Section: 反反变化 | 5.9 反反变化

Section outline

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?

lesson content

Inverse Variation
::反反变化

Inverse variation means that the product of x and y is a constant . We still call the constant the constant of proportionality or the constant of variation .
::反向变异意味着 x 和 y 的产物是一个常数。我们仍称常数为相称性常数或变数常数。

Inverse Variation
::反反变化

\begin{aligned} x y = k \Rightarrow y = \frac{k}{x} or y = \frac{k}{x^{n}}, k \neq 0, & n > 0 \end{aligned}

k is called the constant of proportionality or the constant of variation. We say, " y varies inversely as

\begin{aligned} x^{n} \end{aligned}

" or " y is inversely proportional to x. "

We say that " y varies inversely as x, " or " y is inversely proportional to x ."
::我们说,“与x的反差是相反的,”或者“与x成反比。”

Example 1
::例1

Determine if the set of data varies directly, inversely, or neither. Find the equation if possible.
::确定数据集是否直接、反向变化或两者无变化。可能的话找到方程式。

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

Solution: Looking at the set of data, you cannot get the y -values from the x -values by multiplying by the same constant: $\begin{aligned} 1 \cdot 12 = 12, 2 \cdot 3 = 6 \end{aligned}$ . So the data is not directly proportional.
::解决方案 : 查看数据集时, 您无法通过乘以相同的常数来从 x 值中获取 Y 值 : 1 = 12= 12 2 3 = 6 。因此, 数据不直接成比例。

To determine if the data is inversely proportional, we can multiply the x- and y -values to see if the product is constant.
::为了确定数据是否反比,我们可以乘以 x 和 Y 值,看产品是否不变。

\begin{aligned} k & = 1 \cdot 12 = 12 \\ k & = 2 \cdot 6 = 12 \\ k & = 3 \cdot 4 = 12 \\ k & = 4 \cdot 3 = 12 \end{aligned}

::k=1*12=12k=2*6=12k=3*4=12k=4*3=12

For each set of points, $\begin{aligned} k = 12 \end{aligned}$ . Therefore , the data are inversely proportional and the equation is $\begin{aligned} y = \frac{12}{x} \end{aligned}$ .
::对于每组点, k=12。因此, 数据是反比例的, 方程式是 y= 12x 。

If we cannot find a constant, $\begin{aligned} k \end{aligned}$ for each pair of data points, then this set is neither directly or indirectly proportional.
::如果我们无法为每对数据点找到一个常数 k, 那么这组数据既不直接或间接成比例。

Example 2
::例2

The variables $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ vary inversely, and $\begin{aligned} y = 7 \end{aligned}$ when $\begin{aligned} x = 2 \end{aligned}$ . Write an equation that relates $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ and find $\begin{aligned} y \end{aligned}$ when $\begin{aligned} x = - 6 \end{aligned}$ .
::变量 x 和 y 反向变化, y= 7 当 x= 2 时, y= 7 。写入一个与 x 和 y 相关的方程式, 然后在 x \ 6 时找到 y 。

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

\begin{aligned} k & = x y \\ k & = 2 \cdot 7 \\ k & = 14 \end{aligned}

::k=xyk=2=7k=14

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

by CK-12 explains .
::由 CK-12 解释。

Example 3
::例3

$\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ vary inversely. When $\begin{aligned} x = 3, y = - 5 \end{aligned}$ . Find the equation and determine $\begin{aligned} x \end{aligned}$ when $\begin{aligned} y = 12 \end{aligned}$ .
::x 和 y 反差。当 x= 3, y = 5 时, 查找方程, 确定 x y = 12 时。

Solution: First, solve for $\begin{aligned} k \end{aligned}$ .
::解答: 第一, K的解答。

\begin{aligned} k = x y \to 3 \cdot - 5 & = - 15 \\ y & = - \frac{15}{x} \end{aligned}

::k=xy=3=5=5=15=15x

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

\begin{aligned} 12 & = - \frac{15}{x} \\ 12 x & = - 15 \\ x & = - \frac{5}{4} \end{aligned}

::1215x12x15x54

Example 4
::例4

Solution: We are told the force varies inversely with the board's length, so we can use the inverse variation equation $\begin{aligned} y = \frac{k}{x} \end{aligned}$ . In this case, y equals the force and x equals the length of the board.
::解答: 我们被告知, 力因板的长度而异, 因此我们可以使用反变方程式y=kx。在这种情况下, y等于力, x等于板的长度。

\begin{aligned} F = \frac{k}{L} \\ 21 = \frac{k}{3} \\ k = 63 \end{aligned}

::F=kL21=k3k=63

We have 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

\begin{aligned} F = \frac{63}{2} \\ F = 31.5 \end{aligned}

::F=632F=31.5

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。

by Mathispower4u de fines inverse variation and shows inverse variation examples .
::通过 Mathispower4u 定义反向变异, 并显示反向变异示例。

Example 5
::例5

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. If the time it takes to complete the job is inversely proportional to the number of workers, how long would it take 4 workers to do the same job?
::完成具体工作需要一名工人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,如果完成工作所需的时间与工人人数成反比,4名工人做同样工作需要多长时间?

Solution: Since this is an inverse relationship, we need to find k: $\begin{aligned} k = 12 \cdot 1 = 2 \cdot 6 = 12 \end{aligned}$ and the inverse variation equation is $\begin{aligned} y = \frac{12}{x} \end{aligned}$ .
::解决方案 : 由于这是一个反向关系, 我们需要找到 k: k=121=26=12, 而反向变异方程式是 y=12x 。

For 4 workers to complete the job, it would take $\begin{aligned} y = \frac{12}{4} = 3 \end{aligned}$ hours.
::4名工人完成工作需要y=124=3小时。

Feature: Good Vibrations
::特点:良好振动

by Ellen Lawsky
::埃伦·洛斯基(签名)

Violins and cellos are both string instruments, which means that they both produce sound due to vibrating strings. So how do they still produce such different sounds?
::维伦斯和大提琴都是弦乐器,这意味着它们由于振动弦而产生声音。所以它们如何仍然产生这种不同的声音呢?

A string vibrating at a low frequency will produce a lower pitch than a string vibrating at a high frequency. The tension, length, and thickness of a vibrating string all influence the frequency of that particular string. The length of the string and the thickness of the string are both inversely related to the vibrating frequency. As the length or thickness of the string increases, the frequency of vibration decreases, resulting in a lower pitch.
::在低频下振动的字符串将产生比在高频下振动的字符串更低的音调。振动字符串的张度、长度和厚度都会影响该字符串的频率。字符串的长度和厚度都与振动频率反相相关。随着字符串的长度或厚度增加,振动频率会降低,导致音调降低。

While this relationship was known to Galileo, French mathematician Marin Mersenne ultimately proved this relationship, so it is called Mersenne's Law.
::虽然伽利略知道这种关系, 法国数学家Marin Mersenne最终证明了这种关系, 所以它被称为Mersenne的法律。

How are these inverse relationships used in producing music? If you examine the difference between a violin and a cello, you will see that the strings on the cello are both longer and thicker than those on the violin. Thus, the cello has a lower pitch than the violin. With any given string instrument, a musician can also apply this knowledge of the inverse variation between a string’s length and its vibrating frequency (and thus pitch) as he or she fingers the strings. When different lengths of a certain string are allowed to vibrate, different pitches will be heard.
::这些反向关系是如何用于制作音乐的?如果你检查小提琴和大提琴之间的差别,你会看到大提琴的弦比小提琴的弦长,厚。因此大提琴的弦比小提琴低。有了任何给定的弦乐器,音乐家也可以在他或她指向弦时应用弦长度和振动频率(并由此推高)之间的反向变化知识。当允许某一弦的不同长度振动时,可以听到不同的弦。

by HighSchoolMusicAus explains the string instruments in an orchestra.
::高校音乐Aus解释乐团的弦乐乐器。

Summary
::摘要

The indirect variation equation is: $\begin{aligned} y = \frac{k}{x^{n}} \end{aligned}$ where $\begin{aligned} k \neq 0 \end{aligned}$ and $\begin{aligned} n > 0 \end{aligned}$ .
::间接变异方程式是: y=kxn, 其位置是 k0 和 n>0 。

Review
::回顾

For problems 1-4, the variable $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ vary inversely. Use the given $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ values to write an inverse variation equation and find $\begin{aligned} y \end{aligned}$ given that $\begin{aligned} x = 15 \end{aligned}$ .
::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 。

$\begin{aligned} x = 4, y = 3 \end{aligned}$
::x=4,y=3 x=4,y=3

$\begin{aligned} x = \frac{1}{5}, y = 10 \end{aligned}$
::x=15,y=10 x=15,y=10

$\begin{aligned} x = 8, y = \frac{3}{4} \end{aligned}$
::x=8,y=34x=8,y=34

$\begin{aligned} x = \frac{2}{3}, y = \frac{15}{8} \end{aligned}$
::x=23,y=158

For problems 5-8, the variable $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ vary inversely. Use the given $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ values to write an inverse variation equation and find $\begin{aligned} x \end{aligned}$ given that $\begin{aligned} y = 2 \end{aligned}$ .
::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 。

$\begin{aligned} x = 6, y = \frac{2}{3} \end{aligned}$
::x=6,y=23x=6,y=23

$\begin{aligned} x = 16, y = \frac{3}{8} \end{aligned}$
::x=16,y=38

$\begin{aligned} x = \frac{4}{5}, y = 9 \end{aligned}$
::x=45,y=9

$\begin{aligned} x = \frac{5}{6}, y = \frac{18}{5} \end{aligned}$
::x=56,y=185

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

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

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

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

Explore More
::探索更多

1. 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 $\begin{aligned} x \end{aligned}$ kids at the party.
::1. 在一个聚会上共有3个比萨饼,如果每份比萨饼有8片,确定如果12名儿童参加,每个儿童将得到多少块,如果8名儿童参加,会如何?写反式方程,以确定每个儿童得到多少片,如果在聚会上有x个孩子。

2. 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?
::2. Alena和Estella可以一起工作,在18小时内完成一份工作,如果他们请Tommy帮忙,工作需要多长时间?需要多少朋友一起工作,才能在4小时内完成?

3. As the distance between an object and earth increases, the gravitational force exerted by the Earth on the object decreases. In fact, the gravitational force, F, between an object and the Earth is inversely proportional to the square of the distance, d, from the object to the center of the Earth. ¹ If an astronaut weighs 180 pounds on the surface of Earth, what will he weigh 100 miles above the Earth? Assume that the radius of the Earth is 4,000 miles.
::3. 物体与地球之间的距离随着物体与地球之间的距离增加,地球对物体的引力减少,事实上,物体与地球之间的引力F与从物体到地球中心的距离正方形成反比。d,从物体到地球中心的距离。如果宇航员在地球表面的重量为180磅,那么他将比地球高出100英里?假设地球的半径是4,000英里。

4. Boyle’s Law states that if the temperature of a gas sample remains the same, the pressure of the gas varies inversely to the volume. ² A pressure of 1,600 lb per ft ² is exerted by 2 ft ³ of air in a cylinder. If a piston is pushed into the cylinder until the pressure is 2,400 lb per ft ² , what will be the volume of the air? Round to the nearest tenth.
::4. Boyle的法律规定,如果气体样品的温度保持不变,气体的压力与体积反差。 2 气瓶中的空气压力为每平方英尺1 600磅,每平方英尺1,600磅,如果活塞被推入气瓶,直到压力为每平方英尺2,400磅,空气的体积是多少?圆到最近的十分之一。

5. A scuba diver is using a diving tank that can hold 6 liters of air. If the tank has a pressure rating of 230 bar when full, use Boyle's Law to determine what is the pressure rating when the volume of gas is 4 liters?
::5. 潜水潜水员正在使用一个可容纳6升空气的潜水箱,如果该潜水箱在满时的压力等级为230巴,则使用Boyle法来确定气体量为4升时的压力等级是什么?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

1. "Newton's Law of Universal Gravitation," last edited May 21, 2017,
::1. 《纽顿宇宙引力法》, 2017年5月21日编辑,

2. "Boyle's Law," last edited May 15, 2017, https://en.wikipedia.org/wiki/Boyle%27s_law.
::2. “Boyle's Law”, 2017年5月15日最后编辑, https://en.wikipedia. org/wiki/Boyle%27s_law。

Section outline

Inverse Variation ::反反变化

Inverse Variation ::反反变化

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Feature: Good Vibrations ::特点:良好振动

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Inverse Variation
::反反变化

Inverse Variation
::反反变化

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Feature: Good Vibrations
::特点:良好振动

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX