8.10 棱晶体量

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  棱晶体量

  

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  Carlos is cleaning out his fish tank, so he filled the bathtub to the rim with water for his fish to swim in while he empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how many cubic feet of water can it hold?
  ::Carlos正在清理他的鱼缸,所以他把浴缸装满了水,让鱼在水池里游泳。 如果浴缸长5.5英尺,宽3.3英尺,深2.2英尺,它能支撑多少立方英尺的水?

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  In this concept, you will learn to calculate the of different prisms.
  ::在这个概念中,你会学会计算不同的棱镜。

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  Volume
  ::量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量量

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  The volume of a is the measure of how much three-dimensional space it takes up, or holds. Imagine an aquarium. Its length, width, and height determine how much water the tank will hold. If you fill it with water, the amount of water tells the volume of the tank. Volume is measured in cubic units , because you are multiplying three dimensions : length, width, and height.
  ::a 的体积是测量它吸收或持有的三维空间的大小。 想象一个水族馆。 水族馆的长度、 宽度和高度决定着储水罐的储水量。 如果您用水填充水, 水量会显示储水罐的体积。 量用立方体单位测量, 因为量是乘以三个维度: 长度、 宽度和高度。

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  There are several different ways that you can calculate volume. The first way that you are going to explore is by looking at how you can calculate volume using unit cubes.
  ::您可以用几种不同的方法来计算体积。 您要探索的第一种方式是查看如何使用单位立方体来计算体积。

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  Unit cubes are singular cubes used to represent one unit. When you “fill up” a solid figure with unit cubes, you can see the unit cubes lining up the figure. Then you can count or calculate the number of unit cubes in the solid. The number of unit cubes in the solid figure is the volume of the figure. Let’s examine what this would look like in the following prism .
  ::单位立方体是用于代表一个单位的单立方体。当用单位立方体“填充”一个固态的立方体时,您可以看到单位立方体排列图。然后,您可以计算或计算固体中的单位立方体数量。固方体中的单位立方体数量是图的体积。让我们看看在下面的棱柱中会是什么样子。

  

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  Now you can see that this prism holds unit cubes. It has three unit cubes lined up for the length, it has two unit cubes along the side for the width and it has four unit cubes lined up for the height. If you count all of these unit cubes, then you can see that you have 24 unit cubes.
  ::现在你可以看到,这个棱晶具有单位立方体。它有三个单位立方体的长度排队,它的侧面有两个单位立方体的宽度,它有四个单位立方体的高度排队。如果你将所有这些单位立方体都计算在内,那么你可以看到有24个单位立方体。

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  You would write your answer for volume as 24 cubic units. Notice that you use cubic units because you have length times width times height.
  ::您会以 24 立方单位 写入音量的回答 。 请注意, 您使用立方单位, 因为宽度乘以高度 。

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  The formula for volume is a simplified version of the method above. As you saw, you used length and width to find the number of unit cubes in the first layer of the figure - this is the same as finding the area of its base.
  ::音量公式是以上方法的简化版本。 正如您所看到的, 您使用长度和宽度来查找图中第一个层的单位立方体数量, 这与查找其基数区域相同 。

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  Let’s see how this works.
  ::让我们看看这如何运作。

  

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  In this case, the base of the prism is a rectangle . You can use the area formula for to find the area of the base: $\begin{array}{r}A=lw\end{array}$ . This is the same as counting the number of unit cubes in each row and the number of rows. Once you find the area, you simply multiply it by the height to add the rest of the layers.
  ::在这种情况下,棱晶的底部是一个矩形。您可以使用区域公式来查找底部的区域: A=lw。这与计算每行的单位立方体数和行数相同。一旦找到区域,您只需将区域乘以高度即可添加其余的层。

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  The formula for the volume of a rectangular prism is: $\begin{array}{r}V=B\times h\end{array}$ .
  ::矩形棱柱体积的公式为:V=Bxh。

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  $\begin{array}{r}B\end{array}$  represents the base area of the prism. Remember, a prism can have a base in the shape of any polygon . Therefore, the formula you need to use to find the area of the base will change. But the process stays the same: you find the area of the face that is the base and then multiply this by the height of the prism. Remember, you will need to figure out the area of the base and then multiply it by the height to figure out the volume.
  ::B 代表棱柱的基础区域 。 记住, 棱柱可以以任何多边形的形状为基。 因此, 找到基点区域的公式将会改变 。 但进程保持不变 : 找到基点的面部区域, 然后乘以棱柱的高度 。 记住, 您需要找出基点的面积, 然后乘以高度来计算音量 。

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  Let’s look at an example.
  ::让我们举个例子。

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  Find the volume of the prism below.
  ::查找下面棱晶的体积。

  

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ .
  ::首先,找到基地的面积,B

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  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 16\times 9\\ A& =& 144\end{array}\end{array}$$

  
  ::A=lxwA=16x9A=144 A=lxwA=16x9A=144

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 144\times 4\\ V& =& 576\end{array}\end{array}$$

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  The answer is 576.
  ::答案是576。

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  The volume of the prism is $\begin{array}{r}576c{m}^3\end{array}$ .
  ::棱晶体积为576厘米3。

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  Note: All volume is measured in cubic units, so you will need to use this exponent when working on figuring out the volume of a solid.
  ::注:所有体积均以立方单位计量,因此,在计算固体体积时,您需要使用此指数。

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  If you know the volume and the area of the base of a prism, then you can figure out the missing height dimension of the prism.
  ::如果你知道棱镜底部的体积和面积, 那么你就可以找出棱镜缺失的高度维度。

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  Let’s look at an example.
  ::让我们举个例子。

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  The base of a rectangular prism with a volume of 1,145.52 cubic feet has sides of 17.2 feet and 11.1 feet. What is the height of the prism?
  ::1,144.552立方英尺的长方形棱柱底部有17.2英尺和11.1英尺的两边。棱柱的高度是多少?

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ .
  ::首先,找到基地的面积,B

  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 17.2\times 11.1\\ A& =& 190.92\end{array}\end{array}$$

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  Next, use the volume formula to find the height, $\begin{array}{r}h\end{array}$ .
  ::下一步,使用音量公式查找高度,h。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ 1145.52& =& 190.92\times h\\ \frac{1145.52}{190.92}& =& \frac{190.92\times h}{190.92}\\ h& =& 6\end{array}\end{array}$$

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  The answer is 6.
  ::答案是6个

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  The height of the prism is 6 ft.
  ::棱镜的高度是6英尺

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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  Carlos and his tank cleaning.
  ::早些时候,你对Carlos和他的油箱清洗感到困扰。

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  Carlos wants to find the volume of his temporary bathtub fish tank that is 5.5 feet long, 3.3 feet wide and 2.2 feet deep.
  ::Carlos想找到他临时浴缸鱼缸的体积,该鱼缸长5.5英尺,宽3.3英尺,深2.2英尺。

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ .
  ::首先,找到基地的面积,B

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  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 5.5\times 3.3\\ A& =& 18.15\end{array}\end{array}$$

  
  ::A=lxwA=5.5x3.3A=18.15

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 18.15\times 2.2\\ V& =& 39.93\end{array}\end{array}$$

  
  ::V=BxhV=18.15x2.2V=39.93

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  The answer is 39.93.
  ::答案是39.93

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  The bathtub will hold $\begin{array}{r}39.93f{t}^3\end{array}$  of water.
  ::浴缸将蓄水39.93英尺

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  Example 2
  ::例2

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  What is the volume of the prism below?
  ::下面的棱镜量是多少?

  

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  . Note that the base is a triangle so you need to use the area of a triangle formula. $\begin{array}{r}B\end{array}$ First, find the area of the base,
  ::。请注意,基点是一个三角形,因此您需要使用三角形公式的区域。BFirst,找到基点的区域,

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(16)(6)\\ A& =& 48\end{array}\end{array}$$

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 48\times 10\\ V& =& 480\end{array}\end{array}$$

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  The answer is 480.
  ::答案是480

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  The volume of the prism is $\begin{array}{r}480i{n}^3\end{array}$ .
  ::棱晶体积为480英寸3。

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  Example 3
  ::例3

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  Find the volume of a triangular prism with:
  ::查找三角棱镜的音量 :

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  $\begin{array}{r}b=12in,h=10in,H=15in\end{array}$
  ::b=12 英寸, h=10 英寸, H=15 英寸

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ . Note that the base is a triangle so you need to use the area of a triangle formula.
  ::首先,找到基点的区域, B。 注意, 基点是一个三角形, 所以您需要使用三角形公式的区域 。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(12)(10)\\ A& =& 60\end{array}\end{array}$$

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 60\times 15\\ V& =& 900\end{array}\end{array}$$

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  The answer is 900.
  ::答案是900

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  The volume of the prism is $\begin{array}{r}900i{n}^3\end{array}$ .
  ::棱晶体积为900英寸3。

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  Example 4
  ::例4

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  Find the volume of a rectangular prism with a length of 8 m, width of 7 m, and a height of 3 meters.
  ::查找长8米、宽7米、高度3米的矩形棱柱体积。

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ .
  ::首先,找到基地的面积,B

  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 8\times 7\\ A& =& 56\end{array}\end{array}$$

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 56\times 3\\ V& =& 168\end{array}\end{array}$$

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  The answer is 168.
  ::答案是168个

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  The volume of the prism is $\begin{array}{r}168m^3\end{array}$ .
  ::棱晶体积为168立方米。

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  Example 5
  ::例5

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  Find the volume of a rectangular prism with a length of 10 in, width of 8 in, and a height of 6 inches.
  ::查找长10英寸、宽8英寸、高6英寸的矩形棱柱体积。

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  First, find the area of the base, $\begin{array}{r}B\end{array}$ .
  ::首先,找到基地的面积,B

  $$\begin{array}{r}\begin{array}{rcl}A& =& l\times w\\ A& =& 10\times 8\\ A& =& 180\end{array}\end{array}$$

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  Next, put the area of the base in the volume formula.
  ::下一步,将基数区域置于音量公式中。

  $$\begin{array}{r}\begin{array}{rcl}V& =& B\times h\\ V& =& 80\times 6\\ V& =& 480\end{array}\end{array}$$

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  The answer is 480.
  ::答案是480

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  The volume of the prism is $\begin{array}{r}480i{n}^3\end{array}$ .
  ::棱晶体积为480英寸3。

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  Review
  ::回顾

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  Look at each figure and then answer the questions that follow.
  ::看看每个数字 然后回答接下来的问题

  

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  1. Name the figure pictured above.
  ::1. 说明上述数字。

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  2. What is the length of the figure?
  ::2. 数字的长度是多少?

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  3. What is the width of the figure?
  ::3. 数字的宽度是多少?

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  4. What is the height of the figure?
  ::4. 数字的高度是多少?

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  5. What is the volume of the figure?
  ::5. 数字的数量是多少?

  

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  6. What is the name of the figure pictured above?
  ::6. 上述数字的名称是什么?

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  7. What is the shape of the bases?
  ::7. 基地的形状是什么?

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  8. What is the shape of the side faces?
  ::8. 侧面面的形状是什么?

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  9. What is the volume of the figure?
  ::9. 数字的数量是多少?

  

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  10. What is the name of the figure pictured?
  ::10. 所描绘的数字叫什么?

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  11. What is the volume of the figure?
  ::11. 数字的数量是多少?

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  Use what you have learned about volume to solve each problem.
  ::使用你所学到的关于体积的知识来解决每一个问题。

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  12. A rectangular prism has a base measuring 16.2 by 14.8 feet. If its volume is 2,877.12 cubic feet, what is its height?
  ::12. 矩形棱柱的基数为16.2乘以14.8英尺,如果体积为2,877.12立方英尺,高度是多少?

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  13. Kelly is using a rectangular container to fill up a bucket of water. The container is 3.8 inches long, 2.5 inches wide, and 7.2 inches tall. If the bucket holds 1,368 cubic inches of water, how many times will Kelly have to fill the cup in order to fill the bucket?
  ::13. Kelly正在使用一个长方形容器来装满一桶水,容器长3.8英寸,宽2.5英寸,高7.2英寸,如果桶装着1 368立方英寸的水,Kelly要装满多少次才能装满水桶?

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  14. True or false. Volume can be used to measure the amount of water in a pool.
  ::14. 体积可用于测量池内水量。

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  15. True or false. Surface area and volume measure the same thing.
  ::15. 真实或虚假,表面面积和体积测量相同。

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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 。