9.2 反向变化

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  反反变化

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  The force, F , required for a karate student to break a board varies inversely with the board's length, L . It takes 21 pounds of pressure to break a board that is 3 feet long. How many pounds of pressure does it take to break a board that is 2 feet long?
  ::空手道学生打破板的力度 F 需要21磅的压力才能打破3英尺长的板。打破2英尺长的板需要多少磅的压力?

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  Inverse Variation
  ::反反变化

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  We say that a set of data is related inversely if the independent increases and dependent variables decreases or vice versa. For example, the further away from an object that you are, the smaller it appears. In inverse variation, the variables are related inversely. As $\begin{align*}x\end{align*}$ gets bigger, $\begin{align*}y\end{align*}$ would get smaller. The inverse variation equation is $\begin{align*}y=\frac{k}{x};k,x \ne 0\end{align*}$ . We still call $\begin{align*}k\end{align*}$ the constant of variation and $\begin{align*}y\end{align*}$ is said to vary inversely with $\begin{align*}x\end{align*}$ . $\begin{align*}k\end{align*}$ can also be written $\begin{align*}k=xy\end{align*}$ .
  ::我们说,如果独立增长和依附变量减少,或者反之亦然,一组数据是反向关联的。例如,离对象越远,越看越看越看越看越看越看越看越看越看越看。反向变量是反向关联的。随着 x 越大,y 就会越小。反向变量方程式是 y=kx;k,x0。我们仍称 k 为变量常数, y 据说与 x. k 的反向变化也会被写为 k=xy 。

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  The variables $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ vary inversely, and $\begin{align*}y=7\end{align*}$ when $\begin{align*}x=2\end{align*}$ . Let's write an equation that relates $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ and find $\begin{align*}y\end{align*}$ when $\begin{align*}x=-6\end{align*}$ .
  ::变量 x 和 y 反向变化, y= 7 当 x= 2 时, y= 7 。 让我们写一个与 x 和 y 相关的方程式, 然后在 x \ 6 时找到 y 。

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  Using the inverse variation equation, we can substitute in $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ and solve for $\begin{align*}k\end{align*}$ .
  ::使用反向变异方程式, 我们可以替换 x 和 y , 并解决 k 。

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  $$\begin{align*}k&=yx \\ k&=7 \cdot 2 \\ k&=14\end{align*}$$
  ::kyxk= 72k=14

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  Therefore , the equation is $\begin{align*}y=\frac{14}{x}\end{align*}$ . To find $\begin{align*}y\end{align*}$ when $\begin{align*}x\end{align*}$ is -6, we have $\begin{align*}y=\frac{14}{-6}=- \frac{7}{3}\end{align*}$ .
  ::因此,公式是y=14x。在 x - 6 时找到y,我们有y=14-673。

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  Now, let's determine if the following set of data varies directly, inversely, or neither and find the equation if possible.
  ::现在,让我们来决定下一组数据是否直接、反向或两者无差, 并在可能的情况下找到方程。

  $\begin{align*}x\end{align*}$	1	2	3	4
  $\begin{align*}y\end{align*}$	12	6	4	3

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  Looking at the set of data, the $\begin{align*}x\end{align*}$ values increase. For the data to vary directly, the $\begin{align*}y\end{align*}$ values would also have to increase, and they do not. So, it could be an inverse relationship. Let’s see if $\begin{align*}k\end{align*}$ is the same for every set of points.
  ::查看数据集后, x 值会增加。 要直接改变数据, Y 值也会增加,而y 值不会增加。因此,它可能是一种反向关系。让我们看看 k 是否对每组点都一样。

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  $$\begin{align*}k&=1 \cdot 12=12 \\ k&=2 \cdot 6 \ =12 \\ k&=3 \cdot 4 \ =12 \\ k&=4 \cdot 3 \ =12\end{align*}$$
  ::k=1*12=12k=2*6=12k=3*4=12k=4*3=12

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  So, for each set of points, $\begin{align*}k = 12\end{align*}$ . Therefore, the equation is $\begin{align*}y=\frac{12}{x}\end{align*}$ . If $\begin{align*}k\end{align*}$ is not the same, then the answer would have been neither.
  ::所以, 对于每组点, k=12。 因此, 方程式是 Y=12x。 如果 k 不同, 那么答案就不会相同 。

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  Finally, let's solve the following problem.
  ::最后,让我们解决以下问题。

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  Sherry is driving from San Francisco to Los Angeles (380 miles). How long does it take her if she drives 65 miles per hour (the speed limit)? How fast does she have to drive to get to LA in five and a half hours?
  ::雪利正从旧金山开往洛杉矶(380英里 ) 。 如果她每小时驾驶65英里( 车速限制 ) , 她需要多久才能开到洛杉矶? 她要开多快才能在5个半小时内到达洛杉矶?

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  The faster Sherry drives, the less time it will take her to get to LA. Therefore, this is an inverse relationship. $\begin{align*}y\end{align*}$ is the time driving, $\begin{align*}k\end{align*}$ is the 380 miles between LA and San Francisco and $\begin{align*}x\end{align*}$ is the speed.
  ::雪利车程越快,她到洛杉矶的时间就越少。 因此,这是一个反向的关系。y是时间驾驶,k是洛杉矶和旧金山之间的380英里,x是速度。

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  $\begin{align*}y=\frac{380}{x}\end{align*}$
  ::y=380x y=380x

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  So, it is going to take her $\begin{align*}y=\frac{380}{65} \approx 5.85\end{align*}$ hours, which is 5 hours and 51 minutes. For her to get there in 5.5 hours, she would have to drive $\begin{align*}5.5=\frac{380}{x} \rightarrow 5.5x=380 \rightarrow x=69.1\end{align*}$ miles per hour.
  ::因此,她需要5小时51分钟的y=380655.85小时。如果她要在5.5小时之内到达,她必须驾驶5.5=380x=380x5.5x=380x=69.1英里每小时。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to determine the pounds of pressure it takes to break a board that is 2 feet long. 
  ::早些时候,你被要求确定 打破两英尺长的板子 需要多少压力

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  We are told this is an inverse variation, so we can use the inverse variation equation $\begin{align*}y=\frac{k}{x}\end{align*}$ . In this case, y equals the force and x equals the length of the board.
  ::我们被告知这是一个反向变异, 所以我们可以使用反向变异方程式y=kx。 在这种情况下, y等于力, x等于板的长度 。

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  $$\begin{align*}F = \frac{k}{L}\\ 21 = \frac{k}{3}\\ k = 63\end{align*}$$
  ::F=kL21=k3k=63

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  We've found the constant of variation, so now we use the equation a second time to find the force when the length of the board is 2 feet.
  ::我们发现了变异的常数 所以现在我们第二次使用方程 来找到当板的长度是2英尺时的力量

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  $$\begin{align*}F = \frac{63}{2}\\ F = 31.5\end{align*}$$
  ::F=632F=31.5

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  Therefore 31.5 pounds of pressure are needed to break a board that is 2 feet long.
  ::因此,需要31.5磅的压力才能打破两英尺长的板块。

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  Example 2
  ::例2

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  The variables  $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ vary inversely, and  $\begin{align*}y=-5\end{align*}$  when $\begin{align*}x = 3\end{align*}$ . Find the equation and determine $\begin{align*}x\end{align*}$ when $\begin{align*}y = 12\end{align*}$ .
  ::变量 x 和 y 反向变化, 当 x= 3 时为 y = 5 。 查找方程并确定 x y = 12 时为 y = 。

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  First, solve for $\begin{align*}k\end{align*}$ .
  ::首先,解决k。

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  $$\begin{align*}k=xy \rightarrow 3 \cdot -5&=-15 \\ y&=- \frac{15}{x}\end{align*}$$
  ::k=xy=3=5=5=15=15x

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  Now, substitute in 12 for $\begin{align*}y\end{align*}$ and solve for $\begin{align*}x\end{align*}$ .
  ::现在,以 y 12 取代 y , 用 x 解析 。

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  $$\begin{align*}12&=- \frac{15}{x} \\ 12x&=-15 \\ x&=- \frac{5}{4}\end{align*}$$
  ::1215x12x15x54

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  Example 3
  ::例3

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  Determine if the set below varies directly or inversely.
  ::确定以下集是直接变化还是反变化。

  $\begin{align*}x\end{align*}$	1	2	3	4	5
  $\begin{align*}y\end{align*}$	2	6	12	24	36

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   At first glance, it looks like both values increase together, so we know the set does not vary inversely. Let’s check for by determining if $\begin{align*}k\end{align*}$ is the same for each set of points.
  ::乍一看,它看起来似乎两种数值都同时增长,所以我们知道这组数值并没有反向变化。 让我们通过确定每组点的 k 是否相同来检查一下。

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  $\begin{align*}k=xy=2 \ne 3 \ne 4 \ldots\end{align*}$
  ::k=xy=234...

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  None of these points have the same ratio; therefore the data set does not vary inversely or directly.
  ::这些点中没有一个点的比重相同;因此,数据集没有反差或直接差。

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  Example 4
  ::例4

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  It takes one worker 12 hours to complete a specific job. If two workers do the same job, it takes them 6 hours to finish the job. What type of relationship is this? How long would it take 6 workers to do the same job?
  ::完成一项具体的工作需要一名工人12小时。 如果两名工人做同样的工作,他们需要6小时才能完成工作。这是哪一种关系?6名工人做同样的工作需要多长时间?

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  This is an inverse relationship because as the number of workers goes up, the number of hours it takes to complete the job goes down. $\begin{align*}k=12 \cdot 1=2 \cdot 6=12\end{align*}$ and the inverse variation equation is $\begin{align*}y=\frac{12}{x}\end{align*}$ . For 6 workers to complete the job, it would take $\begin{align*}y=\frac{12}{6}=2 \ hours\end{align*}$ .
  ::这是一种反向关系,因为随着工人数量的增加,完成工作所需的小时数会下降。 k=121=26=12, 反向变异方程式是y=12x。对于6名工人来说,完成工作需要y=126=2小时。

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  Review
  ::回顾

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  For problems 1-4, the variable $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ vary inversely. Use the given $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ values to write an inverse variation equation and find $\begin{align*}y\end{align*}$ given that $\begin{align*}x =15\end{align*}$ .
  ::对于问题 1-4, 变量 x 和 y 反向变化。 使用给定的 x 和 y 值来写入反向变异方程, 并找到 y, 给定的 x 和 y 值 x= 15 。

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  1. $\begin{align*}x=4,y=3\end{align*}$
     ::x=4,y=3 x=4,y=3
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  2. $\begin{align*}x=\frac{1}{5},y=10\end{align*}$
     ::x=15,y=10 x=15,y=10
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  3. $\begin{align*}x=8,y=\frac{3}{4}\end{align*}$
     ::x=8,y=34x=8,y=34
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  4. $\begin{align*}x=\frac{2}{3},y=\frac{15}{8}\end{align*}$
     ::x=23,y=158

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  For problems 5-8, the variable $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ vary inversely. Use the given $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ values to write an inverse variation equation and find $\begin{align*}x\end{align*}$ given that $\begin{align*}y = 2\end{align*}$ .
  ::对于问题 5-8, 变量 x 和 y 反向变化。 使用给定的 x 和 y 值来写入反向变异方程, 并查找 x , 前提是 y= 2 。

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  5. $\begin{align*}x=6,y=\frac{2}{3}\end{align*}$
     ::x=6,y=23x=6,y=23
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  6. $\begin{align*}x=16,y=\frac{3}{8}\end{align*}$
     ::x=16,y=38
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  7. $\begin{align*}x=\frac{4}{5},y=9\end{align*}$
     ::x=45,y=9
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  8. $\begin{align*}x=\frac{5}{6},y=\frac{18}{5}\end{align*}$
     ::x=56,y=185

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  Determine if the following data sets vary inversely.
  ::确定以下数据集是否反向变化。

  9. .

  $\begin{align*}x\end{align*}$	12	6	9	2
  $\begin{align*}y\end{align*}$	3	6	4	18

  10. .

  $\begin{align*}x\end{align*}$	4	7	2	8
  $\begin{align*}y\end{align*}$	10	6	20	5

  11. .

  $\begin{align*}x\end{align*}$	9	6	12	21
  $\begin{align*}y\end{align*}$	28	42	21	12

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  Solve the following word problems using an inverse variation equation.
  ::使用反向变异方程式解决以下字词问题 。

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  12. At a party there are 3 pizzas to share. If each pizza has 8 slices, determine how many pieces each child will receive if 12 kids attend the party. What if 8 children attend? Write an inverse variation equation to determine how many slices each child receives if there are $\begin{align*}x\end{align*}$ kids at the party.
      ::聚会上共有3个比萨饼可以分享。 如果每份比萨饼有8片, 确定如果有12个孩子参加, 每个孩子会收到多少块。 如果有8个孩子参加, 写一个反差方程, 以确定每个孩子会收到多少块。 如果有x个孩子参加, 则决定每个孩子会收到多少块。
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  13. When Lionel drives from Barcelona to Madrid, 390 miles, it takes him about 6.5 hours. How fast will he have to drive in order to make the trip in 5 hours?
      ::当莱昂内尔从巴塞罗那开往马德里390英里时,他大约需要6.5个小时。 他必须开多快才能在5小时内出行?
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  14. Alena and Estella can complete a job in 18 hours when they work together. If they invite Tommy to help, how long will the job take? How many friends need to work together on the job to complete it in 4 hours?
      ::Alena和Estella可以一起工作,在18小时内完成一份工作。如果他们邀请Tommy帮忙,工作需要多长时间?有多少朋友需要一起工作,才能在4小时内完成?
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  15. The temperature of the Pacific Ocean varies inversely with the depth. If the temperature at 2000 m is 2.2 degrees Celsius, what is the temperature at a depth of 4000 m?
      ::太平洋的温度随深度变化而变化,如果2000米的温度为2.2摄氏度,那么4000米的深度的温度是多少?

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。