9.6 利用最不共同的分数来解决逻辑赤道

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  使用最不常见的分数符号解决合理等号

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  When an emergency vehicle passes while you are standing on the sidewalk, it sounds like the siren makes two different sounds—one as it approaches you, and another as it goes away. This change can be explained by the Doppler Effect ¹ , which says that as the vehicle approaches you, the sound waves bunch together so you hear them at a higher frequency, and as the vehicle moves away, the sounds waves are further apart so you hear them at a lower frequency. We can measure these frequencies with a formula ,  $\begin{align*}f=\bigg(\frac{c}{c+v_s} \bigg) f_0\end{align*}$ .
  ::当你站在人行道上时,一辆紧急车辆经过时,听起来就像警笛发出两种不同的声音——一种是接近你的声音,另一种是消失的声音。这个变化可以用多普勒效应1 来解释。它说,当车辆靠近你时,声波会聚集在一起,这样你就能听到更高频率的声音,随着车辆移动,声波会进一步分开,这样你就能以更低的频率听到声音。我们可以用一种公式(f=(cc+vs)f0)测量这些频率。

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  In this section, we discuss how to solve equations with rational expressions by using the LCD.  
  ::在本节中,我们讨论如何通过使用液晶显示器以理性表达方式解决方程式问题。

  

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  Solving Rational Equations Using the LCD
  ::使用液晶解析理衡

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  W e can also use the LCD of all the rational expressions within the equation and to eliminate the fractions.
  ::我们还可以使用方程式中所有合理表达方式的液晶,消除分数。

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     How to Solve Rational Equations Using the LCD
  ::如何使用液晶解析法解析合理等式

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  If  $\begin{align*}\frac{A}{C}=\frac{B}{C}\end{align*}$ , then  $\begin{align*}A=B\end{align*}$ .
  ::如果AC=BC,那么A=B

   

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  1. Find the LCD.
  ::1. 找到液晶显示器。

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  2. Multiply both sides of the equation by the LCD in factored form .
  ::2. 将等式的两侧乘以因子形式液晶体。

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  3. Solve the equations.
  ::3. 解决方程式问题。

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  4.  Check for extraneous solutions—solutions we found that cannot be substituted into the equation.
  ::4. 检查不相干的解决办法——我们发现无法替代为等式的解决办法。

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  To demonstrate, we will look at  a few examples.
  ::为了表明这一点,我们将举几个例子。

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  Example 1
  ::例1

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  Solve  $\begin{align*}\frac{2x}{x-3}=\frac{3x}{x^2-9}\end{align*}$ . 
  ::溶解 2xx-3=3x22-9。

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  Solution: The LCD is $\begin{align*}x^2-9,\end{align*}$  since  $\begin{align*}x-3\end{align*}$  is a factor of  $\begin{align*}x^2-9\end{align*}$ . Next, we multiply both sides of the equation by the LCD in factored form,  $\begin{align*}(x - 3) (x +3) \end{align*}$ . Notice how when we multiply both sides by the LCD, we eliminate the fractions from our equation. 
  ::解析度:LCD是 x2-9, 因为X-3 是 x2- 9. 系数。 其次, 我们以因素化的形式将方程式的两边乘以LCD, (x-3 (x+3)) 注意当我们将两边乘以LCD时, 我们如何消除方程式中的分数 。

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  $$\begin{align*}\frac{2x}{x-3}&=\frac{3x}{x^2-9} \\ {\color{green}\frac{\cancel{\left(x-3\right)}\left(x+3\right)}{1}} \cdot \frac{2x}{\cancel{x-3}}&= {\color{green}\frac{\cancel{\left(x-3\right)\left(x+3\right)}}{1}\cdot} \frac{3x}{\cancel{x^2-9}} \\ 2x(x+3)&=3x \\ 2x^2+6x&=3x\\ 2x^2+3x&=0 \\ x(2x+3)&=0 \\ x=0 & \quad x=\text{-}\frac{3}{2} \\\end{align*}$$
  ::2xx-3=3x2-9(x-3)(x+3)1x2x-3=(x-3)(x+3)3=(x-3)(x+3)1x3x3xxx-2-92x(x+3)=3x2x2+6x=3x2x2x2+3xx=0xx3=0x3)=0x=0x=-32

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  Checking our answers, we have
  ::检查我们的答案,我们有

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  $$\begin{align*}x=0 && x=\text{-}\frac{3}{2} \\ \\ \frac{2(0)}{0-3}=\frac{3(0)}{0^2-9} && \frac{2(\text{-}\frac{3}{2})}{\text{-}\frac{3}{2}-3}=\frac{3(\text{-}\frac{3}{2})}{(\text{-}\frac{3}{2})^2-9}\\ \frac{0}{\text{-}3}=\frac{0}{\text{-}9} && \frac{\text{-}3}{\text{-}\frac{9}{2}}=\frac{\text{-}\frac{9}{2}}{\text{-}\frac{27}{4}} \\ 0=0 && \frac{6}{9}=\frac{18}{27}\\ &&\frac{2}{3}=\frac{2}{3}\end{align*}$$
  ::x0x=- 322(0)0-3=-33(0)02-92(32-32-3=3(32)(32)(32)2-90-3=-0-9-3-92=-9-22740=-069=182723=23)

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  So, indeed, both are solutions to the rational equation .
  ::因此,事实上,两者都是理性等式的解决办法。

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  Example 2
  ::例2

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  Solve $\begin{align*}\frac{5x}{x-2}=\frac{10}{x-2}\end{align*}$ .
  ::解决 5xx-2=10x-2。

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  Solution: Because the denominators are the same, we need to multiply both expressions by $\begin{align*}x-2\end{align*}$ .
  ::解答: 因为分母是相同的, 我们需要将两个表达式乘以 x-2 。

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  $$\begin{align*}\frac{5x}{x-2}&=\frac{10}{x-2} \\ {\color{green}\cancel{(x-2)}} \cdot \frac{5x}{\cancel{x-2}}&={\color{green}\cancel{(x-2)}} \cdot \frac{10}{\cancel{x-2}} \\ 5x&=10 \\ x&=2\end{align*}$$
  ::5x-2=10x-2(x-2)5x-2(x-2)5x-2=(x-2)10x-25x=10x=2

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  Checking our answer, we have  $\begin{align*}\frac{5 \cdot 2}{2-2}=\frac{10}{2-2} \rightarrow \frac{10}{0}=\frac{10}{0}\end{align*}$ . Ten divided by 0 is undefined , so $\begin{align*}x=2\end{align*}$  cannot be a solution. Notice that 2 is the value that makes the denominator equal to 0. If we solve a rational equation and one of the "solutions" makes the denominator of any of the fractions equal 0, we reject it because it is an  extraneous solution . Therefore , there is no solution to this problem.
  ::检查我们的答案时, 我们有 5\ 22-2= 102- 2100=100。 10 除以 0 是未定义的, 所以 x=2 不能是一个解决方案 。 注意 2 是使分母等于 0 的值 。 如果我们解决了一个合理方程, 而一个“ 解决方案” 使任何分数的分数等于 0, 我们拒绝它, 因为它是一个外在的解决方案 。 因此, 这个问题没有解决办法 。

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  by Mathispower4u demonstrates how to solve rational equations.
  ::Mathispower4u 演示如何解析理性方程式。

   

   

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  Example 3
  ::例3

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  Solve $\begin{align*}\frac{5}{2}+\frac{1}{x}=3\end{align*}$ .
  ::解决52+1x=3。

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  Solution: The LCD for 2,  $\begin{align*}x\end{align*}$ , and 1 is $\begin{align*}2x\end{align*}$ . Multiply each term by $\begin{align*}2x\end{align*}$ , so the fractions  are eliminated. 
  ::解析度: 2, x 和 1 的LCD 是 2x。 将每个术语乘以 2x, 所以分数将被删除 。

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  $$\begin{align*}\frac{5}{2}+ \frac{1}{x}&=3 \\ {\color{green}\cancel{2}x} \cdot \frac{5}{\cancel{2}}+ {\color{green}2\cancel{x}} \cdot \frac{1}{\cancel{x}}&=2x \cdot 3 \\ 5x+2&=6x \\ 2&=x\end{align*}$$
  ::52+1x=32x=52=2x=1x1x=2x=3x35x+2=6x2=x

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  Checking the answer, we have $\begin{align*}\frac{5}{2}+ \frac{1}{2}=3 \rightarrow \frac{6}{2}=3 \end{align*}$ .
  ::检查答案,我们有52+12=362=3。

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  Example 4
  ::例4

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  Solve   $\begin{align*}\frac{4}{x-3}=\frac{9}{x+2}-5\end{align*}$ .
  ::解决 4x-3=9x+2-5。

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  Solution:  The LCD is $\begin{align*}(x-3)(x+2)\end{align*}$ . Multiply each term by the LCD.
  ::解答:LCD是(x-3)(x+2)。 用LCD乘以每个术语。

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  $$\begin{align*}\frac{4}{x-3}&=\frac{9}{x+2}-5 \\ {\color{green}\cancel{(x-3)}(x+2)} \cdot \frac{4}{\cancel{x-3}}&={\color{green}(x-3)\cancel{(x+2)}} \cdot \frac{9}{\cancel{x+2}} - {\color{green}(x-3)(x+2)} \cdot 5\\ 4(x+2)&=9(x-3) -5(x-3)(x+2)\\ 4x+8&=9x-27-5x^2+5x+30 \\ 5x^2-10x+5&=0 \\ 5(x^2-2x+1)&=0 \\\end{align*}$$
  ::4x-3=9x+2-5(x-3)(x-3)(x+2)4x-3=(x-3)(x+2)(x+2)9x+2(x-3)(x-3)(x-3)(x+3)(x+2)_5(x+2)5(x+2)=9(x-3)(x+2)(x+2)4x8(8)(x+2)4x+8(9x-27-55x2(x)+5x+305-2-10x+5=05(x2-x+1)=0

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  This polynomial factors to be $\begin{align*}5(x-1)(x-1)=0\end{align*}$ , so $\begin{align*}x = 1\end{align*}$ is a repeated solution. Checking our answer, we have $\begin{align*}\frac{4}{1-3}=\frac{9}{1+2} -5 \rightarrow \text{-}2=3 -5 \rightarrow \text{-}2=\text{-}2\end{align*}$ .
  ::5( x-1)( x-1) =0, 所以 x=1 是一个重复的解答 。 检查我们的答案, 我们有 41- 3 = 91+2- 5-2= 3- 5-2= 2 。

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  by CK-12 demonstrates how to solve rational equations.
  ::由 CK-12 演示如何解析理性方程式 。

   

   

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  Example 5
  ::例5

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  Solve $\begin{align*}\frac{3}{x^2+4x+4}+ \frac{1}{x+2}=\frac{2}{x^2-4}\end{align*}$ .
  ::溶解 3x2+4x+4+1x+2=2x2-4。

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  Solution: The LCD is $\begin{align*}(x+2)(x+2)(x-2)\end{align*}$ .
  ::解答:液晶显示器为(x+2)(x+2)(x-2)(x-2)。

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  $$\begin{align*}&\frac{3}{x^2+4x+4}+ \frac{1}{x+2}=\frac{2}{x^2-4} \\ &{\color{green}\cancel{(x+2)(x+2)}(x-2)} \cdot \frac{3}{\cancel{\left(x+2\right)\left(x+2\right)}}+ {\color{green}\cancel{(x+2)}(x+2)(x-2)} \cdot \frac{1}{\cancel{x+2}}\\ &={\color{green}(x+2)\cancel{(x+2)(x-2)}} \cdot \frac{2}{\cancel{\left(x-2\right)\left(x+2\right)}} \\ &3(x-2)+(x-2)(x+2)=2(x+2) \\ &3x-6+x^2-4=2x+4 \\ &x^2+x-14=0\end{align*}$$
  ::3x2+4x+4+4+4+4+1x+2+2=2x2x2-2(x+2)(x-2)(x-2)(x+2)(x+2)(x+2)(x+2)(x+2)(x-2)(x+2)(x+2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x+2)(x)(x2)(x+2)(x+2)(x+2)(x+2)(x+2(x)(x-2)(x-2)(x-2)(x-2)(x+2)(x+2)(x-2)(x+2)(x-2)(x+2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x+2)(x-2)(x+2)(x+2)(x-2)(x+2)(x+2)(x+2)(x-2)(x+2)(x+2)(x)3x-2)(x+2)(x+2)(x+2)(x-2)(x+2)(x-2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x)3+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)(x)3)(x)3)(x+2)3x+2)(x+2)(x+2)3)(3)3)(3x+2)(3)(3)(3)(3)(x)3x)3x)(3x-2)(3)(3x-2)(3x)3x-2)(3x-2)(3x-2)(3)(x-2)(2)3)(3)(3)(3)(3)(3)(3)(3x-2)(3)(3)(3)(3)(3)(3)(3)(2)(2)(2)(2)(2)(2)(2)(2)(2)(x-2)(2)(2)(2)(2)(2)(x

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  This quadratic is not factorable, so let's use the Quadratic Formula to solve for $\begin{align*}x\end{align*}$ .
  ::此二次方位是不可因数的, 所以让我们使用二次方位公式解析 x 。

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  $\begin{align*}x=\frac{\text{-}1 \pm \sqrt{1-4 \left(\text{-}14\right)}}{2}=\frac{\text{-}1 \pm \sqrt{57}}{2} \approx 3.27\end{align*}$ and $\begin{align*}\text{-}4.27\end{align*}$ . We must also check for extraneous solutions using a calculator.
  ::x=-11-4(- 142) =-15723. 27 和- 4. 27。 我们还必须使用计算器检查外部溶液 。

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  Example 6
  ::例6

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  What was the original frequency of the siren if the velocity of the vehicle is 25 miles per hour or 11.176 meters per second, and the observed frequency is 723.77 hertz ¹ ?
  ::如果车辆的速度是每小时25英里或每秒11.176米,而且观测到的频率是723.77赫兹1,那么警报器的最初频率是多少?

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  Solution: Let's substitute our known values into the formula, including the speed of sound:  $\begin{align*}c=340.29\end{align*}$  meters per second. 
  ::解答:让我们用我们已知的值来替代公式, 包括声音的速度: c=340.29米每秒。

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  $$\begin{align*}723.77&=\frac{340.29}{340.29-11.176}f_0 \\ 723.77&=\frac{340.29}{329.114}f_0 \\\end{align*}$$
  ::723.77=340.2939340.29-11.11.176f0723.77=340.299.114f0

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  To solve, we can multiply both sides of the equation by the LCD, 329.114.
  ::为了解决问题,我们可以把等式的两边乘以LCD, 329.114。

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  $$\begin{align*}329.114 \cdot 723.77 &= \cancel{329.114} \cdot \frac{340.29}{\cancel{329.114}}f_0 \\ \frac{329.114 \cdot 723.77}{340.29} &=\frac{\cancel{340.29} f_0}{\cancel{340.29}} \\ 700&=f_0\end{align*}$$  
  ::329.114.7730.29=3404.2900=0

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  The original frequency of the siren was 700 hertz.
  ::警报器的原频率为700赫兹。

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  by hherhold demonstrates a doppler shift using a car horn.
  ::Hherhold用汽车喇叭展示了多普勒的转变。

   

   

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     How To Solve Rational Equations With a Graphing Utility
  ::如何用图形化工具解析合理等同

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  To solve rational equations with a graphing utility, set $\begin{align*}y=\end{align*}$  to each side of the equation, and then find the points of intersection . Let's consider an example,   $\begin{align*}\frac{3}{x}+\frac{4}{5}=\frac{6}{x-2}\end{align*}$ .
  ::要用图形化工具解析理性方程式, 请将 Y= 设置在方程式的每侧, 然后找到交叉点 。 让我们来举一个例子, 3x+45= 6x-2 。

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  Graph $\begin{align*}\frac{3}{x}+ \frac{4}{5}\end{align*}$  (blue) and $\begin{align*}\frac{6}{x-2}\end{align*}$  (red) using your graphing calculator.
  ::图3x+45(蓝色)和图6x-2(红色)使用你的图形计算器。

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  The $\begin{align*}x\end{align*}$ -values of the points of intersection (purple points in the graph) are the solutions to the equation.  The problem is that you may not be able to graph the functions accurately or precisely enough to determine what those x-values are. When it is challenging to determine the solutions from a graph, we can approximate the solutions.
  ::交叉点的 x 值( 图形中的纯点) 是方程式的解决方案。 问题在于您可能无法准确或精确地绘制函数图, 以确定这些 x 值是什么。 当从图形中确定解决方案具有挑战性时, 我们可以比较解决方案 。

  

   

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  Feature: Going Flat
  ::特点: 变平

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  by Deirdre Mundy
  ::由Deirdre Mundy 编辑

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  You walk out to the school parking lot after a long day. You groan as you look at your car: Some of your tires look flat. You are sure you have not driven over any nails, and the tires are pretty new. Before you panic, check the thermometer. The weather may hold the answer to your tire trauma.
  ::一天之后你走到学校停车场。当你看着自己的车时,你呻吟着:有些轮胎看起来是平的。你确信你没有开过任何钉子,轮胎是相当新的。在你惊慌之前,请检查温度计。天气可能维持着你轮胎创伤的答案。

  

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  Temperamental Tires
  ::喜 感 感 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性 性

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  Tire pressure can be expressed by the rational equation, known as Boyle's Law,  $\begin{align*}P=\frac{nRT}{V}\end{align*}$ , where n is the amount of air in the tire, R is the universal gas constant , T is the temperature, and V is the volume of the tire. Assume that a tire has no leaks. When the temperature changes, the amount of air inside the tire and the volume of the tire remain the same. This means that the tire pressure will increase when the temperature goes up, and decrease when the temperature goes down.
  ::轮胎压力可以通过合理的方程式(称为Boyle's Law, P=nRTV)表示,n是轮胎中的空气量,R是通用气体常数,T是温度,V是轮胎的体积。假设轮胎没有漏气。当温度变化时,轮胎内的空气量和轮胎的体积保持不变。这意味着当温度上升时轮胎压力会增加,温度下降时会减少。

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  If the temperature drops quickly, you will notice a significant change in your tires. Before you call a tow truck, find a tire pressure gauge to measure the air pressure inside your tires. If the pressure is too low, you can increase it by adding air to your tires. Adding air makes n, the amount of air in the tire, larger. If the pressure is too high, you can lower it by releasing air from your tires.
  ::如果温度下降很快, 您就会注意到轮胎中的显著变化。 在调用拖车之前, 请先找到轮胎压力计来测量轮胎内的气压。 如果压力过低, 您可以通过在轮胎上添加空气来增加气压。 添加空气后, 轮胎中的空气量会更大。 如果压力过高, 您可以通过释放轮胎中的空气来降低气压 。

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  It is important to check your tire pressure once a month and after any major temperature changes. If your tires are inflated to the wrong pressure, you will waste gas, your car will have a harder time braking and steering, you will be at greater risk for a blowout, and your tires will wear out more quickly.
  ::重要的是每月检查一次你的轮胎压力,在任何重大温度变化之后检查一次。 如果轮胎被膨胀到错误的压力,你会浪费气体,你的汽车将更难刹车和驾驶,你将面临更大的井喷风险,你的轮胎将很快耗尽。

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  This video by NSF explains the equations related to the feature in this section, and their connection to driving race cars.
  ::NSF的这段影片解释了与本节特征有关的方程式,以及这些方程式与赛车驾驶器的关系。

   

   

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  Summary
  ::摘要

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  • To solve rational equations, find the LCD, multiply both sides of the equation by the LCD, solve the equations, and check for extraneous solutions . 
    ::解开理性方程式,找到液晶, 将方程式的两边乘以液晶, 解开方程式, 并检查不相干的解决办法。

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  Review
  ::回顾

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  Determine if the following values for x are solutions for the given equations:
  ::确定 x 的以下值是否为给定方程式的解决方案:

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  1. $\begin{align*}\frac{4}{x-3} + 2 = \frac{3}{x+4}, \ x=\text{-}1\end{align*}$
  ::1. 4x-3+2=3x+4, x=-1

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  2. $\begin{align*}\frac{2x-1}{x-5} - 3 = \frac{x+6}{2x}, \ x=6\end{align*}$
  ::2. 2x-1x-5-3=x+62x,x=6

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  Solve the following equations:
  ::解决以下方程式:

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  3. $\begin{align*}\frac{6}{x+2}+1=\frac{5}{x}\end{align*}$
  ::3. 6x+2+1=5x

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  4. $\begin{align*}\frac{5}{3x}- \frac{2}{x+1}=\frac{4}{x}\end{align*}$
  ::4. 53x-2x+1=4x

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  5. $\begin{align*}\frac{12}{x^2-9}=\frac{8x}{x-3}- \frac{2}{x+3}\end{align*}$
  ::5. 12x2-9=8xx-3-2x+3

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  6. $\begin{align*}\frac{6x}{x^2-1}+ \frac{2}{x+1}=\frac{3x}{x-1}\end{align*}$
  ::6. 6x2-2-1+2x+1=3xx-1

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  7. $\begin{align*}\frac{5x-3}{4x}- \frac{x+1}{x+2}=\frac{1}{x^2+2x}\end{align*}$
  ::7. 5x-34x-x+1x+2=1x2+2x

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  8. $\begin{align*}\frac{4x}{x^2+6x+9}- \frac{2}{x+3}=\frac{3}{x^2-9}\end{align*}$
  ::8. 4x22+6x+9-2x+3=3x2-9

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  9. $\begin{align*}\frac{x^2}{x^2-8x+16}=\frac{x}{x-4}+\frac{3x}{x^2-16}\end{align*}$
  ::9. x2x2-8x+16=xx-4+3x2-16

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  10. $\begin{align*}\frac{5x}{2x-3}+ \frac{x+1}{x}=\frac{6x^2+x+12}{2x^2-3x}\end{align*}$
  ::10. 5x2x-3+x+1x=6x2+x+122x2-3x

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  11. $\begin{align*}\frac{3x}{x^2+2x-8}=\frac{x+1}{x^2+4x}+\frac{2x+1}{x^2-2x}\end{align*}$
  ::11. 3xx2+2-2x-8=x+1x2+4x+2x+2x+1x2-2x

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  12. $\begin{align*}\frac{x+1}{x^2+7x}+\frac{x+2}{x^2-3x}=\frac{x}{x^2+4x-21}\end{align*}$
  ::12. x+1x2+7x+x2x2x2-3x=xx2+4x-21

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  Explore More
  ::探索更多

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  1.  A right triangle has leg lengths of $\begin{align*}\frac{1}{2}\end{align*}$ and $\begin{align*}\frac{1}{x}\end{align*}$ units. Its hypotenuse is 2 units. What is the value of x ?
  ::1. 右三角形的腿长度为12和1x单位,其下限为2单位。x值是多少?

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  2.  In the figure shown, triangle MNO is similar  to triangle XYZ. Find the lengths of the sides MN and YZ. 
  ::2. 在所示图中,三角MNO与三角XYZ类似。 找出MN和YZ两侧的长度。

  

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  3.  Andrew found three defective cell phones in a sample of 20 cell phones. If this ratio is consistent, how many defective cell phones does he expect to find in a sample of 2,400 cellphones?
  ::3. Andrew在20部手机的抽样中发现了3部有缺陷的手机,如果这一比率一致,他预计在2 400部手机的抽样中会找到多少部有缺陷的手机?

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  4.  A private liberal arts college advertises that it has a ratio of one professor for every 15 undergraduate students. If the college currently has 86 professors, how many students are enrolled?
  ::4. 一所私立自由艺术学院刊登广告说,每15名本科生中有一名教授,如果该学院目前有86名教授,注册的学生人数是多少?

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  5.  On Sacandaga Lake, Liza can row her kayak 5 miles per hour. On the Hudson River, it takes her the same time to row 5 miles downstream as it does to row 3 miles upstream. What is the speed of the river current in miles per hour?
  ::5. 在Sacandaga湖,Liza可以每小时划5英里的独木舟,在Hudson河上,她需要同时在下游划5英里,在上游划3英里。

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  6. If a train travels 3 times as fast as a bus, and the bus takes 4 hours longer to travel 210 miles, what is the speed of the train and the bus?
  ::6. 如果火车的行驶速度是公共汽车的3倍,而公共汽车需要4小时才能行驶210英里,那么火车和公共汽车的速度是多少?

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

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  Please see the Appendix. 
  ::请参看附录。

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  References
  ::参考参考资料

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  1. "Doppler Effect," last edited April 26, 2017,
  ::1. “多普勒效应”, 2017年4月26日最后一次编辑,