Course: 5.10 联合和合并变动

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联合和合并变化
After the dog below rolls around on their bed for awhile, the electrons in their hair transfer to the bed and all of their hair has a positive charge. Since like charges repel, individual strands of hair move away from other strands of hair creating the static electricity hair look below.
::在下面的狗在床上翻滚一段时间后,发型中的电子转移到床上,他们所有的头发都有正电荷。由于像电荷还原一样,单根头发从其他发型中移开,形成静电头发的静电发直观下面。

Using Coulomb's Law, we know that the force exerted on each strand of hair is directly proportional to the charges on each hair and inversely proportional to the square of the distance between the two strands of hair. ¹ Coulomb's Law is an example of combined and joint variation that we discuss in this section.
::使用库伦法,我们知道对每一根头发施加的强度与每根头发的收费直接成比例,与两根头发之间的距离平方反成比例。 1 库伦法是我们在本节讨论的综合和共同变化的例子。

Joint Variation
::联合变动

Joint variation is similar to , but in this case, the dependent variable is equal to product of more than one independent variable and a constant or proportionality or variation.
::联合变异与......类似,但在这种情况下,依赖变量等于一个以上独立变异的产物,以及恒定的或相称的或变异。

Joint Variation
::联合变动

$\begin{aligned} z = k x y & or z = k x^{m} y^{n} \\ k \neq 0, & m > 0, n > 0 \end{aligned}$

::zz=kxyolz=kxmynk0, m>0, n>0

We say " z varies jointly as x and y. " or " z is jointly proportional to x and y ."
::我们说"Z和y是不同的" 或者"Z和y是成比例的"

Example 1
::例1

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 3, y = 8, \end{aligned}$ and $\begin{aligned} z = 6 \end{aligned}$ , find the variation equation . Then, find $\begin{aligned} z \end{aligned}$ when $\begin{aligned} x = - 2 \end{aligned}$ and $\begin{aligned} y = 10 \end{aligned}$ .
::如果 x=3,y=8,z=6,则查找变异方程。然后,在 x=2 和 y=10 时查找z。

Solution: Using the equation when it is solved for $\begin{aligned} k \end{aligned}$ , we have:
::解答: 当 k 解答时使用方程式, 我们有:

$\begin{aligned} k = \frac{z}{x y} = \frac{6}{3 \cdot 8} = \frac{1}{4} \end{aligned}$

::k=zxy=63_8=14

The equation is $\begin{aligned} z = \frac{1}{4} x y \end{aligned}$ .
::方程是z=14xy。

When $\begin{aligned} x = - 2 \end{aligned}$ and $\begin{aligned} y = 10 \end{aligned}$ , then $\begin{aligned} z = \frac{1}{4} \cdot - 2 \cdot 10 = - 5 \end{aligned}$ .
::当 x2 和 y= 10, 然后z= 14 210 5 时。

by Mathispower4u explains joint variation and how to determine the variation constant.
::Mathispower4u 解释联合变异和如何确定变异常数。

Example 2
::例2

The volume of a pyramid varies jointly with the area of the base and the height with a constant of variation of $\begin{aligned} \frac{1}{3} \end{aligned}$ . If the volume is $\begin{aligned} 162 {units}^{3} \end{aligned}$ and the area of the base is $\begin{aligned} 81 {units}^{2} \end{aligned}$ , find the height.
::金字塔的体积与基底面积和高度不一而足,常变13。如果体积为162个单位3,基底面积为81个单位2,则寻找高度。

Solution: Find the joint variation equation first.
::解决方案:先找到联合变异方程式。

$\begin{aligned} V = \frac{1}{3} B h \end{aligned}$

::V=13 小时

Now, substitute in the area of the base to solve for the height.
::现在,在基地区域替换,解决高度问题。

$\begin{aligned} 162 & = \frac{1}{3} \cdot 81 \cdot h \\ 162 & = 27 h \\ 6 & = h \end{aligned}$

::162=1381h162=27 h6=h

Example 3
::例3

Kinetic energy $\begin{aligned} K E \end{aligned}$ (the energy something possesses due to being in motion) varies jointly with the mass $\begin{aligned} m \end{aligned}$ (in kilograms) of that object and the square of the velocity $\begin{aligned} v \end{aligned}$ (in meters per seconds) ² . The constant of variation is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::2. 变化的常数为12。

a. Write the equation for kinetic energy.
::a. 写动能方程式。

b. If a car is travelling 104 kilometers/hour and weighs 8,800 kg, what is its kinetic energy?
::b. 如果汽车每小时行驶104公里,重8 800公斤,其动能如何?

Solution:
::解决方案 :

a. $\begin{aligned} K E = \frac{1}{2} m v^{2} \end{aligned}$
::a. KE=12 mv2

b. First, we have to convert the km/hr into meters per second.
::b. 第一,我们必须将公里/小时转换为每秒每米。

$\begin{aligned} \frac{104 k m}{h r} \cdot \frac{h r}{3600 s} \cdot \frac{1000 m}{k m} = 0.44 \frac{m}{s} \end{aligned}$

::104公里3600秒1000公里=0.44米

Now, plug this into the equation from part a.
::现在,把这个插进A部分的方程中

$\begin{aligned} K E & = \frac{1}{2} \cdot 8800 k g \cdot {(0.44 \frac{m}{s})}^{2} \\ = 1955.56 \frac{k g \cdot m^{2}}{s^{2}} \end{aligned}$

::KE=128800公斤 (0.44毫西米)2=1955.56 kg_m2s2

Typically, the unit of measurement of kinetic energy is called a joule. A joule is $\begin{aligned} \frac{k g \cdot m^{2}}{s^{2}} \end{aligned}$ .
::通常,动能测量单位称为焦耳,焦耳为kgm2s2。

Combined Variation
::合并变化

Many situations involve more than one type of variation. Combined variation is a combination of direct, inverse , and joint variation. For example, the sales of a product may be directly proportional to the amount of money spent on advertising the product, but inversely proportional to the price of the product.
::许多情况涉及一种以上的变异,合并变异是直接的、反的和联合变异的组合,例如,产品的销售可能直接与广告产品所花钱的数额成正比,但与产品价格成反比。

Example 5
::例5

Write an equation for the given relationships.
::写入给定关系的方程。

a. $\begin{aligned} y \end{aligned}$ varies inversely with the square of $\begin{aligned} x \end{aligned}$ .
::a. y 与 x 平方反差。

b. $\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and the square root of $\begin{aligned} y \end{aligned}$ .
::b. z 与x和y的平方根并存。

c. $\begin{aligned} z \end{aligned}$ varies directly with $\begin{aligned} x \end{aligned}$ and inversely with $\begin{aligned} y \end{aligned}$ .
::c. z 与 x 直接不同,y 反差。

Solution:
::解决方案 :

a. $\begin{aligned} y = \frac{k}{x^{2}} \end{aligned}$
::a. y=kx2

b) $\begin{aligned} z = k x \sqrt{y} \end{aligned}$
::b) z =kxy

c) $\begin{aligned} z = \frac{k x}{y} \end{aligned}$
::c) z=kxy

by Barb Koehler explains combined variation .
::Barb Koehler解释综合变异。

by Joshua Helston shows a real-world example of combined variation.
::Joshua Helston展示了一个世界性综合变异的例子。

Summary
::摘要

Combined variation is a mix of direct and indirect variation.
::合并变异是直接和间接变异的混合体。

The joint variation equation is $\begin{aligned} z = k x^{m} y^{n} \end{aligned}$ where $\begin{aligned} k \neq 0 \end{aligned}$ and $\begin{aligned} m > 0, n > 0 \end{aligned}$ .
::联合变异方程式是 z=kxmyn, k0 和 m>0,n>0 。

Review
::回顾

For questions 1-5, write an equation that represents relationship between the variables.
::对于问题1-5,写一个代表变量之间关系的方程式。

$\begin{aligned} w \end{aligned}$ varies inversely with respect to $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::x 和 y 的逆差。

$\begin{aligned} r \end{aligned}$ varies inversely with the square of $\begin{aligned} q \end{aligned}$ .
::r 与q平方反差。

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ and inversely with $\begin{aligned} w \end{aligned}$ .
::z 与 x 和 y 合并变化,与 w 反向变化。

$\begin{aligned} a \end{aligned}$ varies directly with $\begin{aligned} b \end{aligned}$ and inversely with $\begin{aligned} c \end{aligned}$ and the square root of $\begin{aligned} d \end{aligned}$ .
::a 与 b 直接不同,与 c 和 d 的平方根相反。

$\begin{aligned} l \end{aligned}$ varies directly with $\begin{aligned} m \end{aligned}$ , and inversely with $\begin{aligned} p \end{aligned}$ .
::我与M直接不同,与p相反。

Write the variation equation and answer the given question in each problem.
::写下变异方程并回答每个问题中的问题。

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 2, y = 3 \end{aligned}$ and $\begin{aligned} z = 4 \end{aligned}$ , write the variation equation and find $\begin{aligned} z \end{aligned}$ when $\begin{aligned} x = - 6 \end{aligned}$ and $\begin{aligned} y = 2 \end{aligned}$ .
::如果 x=2,y=3 和 z=4, 则在 x=6 和 y=2 时, 刻写变异方程并找到 z。

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 5 \end{aligned}$ , $\begin{aligned} y = - 1 \end{aligned}$ and $\begin{aligned} z = 10 \end{aligned}$ , write the variation equation and find $\begin{aligned} z \end{aligned}$ when $\begin{aligned} x = - \frac{1}{2} \end{aligned}$ and $\begin{aligned} y = 7 \end{aligned}$ .
::z 与 x 和 y 相异。如果 x= 5, y 1 和 z= 10, 请写入变异方程, 并在 x 12 和 y = 7 时找到 z 。

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 7 \end{aligned}$ , $\begin{aligned} y = 3 \end{aligned}$ and $\begin{aligned} z = - 14 \end{aligned}$ , write the variation equation and find $\begin{aligned} y \end{aligned}$ when $\begin{aligned} z = - 8 \end{aligned}$ and $\begin{aligned} x = 3 \end{aligned}$ .
::z 与 x 和 y 相异。如果 x= 7, y= 3 和 z 14, 请写变量方程, 并在 z 8 和 x = 3 时查找 y 。

$\begin{aligned} z \end{aligned}$ varies jointly with $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 8 \end{aligned}$ , $\begin{aligned} y = - 3 \end{aligned}$ and $\begin{aligned} z = - 6 \end{aligned}$ , write the variation equation and find $\begin{aligned} x \end{aligned}$ when $\begin{aligned} z = 12 \end{aligned}$ and $\begin{aligned} y = - 16 \end{aligned}$ .
::z 与 x 和 y 相异。如果 x= 8, y 3 和 z 6, 写变量方程, 并在 z = 12 和 y = 16 时找到 x 。

$\begin{aligned} z \end{aligned}$ varies inversely with $\begin{aligned} x \end{aligned}$ and directly with $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = 4 \end{aligned}$ , $\begin{aligned} y = 48 \end{aligned}$ and $\begin{aligned} z = - 2 \end{aligned}$ , write the variation equation and find $\begin{aligned} x \end{aligned}$ when $\begin{aligned} z = 8 \end{aligned}$ and $\begin{aligned} y = 96 \end{aligned}$ .
::如果 x=4, y=48 和 z2, 写变量方程式, 然后在 z=8 和 y=96 时找到 x 。

$\begin{aligned} z \end{aligned}$ varies inversely with $\begin{aligned} x \end{aligned}$ and directly with $\begin{aligned} y \end{aligned}$ . If $\begin{aligned} x = \frac{1}{2} \end{aligned}$ , $\begin{aligned} y = 5 \end{aligned}$ and $\begin{aligned} z = 20 \end{aligned}$ , write the variation equation and find $\begin{aligned} x \end{aligned}$ when $\begin{aligned} z = - 4 \end{aligned}$ and $\begin{aligned} y = 8 \end{aligned}$ .
::z 和 x 和 y 的反向变化。如果 x=12, y=5 和 z=20, 请写入变异方程, 并在 z 4 和 y=8 时找到 x 。

Explore More
::探索更多

1. If 20 volunteers can wash 100 cars in 2.5 hours, find the constant of variation and find out how many cars 30 volunteers can wash in 3 hours.
::1. 如果20名志愿人员能够在2.5小时内洗100辆汽车,找到变化的常数,并找出3小时内30名志愿人员可以洗多少辆汽车。

2. If 10 students from the environmental club can clean up trash on a 2-mile stretch of road in 1 hour, find the constant of variation and determine how low it will take to clean the same stretch of road if only 8 students show up to help.
::2. 如果环境俱乐部的10名学生能在1小时内在2英里长的公路上清理垃圾,发现变异的常态,并确定如果只有8名学生到场帮忙,那么清理同一路段需要多少时间。

3. The work $\begin{aligned} W \end{aligned}$ (in joules) done when lifting an object varies jointly with the mass $\begin{aligned} m \end{aligned}$ (in kilograms) of the object and the height $\begin{aligned} h \end{aligned}$ (in meters) that the object is lifted ³ . The work done when a 100 kilogram object is lifted 1.5 meters is 1,470 joules. Write an equation that relates $\begin{aligned} W, m, \end{aligned}$ and $\begin{aligned} h \end{aligned}$ . How much work is done when lifting a 150 kilogram object 2 meters?
::3. 物体升起时完成的工作W(约尔斯)与物体质量m(公斤)和物体升起时的高度h(米)不尽相同。100公斤物体升起1.5米时完成的工作为1,470焦耳。写一个与W,m和h有关的方程式。举起150公斤物体2米时做了多少工作?

4. The intensity $\begin{aligned} I \end{aligned}$ of a sound (in watts per square meter) varies inversely with the square of the distance $\begin{aligned} d \end{aligned}$ (in meters) from the sound’s source ⁴ . At a distance of 1.5 meters from the stage, the intensity of the sound at a rock concert is about 9 watts per square meter. Write an equation relating $\begin{aligned} I \end{aligned}$ and $\begin{aligned} d \end{aligned}$ . If you are sitting 10 meters back from the stage, what is the intensity of the sound you hear?
::4. 声音的强度I(按每平方米瓦特计算)与声源的距离(以米计)4 的正方形反差4,距离舞台1.5米,摇滚音乐会的声音强度约为每平方米9瓦特。写一个与I和d有关的方程式。如果你坐在距离舞台10米的后方,听到的声音的强度是多少?

5. The volume of a cylinder varies jointly as its height and the square of its radius . What is the value of the constant of variation if a cylinder has the following approximate measurements: V = 754 ft ³ , r = 4 ft, h = 15 ft. Round your answer to two decimal places.
::5. 圆柱体的体积随其高度和半径的方形而异,如果圆柱体有以下近似测量值:V=754平方英尺,r=4平方英尺,h=15平方英尺。

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. "Coulomb's Law," last edited May 16, 2017,
::1. 2017年5月16日编辑的《库伦姆法》,

2. "Kinetic Energy," last edited May 15, 2017,
::2. 2017年5月15日上一次编辑的"营养能源"

3."Work (physics)," last edited May 22, 2017,
::"3Work(物理),"上次编辑 2017年5月22日,2017年5月22日,

4. "Sound Intensity," last edited February 5, 2017, https://en.wikipedia.org/wiki/Sound_intensity.
::4. “健康强度”, 2017年2月5日编辑, https://en.wikipedia.org/wiki/Sound_intensity。

Section outline

General

联合和合并变化

Joint Variation ::联合变动

Joint Variation ::联合变动

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Combined Variation ::合并变化

Example 5 ::例5

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

References ::参考参考资料

Joint Variation
::联合变动

Joint Variation
::联合变动

Example 1
::例1

Example 2
::例2

Example 3
::例3

Combined Variation
::合并变化

Example 5
::例5

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

References
::参考参考资料