8.9比例查找比例因子

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  比例查找比例因子

  

  [Figure 1]

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  Peter is working at the miniature museum. While cleaning the miniatures, he accidently mixes up the scale  factor  labels. Luckily, the  dimensions  of the actual building and the model are written on the display. How can Peter find which label goes to which model?
  ::Peter在微型博物馆工作。 在清理微型博物馆时, 他偶然混淆了比例系数标签。 幸运的是, 实际建筑的尺寸和模型都写在显示器上。 Peter如何找到哪个标签去到哪个模型?

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  In this concept, you will learn how to find the scale factor and use proportions to find scaled measurements.
  ::在此概念中,您将学会如何找到比例因素,并使用比例来找到比例衡量标准。

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  Using Proportions to Find Scale Factor
  ::使用比例来查找比例系数

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  To make realistic models, the scale factor is used to find proportional measurements.  Proportional  means having the same ratio. A  scale factor  is the ratio of the model measurement to the actual measurement in simplest form. Unlike a scale ratio, the scale factor does not compare different units of measurement. A model car with the scale factor of 1 : 20 means that the car is  $\begin{array}{r}\frac{1}{20}\end{array}$  times the size of the actual car. It also means that the car is 20 times the size of the model.
  ::为了建立现实的模型,使用比例系数来寻找比例测量。比例系数是指具有相同比例的比例。一个比例系数是指模型测量与以最简单形式实际测量的比例。与比例比率不同,比例系数并不比较不同的测量单位。规模系数为1:20的模型汽车意味着汽车是实际汽车大小的120倍。它还意味着汽车是模型大小的20倍。

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  A fence is actually 16 feet long. On a map, the fence is drawn as 4 inches. To find the scale factor of the drawing to the actual fence, first write a ratio that compares of the two quantities.
  ::栅栏实际上长16英尺。在地图上,栅栏划为4英寸。要找到实际栅栏的绘制比例系数,首先写出与这两个数量相比的比率。

  $$\begin{array}{r}\frac{4in.}{16ft.}\end{array}$$

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  Next, compare the quantities using the same units. Convert feet to inches by multiplying the ratio by the conversion  rate  of 1 ft = 12 in.
  ::接下来,用相同的单位比较数量。将英尺转换为英寸,将比率乘以1英尺=12英寸的换算率。

  $$\begin{array}{r}\frac{4\cancel{in.}}{16\cancel{ft.}}\cdot \frac{1\cancel{ft.}}{12\cancel{in.}}=\frac{4}{192}\end{array}$$

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  Then, simplify the  fraction .
  ::然后,简化分数。

  $$\begin{array}{r}\frac{4÷4}{192÷4}=\frac{1}{48}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{48}\end{array}$ . The drawing is  $\begin{array}{r}\frac{1}{48}\end{array}$  times the size of the actual fence.
  ::比例系数为148。绘图是实际围栏大小的148倍。

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  Use the scale factor to another length of the fence that is actually 8 ft long.
  ::将比例系数乘以围栏的另一长,实际长度为8英尺。

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  First, write a  proportion  to find the equivalent length of 8 ft on the drawing. Use  $\begin{array}{r}x\end{array}$  to represent the drawing length.
  ::首先,在绘图中写一个比例以找到相等的长度为 8 英尺。使用 x 表示绘图长度 。

  $$\begin{array}{r}\frac{1}{48}=\frac{x}{8ft}\end{array}$$

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  Next, cross multiply and simplify to solve for  $\begin{array}{r}x\end{array}$ .
  ::下一步,交叉倍数和简化以解析 x 。

  $$\begin{array}{r}\begin{array}{rcl}48x& =& 8\\ x& =& \frac{8}{48}=\frac{1}{6}\end{array}\end{array}$$

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  The drawing will be  $\begin{array}{r}\frac{1}{6}\end{array}$  ft long. To find  $\begin{array}{r}\frac{1}{6}\end{array}$  of 1 foot, multiply  $\begin{array}{r}\frac{1}{6}\end{array}$  by 12 inches. Remember, 1 ft = 12 in.
  ::绘图将长16英尺。 要找到1英尺的16英尺, 乘以16乘以12英寸。 记住, 1英尺=12英寸。

  $$\begin{array}{r}\frac{1}{6}\times 12in.=2in.\end{array}$$

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  An 8 ft fence will be 2 inches long on the map with a scale factor of  $\begin{array}{r}\frac{1}{48}\end{array}$ .
  ::8英尺长的围栏在地图上长2英寸,比例系数为148。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Peter at the miniature museum.
  ::早些时候,你在迷你博物馆 遇到彼得的问题

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  He needs to match the scale factor to the correct model. The sign says that the Empire State Building is 1,250 feet tall and the model is 25 inches tall. To find the scale factor, Peter can use the ratio of the model dimension to the  actual dimension .
  ::他需要将比例系数与正确的模型相匹配。 标志上写着帝国大厦是1,250英尺高,模型是25英寸高。 要找到比例系数, Peter可以使用模型维度与实际维度的比率 。

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  First, write the ratio of model dimension to actual dimension.
  ::首先,写下模型维度与实际维度的比率。

  $$\begin{array}{r}\frac{25in.}{1250ft.}\end{array}$$

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  Next, convert the quantities to the same unit.
  ::下一步,将数量转换为同一单位。

  $$\begin{array}{r}\frac{25\cancel{in.}}{1250\cancel{ft}}\cdot \frac{1\cancel{ft}}{12\cancel{in.}}=\frac{25}{15000}\end{array}$$

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  Next, simplify to find the scale factor.
  ::下一步,简化以找到比额表系数。

  $$\begin{array}{r}\frac{25÷5}{15000÷5}=\frac{1}{600}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{600}\end{array}$ .
  ::比额表系数为1 600。

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  Peter finds the label that says “scale factor = 1 : 600” and sticks that to the model. He can repeat the same process for the other labels.
  ::Peter发现标签上写着“ 比例系数= 1: 600 ” , 并粘贴在模型上。 他可以为其他标签重复同样的程序 。

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  Example 2
  ::例2

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  Find the scale factor for each ratio.
  ::找出每个比率的比额系数。

  $$\begin{array}{r}\frac{3\text{inches}}{12\text{feet}}\end{array}$$

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  First, convert the quantities to the same unit.
  ::首先,将数量转换为同一单位。

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  Next, simplify to find the scale factor.
  ::下一步,简化以找到比额表系数。

  $$\begin{array}{r}\frac{3÷3}{12÷3}=\frac{1}{4}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{4}\end{array}$ .
  ::比额表系数为14。

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  Example 3
  ::例3

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  Find the scale factor for each ratio.
  ::找出每个比率的比额系数。

  $$\begin{array}{r}\frac{5^{\mathrm{\prime }\mathrm{\prime }}}{{25}^{\mathrm{\prime }}}\end{array}$$

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  First, convert the quantities to the same unit.
  ::首先,将数量转换为同一单位。

  $$\begin{array}{r}\frac{5\cancel{in.}}{25\cancel{ft}}\cdot \frac{1\cancel{ft}}{12\cancel{in.}}=\frac{5}{300}\end{array}$$

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  Next, simplify to find the scale factor.
  ::下一步,简化以找到比额表系数。

  $$\begin{array}{r}\frac{5÷5}{300÷5}=\frac{1}{60}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{60}\end{array}$ .
  ::比额表系数为160。

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  Example 4
  ::例4

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  Find the scale factor for each ratio.
  ::找出每个比率的比额系数。

  $$\begin{array}{r}\frac{2^{\mathrm{\prime }\mathrm{\prime }}}{{50}^{\mathrm{\prime }}}\end{array}$$

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  First, convert the quantities to the same unit.
  ::首先,将数量转换为同一单位。

  $$\begin{array}{r}\frac{2\cancel{in.}}{50\cancel{ft}}\cdot \frac{1\cancel{ft}}{12\cancel{in.}}=\frac{2}{600}\end{array}$$

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  Next, simplify to find the scale factor.
  ::下一步,简化以找到比额表系数。

  $$\begin{array}{r}\frac{2÷2}{600÷2}=\frac{1}{300}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{300}\end{array}$ .
  ::比额表系数为1 300。

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  Example 5
  ::例5

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  Find the scale factor for each ratio.
  ::找出每个比率的比额系数。

  $$\begin{array}{r}\frac{{12.5}^{\mathrm{\prime }\mathrm{\prime }}}{{25}^{\mathrm{\prime }}}\end{array}$$

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  First, convert the quantities to the same unit.
  
  
  ::首先,将数量转换为同一单位。

  $$\begin{array}{r}\frac{12.5\cancel{in.}}{25\cancel{ft}}\cdot \frac{1\cancel{ft}}{12\cancel{in.}}=\frac{12.5}{300}\end{array}$$

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  Next, simplify to find the scale factor.
  ::下一步,简化以找到比额表系数。

  $$\begin{array}{r}\frac{12.5÷12.5}{300÷12.5}=\frac{1}{24}\end{array}$$

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  The scale factor is  $\begin{array}{r}\frac{1}{24}\end{array}$ .
  
  ::比额表系数为1 24。

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  Review
  ::回顾

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  Find the scale factor for 1 scale unit.
  ::查找一个比例单位的比额表因数。

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  1. $\begin{array}{r}\frac{3^{\mathrm{\prime }\mathrm{\prime }}}{6ft}\end{array}$
     ::3 6 f t
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  2. $\begin{array}{r}\frac{{12}^{\mathrm{\prime }\mathrm{\prime }}}{24ft}\end{array}$
     ::12 24 f t
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  3. $\begin{array}{r}\frac{6^{\mathrm{\prime }\mathrm{\prime }}}{18ft}\end{array}$
     ::6 18 f t
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  4. $\begin{array}{r}\frac{9^{\mathrm{\prime }\mathrm{\prime }}}{27ft}\end{array}$
     ::9 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  5. $\begin{array}{r}\frac{4^{\mathrm{\prime }\mathrm{\prime }}}{16ft}\end{array}$
     ::4 16 f t
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  6. $\begin{array}{r}\frac{5^{\mathrm{\prime }\mathrm{\prime }}}{30ft}\end{array}$
     ::5 _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  7. $\begin{array}{r}\frac{3^{\mathrm{\prime }\mathrm{\prime }}}{30ft}\end{array}$
     ::3 _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  8. $\begin{array}{r}\frac{3^{\mathrm{\prime }\mathrm{\prime }}}{60miles}\end{array}$
     ::3 60米 i l e s
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  9. $\begin{array}{r}\frac{4^{\mathrm{\prime }\mathrm{\prime }}}{100miles}\end{array}$
     ::4 100米 i l e s
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  10. $\begin{array}{r}\frac{5^{\mathrm{\prime }\mathrm{\prime }}}{1000km}\end{array}$
      ::5 1000公里米
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  11. $\begin{array}{r}\frac{6^{\mathrm{\prime }\mathrm{\prime }}}{1200km}\end{array}$
      ::6 1200公里米
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  12. $\begin{array}{r}\frac{8^{\mathrm{\prime }\mathrm{\prime }}}{24000m}\end{array}$
      ::8 24000米
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  13. $\begin{array}{r}\frac{{11}^{\mathrm{\prime }\mathrm{\prime }}}{11,000km}\end{array}$
      ::11 11 000公里
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  14. $\begin{array}{r}\frac{{15}^{\mathrm{\prime }\mathrm{\prime }}}{3000km}\end{array}$
      ::15 3000千米
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  15. $\begin{array}{r}\frac{{15}^{\mathrm{\prime }\mathrm{\prime }}}{45,000m}\end{array}$
      ::45 000米 45 000米

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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 。

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  Resources
  ::资源

   

  Image Attributions