Course: 6.8 简单理性模型-interactive

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简单理性模型

Lesson Objectives
::经验教训目标

Create rational equations in one variable and use them to solve problems.
::在一个变量中创建理性方程式, 并将其用于解决问题。

Identify the effect on the graph of a rational function of replacing $\begin{aligned} f (x) \end{aligned}$ by $\begin{aligned} f (x) + k \end{aligned}$ , $\begin{aligned} k f (x) \end{aligned}$ , $\begin{aligned} f (k x) \end{aligned}$ , and $\begin{aligned} f (x + k), \end{aligned}$ for specific values of $\begin{aligned} k \end{aligned}$ (both positive and negative) .
::以 f(x)+k、 kf(x)、 f(kx) 和 f(x+k) 替换 f(x) 和 f(x+k) 的逻辑函数, 以 k( 正和负) 具体值替换 f(x) 和 f(x+k) 。

Introduction: Harmonic Mean Revisited
::一. 导言:重新审视调和平均值

The section Solving Rational Equations explored the link between the harmonic mean and applications of rational functions. T he following two formulas are used to define the harmonic mean of two numbers, a and b:
::解析逻辑平方部分探讨了调和平均值与合理函数应用之间的联系。以下两个公式用于界定两个数字(a和b)的调和平均值:

\begin{aligned} Harmonic Mean = \frac{2}{\frac{1}{a} + \frac{1}{b}} or \frac{2 a b}{a + b} \end{aligned}

::热力平均值=21a+1b或2aba+b

The general definition of the harmonic mean for a set of n numbers is the following:
::一组n数字的调和平均值的一般定义如下:

\begin{aligned} Harmonic Mean = \frac{n}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + . . .} \end{aligned}

::调力平均值=n1a+1b+1c+...

T he h armonic mean of two numbers, a and b, is the number which is the same ratio away from a and b. A ssuming that b is the greater number, you can write an equation to represent this definition: "The ratio of the difference between the harmonic mean and a to a is equal to the ratio of the difference between the harmonic mean and b to b."
::两个数字(a和b)的调和平均值是与a和b的相同比率。假设b是较大数字,你可以写一个方程式来表示这个定义:“调和平均值与a之间的差比等于调和平均值与b至b之间的差比。”

\begin{aligned} \frac{m - a}{a} = \frac{b - m}{b} \end{aligned}

::m-aa=b-mb

Discussion Question : The variable $\begin{aligned} m \end{aligned}$ represents the harmonic mean. Can you prove that this definition is the same as the equations used above?
::讨论问题:变量m 代表调和平均值。您能否证明该定义与上述方程式相同?

Extension: Visualizing the Harmonic Mean
::扩展面: 视视波调平均值

The harmonic mean can be displayed on a circle along with the arithmetic mean, geometric mean, and root mean.
::口音平均值可以显示在一个圆圈上,加上算术平均值、几何平均值和根平均值。

Use the interactive below to derive the definition of each mean visually.
::使用下面的交互效果来得出每个平均值的视觉定义。

INTERACTIVE

Visualizing Means

Drag the red point to see how the means change when you change the length of the segments.

::拖曳红色点以查看当您更改区段长度时该值是如何变化的。

Click the buttons to see visual representations of different types of means.
- Note that a circle visual only applies if you are finding the mean of two numbers.
  ::请注意,如果您发现两个数字的平均值,则仅应用圆形视觉。
::单击按钮以查看不同类型手段的直观表达式。请注意,只有当您找到两个数字的平均值时,才适用圆的直观表达式。

Activity 1: Fitting Rational Models to D ata
::活动1:将合理模型与数据匹配

Throughout the chapter Rational Functions , you have explored some of the many applications of rational functions. One of those applications is modeling inversely proportional relationships with the reciprocal function $\begin{aligned} f (x) = \frac{1}{x} . \end{aligned}$ Many real-world studies return data that appears to be the result of a rational relationship. T ransformations of rational functions can be used to fit this data.
::在“理性函数”整个章节中,您已经探索了理性函数的多种应用中的一些应用。其中一种应用是模拟与对等函数f(x)=1x的反比例关系。许多现实世界研究的返回数据似乎是理性关系的结果。理性函数的转变可以用来适应这些数据。

The data in the interactive below can be modeled using the reciprocal function . Transform the reciprocal function to fit the data best.
::以下互动数据可用对等功能建模。将对等功能转换为最适合数据的对等功能。

INTERACTIVE

Rational Model Estimation To Scatter Plot

Move the sliders to manipulate the rational function.
::移动滑动器以操纵理性函数。

Press the Check button in order to compare your estimated rational function with the best fit function.
::按下“勾选”按钮,以便比较您估计的合理函数和最合适的函数。

Press the New Data button to try a new set.
::按新数据按钮尝试新数据集。

If you have wondered how some rational functions take on such a strange shape when graphed , recall that a rational function must be expressed as a ratio between two . A scenario could require either part of the ratio to be transformed.
::如果您想知道,当绘制图表时,一些理性函数如何以这种奇怪的形状出现,请记住,理性函数必须用两种函数之间的比例表示。假设情况可能要求改变该比例的任何一个部分。

Answer the questions below to practice writing transformations of the rational functions.
::回答下述问题,以实践对合理功能的写作转变。

Extension: Win Percentage
::扩展部分: 赢百分比

Example
::示例示例示例示例

Westlake High School and Marshall High School are rival basketball schools. Westlake has won 8 out of 1 5 games against Marshall going back five years. Westlake wants to improve their winning percentage to 80% against M arshall.
::韦斯特莱克高中和马歇尔高中是对手篮球学校。韦斯特莱克赢得了15场反对马歇尔的比赛中的8场比赛。威斯特莱克高中和马歇尔高中是对手篮球学校。韦斯特莱克五年前在与马歇尔的15场比赛中赢得了8场比赛。韦斯特莱克希望将其获胜率提高到80%,而马歇尔则赢得80 % 。

Use the interactive below to explore how wins and losses affect a team's win percentage.
::利用下面的互动来探究赢与亏对团队赢百分比的影响。

INTERACTIVE

Simple Rational Models

The numerator and denominator represent how many games the school's basketball team won and how many games they played respectively. Enter in the team's initial record and click the buttons to see what a win or a loss does to their overall win percentage.
::分子和分母代表学校篮球队赢得了多少场比赛, 以及他们分别玩过多少场比赛。输入球队的初始记录并单击按钮, 以了解一个或一个赢或一个输对总赢的百分比有何影响。

How many games would Westlake High School have to win in a row to get up to 80%? How can you apply a rational model to calculate this for them?
::Westlake高中要连续赢多少场比赛才能达到80%?如何使用理性模型来计算这些比赛?

Let $\begin{aligned} g \end{aligned}$ represent the number of games Westlake needs to win to raise their winning percentage to 80%. Westlake has won 8 out of 15 (or $\begin{aligned} \frac{8}{15} \end{aligned}$ ) so they would need to add $\begin{aligned} g \end{aligned}$ games to both the numerator and denominator. S et this fraction equal to their goal of 80%, and solve for $\begin{aligned} g . \end{aligned}$
::g 代表 Westlake 的游戏数量, 以将赢家百分比提高到80%。 Westlake 赢得了15场游戏中的8场比赛( 或815场比赛 ) 。因此他们需要在分子和分母中加入 g 游戏。设定此分数等于80%的目标, 并解决 g 。

\begin{aligned} \frac{8 + g}{15 + g} = 0.8 \end{aligned}

::8+g15+g=0.8

Write 0.8 as the simplified fraction $\begin{aligned} \frac{4}{5} \end{aligned}$ and cross multiply.
::将0.8写为简化分数45和交叉乘法。

Answer: Westlake High School will have to win the next 20 consecutive games to raise their win percentage to 80%.
::回答:Westlake高中必须连续赢得20场比赛,

Discussion Questions:
::讨论问题:

Marshall has won $\begin{aligned} \frac{7}{15} \end{aligned}$ against Westlake dating back five years and wants to get up to 60%. How many games would Marshall have to win in a row to get up to 60%?
::马歇尔五年前赢得了715场Westlake比赛,并想获得60%。马歇尔要赢得多少场比赛才能赢得60%?

After Westlake reached a winning percentage of 80%, how many games in a row can they lose in a row and still maintain a 5 0 % winning percentage? How would this change the approach used to answer the previous question?
::Westlake达到80%的赢率后,他们连续输输多少场游戏,仍然能保持50%的赢率? 这如何改变用来回答上一个问题的方法呢?

Activity 2: Moving Company
::活动2:移动公司

Example
::示例示例示例示例

Maya, Jason, and Tyrone have a small moving company. Tyrone thinks he's the fastest member of the team. As a group, it takes them 2 hours to fill their moving truck. When Maya works alone, it takes her 4 hours, and when Jason works alone, it takes him 6 hours. Is Tyrone right about his truck-filling skills?
::马雅、杰森和蒂龙有一个小型的移动公司。蒂龙认为他是队伍中最快的成员。作为一个团队,他们需要两个小时来填满他们的移动卡车。当玛雅单独工作时,需要四个小时,杰森单独工作时,需要六个小时。蒂龙是否认为他的卡车装货能力是正确的?

This problem is a variation of the pool problem seen in Solving Rational Equations . Think about how long it would take each person to fill the moving truck in one hour. If Maya can fill the truck in 4 hours, her rate is 1 truck in four hours, or $\begin{aligned} \frac{1}{4} \end{aligned}$ of a truck in one hour. Jason's rate would be $\begin{aligned} \frac{1}{6} \end{aligned}$ of a truck in one hour. You don't know Tyrone's rate, so if $\begin{aligned} t \end{aligned}$ = the number of hours it takes Tyrone to fill the truck, his rate is $\begin{aligned} \frac{1}{t} \end{aligned}$ of a truck per hour. A s a team, their rate is $\begin{aligned} \frac{1}{2} \end{aligned}$ of a truck in one hour. Adding together, Maya, Jason, and Tyrone's rates will be equal to the rate of their group.
::这个问题在解决合理方程中是一个不同的问题。想想每个人用一个小时来填满移动的卡车需要多长时间。如果玛雅能在4小时内填满卡车, 她的车价是4小时内装满1辆卡车, 或1小时内装满14辆卡车。杰森的车价是1小时内一辆卡车的16辆。您不知道泰龙的车价, 所以如果泰龙要填满卡车需要多少小时, 那么他的车价是每小时一辆卡车的1吨。作为团队, 他们每小时的车价是12辆卡车的12辆。加在一起, 玛雅、杰森和蒂龙的车价将相当于他们组的车价。

Activity 3: Mixtures
::活动3:混合物

Marco is conducting an experiment in his chemistry class. He has one solution that contains a 20% solution and another that contains a 30% solution.
::马可正在化学课上做实验。他有一个包含20%解决方案的解决方案,另一个包含30%解决方案的解决方案。

a. How many liters of the 30% solution would need to be added to 1 liter of the 20% solution to create a new mixture that is 25% water?
::a. 创造一种25%的水的新混合物,在20%解决办法的1升中,需要增加多少升的30%解决办法?

Let $\begin{aligned} L \end{aligned}$ equal the number of liters from the 30% solution you'll need to add to the 20% solution. Therefore , the new mixture will be $\begin{aligned} (1 + L) \end{aligned}$ liters which will contain $\begin{aligned} (1 \cdot 20 %) + (L \cdot 30 %) \end{aligned}$ water. S et up and solve this equation for $\begin{aligned} L : \end{aligned}$
::LetL 等量30%溶液的升数, 您需要加到20%溶液中。因此, 新混合的升数将是 1+L 升, 其中将包含 (1+20%) + (L+- 30%) 水。设置并解析 L 的这个方程式 :

\begin{aligned} \frac{0.2 + 0.3 L}{1 + L} & = 0.25 \\ 0.2 + 0.3 L & = 0.25 (1 + L) \\ 0.2 + 0.3 L & = 0.25 + 0.25 L \\ 0.05 L & = 0.05 \\ L & = 1 \end{aligned}

::0.250.2+0.3L1+L=0.250.2+0.3L=0.25(1+L)0.2+0.3L=0.25+0.25L0.05L=0.05L=1

Focusing on the expression $\begin{aligned} \frac{0.2 \cdot 1 + 0.3 \cdot L}{1 + L}, \end{aligned}$ you see that as the value of $\begin{aligned} L \end{aligned}$ increases , both $\begin{aligned} 0.2 + 0.3 L \end{aligned}$ and $\begin{aligned} 1 + L \end{aligned}$ change. Since the numerator is increasing at a rate of $\begin{aligned} .3 L \end{aligned}$ and the denominator is increasing at a rate of $\begin{aligned} L, \end{aligned}$ the value of $\begin{aligned} \frac{0.2 + 0.3 L}{1 + L} \end{aligned}$ will increase as L increases. However, since the denominator increases at a faster rate than the numerator, the rate of increase will decrease as L approaches infinity .
::以表达式0.21+0.3+3.3L1+L为重点,你可以看到,随着L值的增加, 0.2+0.3L和1+L值的变动。由于分子以0.3L的速率增长,分母以0.3L的速率增长,0.2+0.3L1+L值将随着L的增加而增加。然而,由于分母的增速比分子的速率增长更快,随着L接近无限度,增速将下降。

b. Generalize your approach to solving this problem in a way that would enable you to determine the amount of the 30% solution that would need to be added to the 20% solution to produce a new solution that is $\begin{aligned} w \end{aligned}$ % water. Are there any values for $\begin{aligned} w \end{aligned}$ that you could rule out?
::b. 以能够使您确定在20%解决方案中需要增加的30%解决方案的多少,从而产生一种W%水的新解决方案。您能否排除任何价值?

If you wanted an equation for a solution with $\begin{aligned} w \end{aligned}$ % water, you would replace 0.25 (a 25% solution) with $\begin{aligned} \frac{w}{100} \end{aligned}$ or $\begin{aligned} 0.01 w \end{aligned}$ (a $\begin{aligned} w \end{aligned}$ % solution) and solve for $\begin{aligned} w \end{aligned}$ . Additionally, because one solution is 20% water and the other is 30% water, any value you determine for $\begin{aligned} w \end{aligned}$ has to be between these percentages. You'll never get a solution that is less than 20% water nor a solution that is greater than 30% water.
::如果您想要一个以 w% 水溶解的方程式, 您将会用 w100 或 0.01w (a w% 溶解) 替换0. 25 (a 25% 溶解) , 并解决 w。此外, 因为一个溶解是 20% 水, 而另一个溶解是 30% 水, 您决定的任何数值必须介于这些百分比之间。您将得不到一个低于 20% 的水溶解, 或一个超过 30% 的水溶解。

Extension: Plane Ride
::扩展部分: 乘机

A plane is flying into the wind at a speed of 400 miles per hour for the first 2,000 miles of a 3,000-mile trip. What speed does the plane need to average for the remaining 1,000 miles to have an average speed of 500 miles per hour for the entire trip?
::飞机以每小时400英里的速度飞入风中,这是3000英里行程中头2 000英里的行程。飞机平均需要多少速度才能达到其余1 000英里,整个行程的平均速度为每小时500英里?

First, recall that $\begin{aligned} distance = rate \cdot time. \end{aligned}$ The table below compares the distance , rate, and time for the two parts of the flight and the overall trip. Fill in the missing information:
::首先,请记住这个距离=时间。下表比较了飞行的两个部分和整个行程的距离、速度和时间。填写缺失的信息:

INTERACTIVE

Average Flight Speed

Fill in the blanks with the correct values to complete the table.
::填空时填入填表的正确值。

You know that by adding the two parts of the trip, you'll get the overall trip. How can you use this information to set up an equation and solve for $\begin{aligned} x ? \end{aligned}$
::你知道,如果加上行程的两个部分, 你会得到整个行程。你怎么能使用这些信息来设置一个方程和解答 x ?

Summary

The harmonic mean of two numbers $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ is defined by the following formulas: $\begin{aligned} \frac{2 a b}{a + b} \end{aligned}$ and $\begin{aligned} \frac{n}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + . . .} \end{aligned}$ for a set of n numbers.
::两个数字a和b的调和平均值由以下公式定义:对于一组n数字,为 2aba+b 和 n1a+1b+1c+...。

Rational functions can be used to model inversely proportional relationships, such as the reciprocal function $\begin{aligned} y = \frac{1}{x} . \end{aligned}$
::理性函数可用于模拟反比例关系,如对等函数y=1x。

Section outline

General

简单理性模型

Introduction: Harmonic Mean Revisited ::一. 导言:重新审视调和平均值

Extension: Visualizing the Harmonic Mean ::扩展面: 视视波调平均值

Activity 1: Fitting Rational Models to D ata ::活动1:将合理模型与数据匹配

Extension: Win Percentage ::扩展部分: 赢百分比

Activity 2: Moving Company ::活动2:移动公司

Activity 3: Mixtures ::活动3:混合物

Extension: Plane Ride ::扩展部分: 乘机

Wrap-Up: Review Questions ::总结:审查问题

Introduction: Harmonic Mean Revisited
::一. 导言:重新审视调和平均值

Extension: Visualizing the Harmonic Mean
::扩展面: 视视波调平均值

Activity 1: Fitting Rational Models to D ata
::活动1:将合理模型与数据匹配

Extension: Win Percentage
::扩展部分: 赢百分比

Activity 2: Moving Company
::活动2:移动公司

Activity 3: Mixtures
::活动3:混合物

Extension: Plane Ride
::扩展部分: 乘机

Wrap-Up: Review Questions
::总结:审查问题