11.9 类似固体的面积和数量

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  类似固体的面积和数量

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  Area and Volume of Similar Solids
  ::类似固体的面积和数量

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  Two shapes are similar if all their are congruent and all their corresponding sides are proportional. Two solids are similar if they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional.
  ::两个形状相似,如果所有形状都一致,并且所有对应的两边均成正比。两个固体相似,如果是同一类型的固体,相应的弧度、高度、基长、宽度等也成正比。

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  Surface Areas of Similar Solids
  ::类似固体表面面积

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  In two dimensions , when two shapes are similar, the ratio of their areas is the square of the scale factor . A comparable  relationship holds in three dimensions as well.
  ::在两个维度中,当两个形状相似时,其区域的比例是比额表系数的平方,类似关系也包含三个维度。

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  Ratio: If two solids are similar with a scale factor of $\begin{array}{r}\frac{a}{b}\end{array}$ , then the surface areas are in a ratio of $\begin{array}{r}{\left(\frac{a}{b}\right)}^2.\end{array}$
  ::比率:如果两个固体与 ab 比例系数相似,那么表面积的比重为(ab)2。

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  Volumes of Similar Solids
  ::类似固体体积

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  Just like surface area, volumes of similar solids have a relationship that is related to the scale factor.
  ::正如表面积一样,类似的固体数量与比例系数有关。

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  Ratio: If two solids are similar with a scale factor of $\begin{array}{r}\frac{a}{b}\end{array}$ , then the volumes are in a ratio of $\begin{array}{r}{\left(\frac{a}{b}\right)}^3.\end{array}$
  ::比率:如果两个固体与 ab 的比重系数相似,则体积的比重为(ab)3。

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  Summary
  ::摘要

  	Ratios	Units
  Scale Factor	$\begin{array}{r}\frac{a}{b}\end{array}$	$\begin{array}{r}\text{in, ft, cm, m, etc.}\end{array}$
  Ratio of the Surface Areas	$\begin{array}{r}{\left(\frac{a}{b}\right)}^2\end{array}$	$\begin{array}{r}\mathrm{i}{\mathrm{n}}^2,\mathrm{f}{\mathrm{t}}^2,\mathrm{c}{\mathrm{m}}^2,{\mathrm{m}}^2,\mathrm{e}\mathrm{t}\mathrm{c}.\end{array}$
  Ratio of the Volumes	$\begin{array}{r}{\left(\frac{a}{b}\right)}^3\end{array}$	$\begin{array}{r}\mathrm{i}{\mathrm{n}}^3,\mathrm{f}{\mathrm{t}}^3,\mathrm{c}{\mathrm{m}}^3,{\mathrm{m}}^3,\mathrm{e}\mathrm{t}\mathrm{c}.\end{array}$

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  Suppose you were given two similar square prisms and told the scale factor of their sides . How do you think  you might find the ratio of their surface areas and the ratio of their volumes?
  ::假设你们有两种相似的平方棱镜,并且告诉了他们两侧的比重。你们怎么发现它们的表面积和体积的比例?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Determine if the two triangular pyramids are similar.
  ::确定两个三角金字塔是否相似。

  

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  Match up the corresponding parts.
  ::匹配相应的部分。

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  $\begin{array}{r}\frac{6}{8}=\frac{3}{4}=\frac{12}{16}\end{array}$ however, $\begin{array}{r}\frac{8}{12}=\frac{2}{3}\ne \frac{3}{4}\end{array}$ .
  ::68=34=1216 然而 812=23=34

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  These triangle pyramids are not similar.
  ::这些三角金字塔不相似

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  Example 2
  ::例2

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  Two similar triangular prisms are below. T he ratio of their volumes is 343 : 125 . F ind the missing sides in both triangles.
  ::下面是两个相似的三角棱镜。它们的体积比例是343:125,在两个三角形中找出缺失的两边。

  

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  The scale factor is equal to 7:5, since that is the cube root of 343:125. With the scale factor, we can now set up several proportions.
  ::比额表系数等于7:5,因为这是343:125的立方根。有了比额表系数,我们现在可以确定几个比例。

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  $$\begin{array}{rlrlrlrlrl}& \frac{7}{5}=\frac{7}{y}& & \frac{7}{5}=\frac{x}{10}& & \frac{7}{5}=\frac{35}{w}& & 7^2+x^2=z^2& & \frac{7}{5}=\frac{z}{v}\\ & y=5& & x=14& & w=25& & 7^2+{14}^2=z^2\\ & & & & & & & z=\sqrt{245}=7\sqrt{5}& & \frac{7}{5}=\frac{7\sqrt{5}}{v}\to v=5\sqrt{5}\end{array}$$

  
  ::75=7y75=x1075=35w72+x2=z275=z275=zv=5x14x=14w=2572+142=z2z=245=7575=75vvv=55

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  Example 3
  ::例3

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  Are the two rectangular prisms similar? How do you know?
  ::两个长方形棱镜相似吗?

  

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  Match up the corresponding heights, widths, and lengths.
  ::匹配相应的高度、宽度和长度。

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  $$\begin{array}{r}\frac{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{m}}{\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{m}}:\frac{3}{4.5}=\frac{4}{6}=\frac{5}{7.5}\end{array}$$

  
  ::小棱晶大棱晶: 34.5=46=57.5

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  The congruent ratios tell us the two prisms are similar.
  ::一致比率告诉我们,两个棱镜是相似的。

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  Example 4
  ::例4

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  Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller cylinder?
  ::如果区域比例是16:25,那么高的气瓶的高度是多少?

  

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  First, we need to take the square root of the area ratio to find the scale factor, $\begin{array}{r}\sqrt{\frac{16}{25}}=\frac{4}{5}\end{array}$ . Set up a proportion to find $\begin{array}{r}h\end{array}$ .
  ::首先,我们需要从面积比的平方根找到比例系数1625=45。 设定比例以找到 h。

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  $$\begin{array}{rl}\frac{4}{5}& =\frac{24}{h}\\ 4h& =120\\ h& =30\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}\end{array}$$

  
  ::45=24h4h=120h=30单位

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  Example 5
  ::例5

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  Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes?
  ::两个领域以3:4的比例存在半径。 其数量的比例是多少?

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  If we cube 3 and 4, we will have the ratio of the volumes. $\begin{array}{r}{3}^3:4^3=27:64\end{array}$ .
  ::如果我们是立方体3和4, 我们就会拥有数量的比例。 33: 43=27: 64。

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  Review
  ::回顾

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  Determine if each pair of right solids are similar.
  ::确定每对右固体是否相似。

  1. 
  2. 
  3. 
  4. 
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  5. Are all cubes similar? Why or why not?
     ::所有立方体都相似吗?
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  6. Two prisms have a scale factor of 1:4. What is the ratio of their surface areas?
     ::两个棱晶的比重系数为1:4。其表面积的比例是多少?
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  7. Two pyramids have a scale factor of 2:7. What is the ratio of their volumes?
     ::两个金字塔的比重系数为2:7. 其体积的比例是多少?
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  8. Two spheres have radii of 5 and 9. What is the ratio of their volumes?
     ::两个领域有5和9的半径,其数量的比例是多少?
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  9. The surface area of two similar cones is in a ratio of 64:121. What is the scale factor?
     ::两个相似锥体的表面积比率为64:121。 比例系数是多少?
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  10. The volume of two hemispheres is in a ratio of 125:1728. What is the scale factor?
      ::两个半球的体积比率为125:1728。
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  11. A cone has a volume of $\begin{array}{r}15\pi \end{array}$ and is similar to another larger cone. If the scale factor is 5:9, what is the volume of the larger cone?
      ::锥体的体积为15,与另一个较大的锥体类似。如果比额表系数是5:9,则较大的锥体的体积是多少?
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  12. The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
      ::两个类似金字塔的体积比例是8:27,其总面积的比例是多少?
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  13. The ratio of the volumes of two tetrahedrons is 1000:1. The smaller tetrahedron has a side of length 6 cm. What is the side length of the larger tetrahedron?
      ::两个四面形体的体积比是1000:1。较小的四面形体的长度为6厘米。较大的四面形体的体积是多少?
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  14. The ratio of the surface areas of two cubes is 64:225. What is the ratio of the volumes?
      ::两个立方体的表面积比例为64:225。 体积的比例是多少?

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  Below are two similar square pyramids with a volume ratio of 8:27. The base lengths are equal to the heights. Use this to answer questions 15-18.
  ::下面是两个相似的平方金字塔,其体积比率为8:27。基长等于高度。用它回答问题15-18。

  

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  15. What is the scale factor?
      ::比额表因素是什么?
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  16. What is the ratio of the surface areas?
      ::地表面积的比例是多少?
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  17. Find $\begin{array}{r}h,x\end{array}$ and $\begin{array}{r}y\end{array}$ .
      ::寻找h,x和y。
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  18. Find the volume of both pyramids.
      ::找到两个金字塔的体积

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  Use the hemispheres below to answer questions 19-20.
  ::用下面的半球回答问题19 -20

  

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  19. Are the two hemispheres similar? How do you know?
      ::两个半球相似吗?
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  20. Find the ratio of the surface areas and volumes.
      ::查找表面面积和体积的比例。

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  21. The ratio of the surface areas of two similar cylinders is 16:81. What is the ratio of the volumes?
      ::两个类似圆柱体的表面积比例为16:81。 体积的比例是多少?

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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 。