latitude. The second component is longitude, which is the angular measure along the circumference of the earth in a clockwise direction as referenced as Greenwich, England at zero degrees. Latitude and longitude are expressed in degrees (often degrees, minutes, and seconds) and followed by a letter that designates as being north of the equator and west of Greenwich, England.<|endoftext|>
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