4.11 理解规模关系

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  理解缩缩缩关系

  

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  Tim has a cube with a side length of 4 inches. He has a similar cube with dimensions that are twice the first cube. How does the of the larger cube compare to the volume of the smaller cube?
  ::蒂姆有一个侧长为4英寸的立方体。他有一个相似的立方体,其尺寸是第一个立方体的两倍。较大的立方体与较小立方体的体积相比如何?

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  In this concept, you will learn to understand scale relationships of area and volume.
  ::在这个概念中,你们将学会理解面积和体积的规模关系。

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  Scale Relationships
  ::比例关系

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  You can compare the scale relationships of distance , area and volume when looking at three–dimensional figures. Some examples of three–dimensional figures include a prism or a pyramid . When you compare different measurements, you will see the proportional relationships between them.
  ::在查看三维数字时,您可以比较距离、面积和体积的尺度关系。 三维数字的一些例子包括棱镜或金字塔。 在比较不同的测量时,您可以看到它们之间的比例关系。

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  Let’s look at a situation involving volume.
  ::让我们看看涉及数量的情况。

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  Brooke has a scale model of a warehouse. A storage unit is shaped like a rectangular prism and has the dimensions 4 in. by 3 in. by 6 in. If the scale of the model is $\begin{array}{r}0.5\mathrm{i}\mathrm{n}=2\mathrm{f}\mathrm{t},\end{array}$  what are the actual dimensions of the storage unit? What is the volume?
  ::Brooke有一个仓库规模模型。 存储器的形状像矩形棱柱, 尺寸为 4 乘 3 乘 乘 6 英寸。 如果模型的大小是 0.5 英寸=2 平方英尺, 那么存储器的实际尺寸是多少? 数量是多少 ?

  

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  First, notice that there are two parts to this problem. The first part is figuring out the actual dimensions given that Brooke has a scale model. The second part is figuring out the volume. Start by using the scale to write a proportion for the length, width, and height .
  ::首先,请注意这个问题有两个部分。 第一部分正在根据布鲁克有一个比例尺模型来了解实际的尺寸。 第二部分正在计算音量。 开始使用比例尺来刻出长度、 宽度和高度的比例 。

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{length}\text{width}\text{height}\\ \\ & & \frac{0.5\mathrm{i}\mathrm{n}}{2\mathrm{f}\mathrm{t}}=\frac{4\mathrm{i}\mathrm{n}}{x\mathrm{f}\mathrm{t}}\frac{0.5\mathrm{i}\mathrm{n}}{2\mathrm{f}\mathrm{t}}=\frac{3\mathrm{i}\mathrm{n}}{x\mathrm{f}\mathrm{t}}\frac{0.5\mathrm{i}\mathrm{n}}{2\mathrm{f}\mathrm{t}}=\frac{6\mathrm{i}\mathrm{n}}{x\mathrm{f}\mathrm{t}}\end{array}\end{array}$$

  
  ::长宽高度 0.5 英寸 2 英尺= 4 英寸 ft0.5 英寸 2 英尺= 3 英寸 ft0.5 英寸 2 英尺= 6 英寸

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  Next, cross multiply for each dimension.
  ::下一个,每个维度的交叉乘法。

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{length}\text{width}\text{height}\\ \\ & & \frac{0.5}{2}=\frac{4}{x}\frac{0.5}{2}=\frac{3}{x}\frac{0.5}{2}=\frac{6}{x}\\ & & 0.5x=2\times 40.5x=2\times 30.5x=2\times 6\\ & & 0.5x=80.5x=60.5x=12\end{array}\end{array}$$

  
  ::宽度高度 052=4x0.52=3x0.52=6x0.5x=2x40.5x=2x30.5x=2x30.5x=2x60.5x80.5x=60.5x=12

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  Then, divide both sides by 0.5 to solve for $\begin{array}{r}x\end{array}$ . 
  ::然后将两边除以0.5 解决x

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{length}\text{width}\text{height}\\ \\ & & 0.5x=80.5x=60.5x=12\\ & & \frac{0.5x}{0.5}=\frac{8}{0.5}\frac{0.5x}{0.5}=\frac{6}{0.5}\frac{0.5x}{0.5}=\frac{12}{0.5}\\ & & x=16x=12x=24\end{array}\end{array}$$

  The answers are 16, 12, and 24.
  ::宽度 0.5x=8 0.5x=6 0.5x=12 0.5x0.5=80.50.5x0.5x0.5=60.55.5x0.5x120.5x=16x=12x=24 答案是16、12和24。

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  The length of the storage unit is 16 ft, the width is 12 ft, and the height is 24 ft.
  ::储存装置的长度为16英尺,宽度为12英尺,高度为24英尺。

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  Then, you need to calculate the volume of the storage unit.
  ::然后,你需要计算存储器的体积。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& l\times w\times h\\ V& =& 16\times 12\times 24\\ V& =& 4608\end{array}\end{array}$$

  The answer is 4608.
  ::V=lxwxhV=16x12x24V=4608 答案是4608。

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  The volume of the storage unit is $\begin{array}{r}4608{\mathrm{f}\mathrm{t}}^3\end{array}$  or 4608 cubic feet.
  ::储存单位的容量为4608英尺或4608立方英尺。

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  There is a relationship between the area of the base of the prism and the volume of the prism. Let’s take a look at how the area of the base of the prism relates to the volume of the prism using the storage unit problem. 
  ::棱柱底部与棱柱体积之间存在某种关系。 让我们看看棱柱基部与使用存储单元问题的棱柱体积之间的关系。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 16\times 12\\ A& =& 192\end{array}\end{array}$$

  
  ::A=lxwA=16x12A=192

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  The area of the prism is $\begin{array}{r}192{\mathrm{f}\mathrm{t}}^2\end{array}$  or 192 square feet.
  ::棱晶区域为192平方英尺或192平方英尺。

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  Now, to see the relationship between the volume of a prism and the area of a prism, divide the volume by the area.
  ::现在,看看棱晶体积与棱晶体面积之间的关系, 将体积除以面积。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{V}{A}& =& \frac{4608}{192}\\ \frac{V}{A}& =& 24\end{array}\end{array}$$

  
  The answer is 24.
  ::VA=46008192VA=24 答案是24

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  Notice that this is the height of the prism.
  ::注意这是棱镜的高度

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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  Tim’s cubes.
  ::Tim的立方体问题。

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  Tim has two cubes where the larger one is twice the size of the smaller one. This means that there is a scale factor of 2.
  ::蒂姆有两个立方体, 大的立方体是小的立方体的两倍。 这意味着比例系数为 2 。

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  First, find the dimensions of the larger cube.
  ::首先,找到更大的立方体的维度。

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  The smaller cube has a side measure of 4 inches. Since this a cube, the  $\begin{array}{r}\text{length}=\text{width }=\text{ height }=4\text{inches}\end{array}$ .
  ::小方块的侧边测量量为 4 英寸。 由于此方块, 长度= 宽= 高度= 4 英寸 。

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  The larger cube will have a side measure of $\begin{array}{r}4\mathrm{i}\mathrm{n}\times 2=8\mathrm{i}\mathrm{n}\end{array}$ . Therefore the  $\begin{array}{r}\text{length}=\text{width }=\text{ height }=8\text{inches}\end{array}$ .
  ::较大的立方体的侧边测量值为 4 英寸x2=8。 因此, 长度 = 宽 = 高度 = 8 英寸 。

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  Next find the volume of both cubes and compare.
  ::下一步找到两个立方体的体积并进行比较。

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{Volume of smaller cube}\text{Volume of larger cube}\\ \\ & & V=l\times w\times hV=l\times w\times h\\ & & V=4\mathrm{i}\mathrm{n}\times 4\mathrm{i}\mathrm{n}\times 4\mathrm{i}\mathrm{n}V=8\mathrm{i}\mathrm{n}\times 8\mathrm{i}\mathrm{n}\times 8\mathrm{i}\mathrm{n}\\ & & V=64{\mathrm{i}\mathrm{n}}^{3}V=512{\mathrm{i}\mathrm{n}}^3\end{array}\end{array}$$

  
  ::小立方体积小的立方体体积 较大的立方体V=lxwxhV=lxwxhV=lxwxhV=4 inx4 inx4 V=8 inx8 inx8 inx8 V=64 in3 V=512 in3

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  Then, compare the volume of the larger cube to the smaller cube.
  ::然后,比较大立方体的体积和小立方体的体积。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{\text{Volume of larger cube}}{\text{Volume of smaller cube}}& =& \frac{512}{64}\\ \frac{\text{Volume of larger cube}}{\text{Volume of smaller cube}}& =& 8\end{array}\end{array}$$

  
  ::较大立方体体积 较大立方体体积 = 51264 小立方体体量 = 8

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  The answer is 8.
  ::答案是8岁

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  The volume of the larger cube is 8 times the volume of the smaller cube.
  ::大方块的体积是小方块的8倍。

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  Example 2
  ::例2

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  Prove that the height of the following prism can be found by using a ratio of volume to area for a prism with a length of 6 inches, a width of 5 inches and a height of 9 inches.
  ::证明以下棱柱的高度可以通过对长6英寸、宽5英寸和高9英寸的棱柱使用体积与面积之比来找到。

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  First, calculate the volume of the prism.
  ::首先,计算棱镜的体积。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& l\times w\times h\\ V& =& 6\times 5\times 9\\ V& =& 270\end{array}\end{array}$$

  
  ::V=lxwxhV=6x5xx9V=270

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  Next, calculate the area of the prism.
  ::接下来,计算棱镜的面积。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 6\times 5\\ A& =& 30\end{array}\end{array}$$

  
  ::A=lxwA=6x5A=30

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  Then, divide the volume by the area.
  ::然后,将体积除以面积。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{V}{A}& =& \frac{270}{30}\\ \frac{V}{A}& =& 9\end{array}\end{array}$$

  
  The answer is 9.
  ::VA=27030VA=9 答案是9

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  Notice that this is the height of the prism.
  ::注意这是棱镜的高度

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  Example 3
  ::例3

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  A prism has a length of 16 feet, a width of 12 feet, and a height of 18 feet . F ind the volume of the prism. 
  ::棱镜的长度为16英尺,宽度为12英尺,高度为18英尺。找出棱镜的体积。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& l\times w\times h\\ V& =& 16\times 12\times 18\\ V& =& 3456\end{array}\end{array}$$

  
  ::V=lxwxhV=16x12x18V=3456

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  The answer is 3456.
  ::答案是3456

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  The volume of the prism is $\begin{array}{r}3456\mathrm{f}{\mathrm{t}}^3\end{array}$  or 3456 cubic feet.
  ::棱晶体积为3456平方英尺或3456立方英尺。

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  Example 4
  ::例4

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  A prism has a length of 16 feet, a width of 12 feet, and a height of 18 feet . Find the area of the base.
  ::棱镜长16英尺,宽12英尺,高度18英尺,找到基地区域。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 16\times 12\\ A& =& 192\end{array}\end{array}$$

  
  ::A=lxwA=16x12A=192

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  The answer is 192.
  ::答案是192

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  The area of the prism is $\begin{array}{r}192{\mathrm{f}\mathrm{t}}^2\end{array}$  or 192 square feet.
  ::棱晶区域为192平方英尺或192平方英尺。

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  Example 5
  ::例5

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  Write a ratio comparing the volume to the area of the prism and simplify.
  ::将体积与棱晶和简化领域作比较的写法比率。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{V}{A}& =& \frac{3456}{192}\\ \frac{V}{A}& =& 18\end{array}\end{array}$$

  
  The answer is 18.
  ::VA=3456192VA=18 答案是18

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  The height of the prism is 18 feet.
  ::棱镜的高度是18英尺

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  Review
  ::回顾

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  Solve each problem and simplify any ratios.
  ::解决每一个问题,简化任何比率。

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  1. A cube measures 8 feet on each side. A similar cube has dimensions that are twice as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
  ::1. 立方体在每侧测量8英尺。相类似的立方体的尺寸是两倍。较大的立方体的体积如何与较小立方体的体积相比较?写一个比方以显示比较。

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  2. A cube measures 3 inches on each side. A similar cube has dimensions that are half that of the other cube. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison
  ::2. 立方体在每一边的方体体体积为3英寸,相近的立方体的尺寸为另一立方体的半数。较大的立方体体体体积与较小立方体体体体体体积相比如何?写一个比来显示比较

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  3. A scale model of a sandbox has dimensions 0.5 inch by 3 inches by 4 inches. If the scale of the model is $\begin{array}{r}1\text{inch}=1\text{foot}\end{array}$ , what is the volume of the actual sandbox?
  ::3. 沙箱规模模型的尺寸为0.5英寸乘3英寸乘4英寸,如果模型的大小为1英寸=1英尺,实际沙箱的体积是多少?

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  4. A cube measures 5 inches on each side. A similar cube has dimensions that are 3 times as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
  ::4. 立方体在每侧测量5英寸,类似立方体的尺寸为3倍。较大的立方体的体积与较小立方体的体积相比如何?写一个比方以显示比较。

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  5. A shipping box measures 16 inches by 12 inches by 8 inches. A second box has a similar size but each dimension is $\begin{array}{r}\frac{1}{4}\end{array}$  as long. How does the volume of the second box compare to the volume of the first box?
  ::5. 海运箱的尺寸为16英寸乘12英寸乘8英寸,第二个箱的尺寸类似,但每个尺寸为14秒,第二个箱的体积与第一个箱的体积相比如何?

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  6. Rina’s fish tank has a volume of 8,000 cubic inches. The dimensions of Ava’s fish tank are all $\begin{array}{r}\frac{1}{2}\end{array}$  the size of Rina’s. What is the volume of Ava’s fish tank?
  ::6. Rina的鱼缸容量为8 000立方英寸,Ava的鱼缸尺寸均是Rina的12个尺寸。 Ava的鱼缸容量是多少?

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  7. A prism has a width of 6 feet, a length of 8 feet and a height of 12 feet. What is the volume of the prism?
  ::7. 棱镜宽度为6英尺,长8英尺,高12英尺,棱镜的体积是多少?

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  8. What is the area of the base of this prism?
  ::8. 这一棱镜的基础领域是什么?

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  9. What would the volume of the prism be if each of its dimensions were $\begin{array}{r}\frac{1}{4}\end{array}$   the size of the one described above?
  ::9. 如果棱镜的每个维度都与上述维度的大小是14,那么棱镜的体积会有多大?

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  10. What would the volume of the prism be if each of its dimensions were $\begin{array}{r}\frac{1}{2}\end{array}$   the size of the one described above?
  ::10. 如果棱镜的每个维度都是上述12个大小的12个,其棱镜的体积会有多大?

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  11. What would the volume of the prism be if each of its dimensions were twice the size of the one described above?
  ::11. 如果棱镜的每个维度是上述维度的两倍,那么棱镜的体积会有多大?

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  12. What ratio can you use to discover the relationship between volume and area?
  ::12. 您用什么比例来发现体积与面积之间的关系?

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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