10.16棱镜体积

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  棱镜体积

  

  [Figure 1]

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  Karen and Lance are building a backyard playground for their two children. They have a swing  set , a merry-go-round, and a sandbox already set up. Lance is going to the store to buy sand for the sandbox and needs to know how much sand it can hold. The sandbox is 6 feet wide, 8 feet long, and 1 feet deep. How can Lance use a formula to calculate the total   of this sandbox?
  ::凯伦和兰斯正在为他们的两个孩子建造一个后院操场。他们有一个秋千、一个旋转木马和一个沙箱。兰斯将去商店买沙箱用的沙子,需要知道有多少沙子。沙箱宽6英尺、8英尺长、1英尺深。兰斯如何使用公式来计算沙箱的总数?

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  In this concept, you will learn to find volumes of rectangular and triangular prisms using formulas.
  ::在此概念中,您将学会使用公式找到大量矩形和三角棱晶。

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  Finding Volume of Prisms
  ::寻找棱晶体量

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  Volume  is the amount of  space  inside a  .
  ::音量是 .

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  Filling  with cubes is a simple, easy way to understand volume. If you can count the cubes, you can figure out the volume. However, sometimes you will have to figure out the volume of a  prism  when there aren’t any cubes drawn inside it.
  ::用立方体填充是一个简单、容易理解音量的方法。 如果您可以计算立方体, 您可以找到音量 。 然而, 有时当没有立方体时, 您必须找出棱晶的体积 。

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  Take a look at the prism below.
  ::看看下面的棱镜

  

  [Figure 2]

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  This  rectangular prism  has a  height  of 5 inches, a width of 3 inches, and a length of 4 inches.
  ::这种矩形棱镜的高度为5英寸,宽度为3英寸,长度为4英寸。

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  Here is a formula for finding the volume of this type prism.
  ::这是找到这种棱柱的量的公式。

  $$\begin{array}{r}V=Bh\end{array}$$

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  $\begin{array}{r}B\end{array}$  means the  area  of the base, which is the length times the width, and  $\begin{array}{r}h\end{array}$  means the height.
  ::B 表示基数的区域,即宽度的长度倍数,h 表示高度。

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  So, first figure out the area of the base.
  ::所以,首先找出基地的面积。

  $$\begin{array}{r}A=3\times 4=12\end{array}$$

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  Next, multiply  $\begin{array}{r}B\end{array}$  by  $\begin{array}{r}h\end{array}$ .
  ::下一个乘以B乘以h

  $$\begin{array}{r}\begin{array}{rcl}h& =& 5\\ V& =& 12\times 5=60\end{array}\end{array}$$

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  The volume is 60 cubic inches or  $\begin{array}{r}i{n}^3\end{array}$ .  Remember, volume is in  cubic units .
  ::音量为 60 立方英寸或 i n 3。 记住, 音量为立方单位 。

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  Let’s look at another example. Find the volume using the volume formula.
  ::让我们看看另一个例子。 用音量公式查找音量 。

  

  [Figure 3]

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, figure the area of the base.
  ::首先,图图基地的面积。

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  The area of the base is  $\begin{array}{r}2\times 8=16\end{array}$
  ::基数区域为 2 × 8 = 16

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  Next, multiply the  $\begin{array}{r}B\end{array}$  by  $\begin{array}{r}h\end{array}$ . The height is 3 inches.
  ::下一个乘以 B 乘以 h 。 高度为 3 英寸 。

  $$\begin{array}{r}\begin{array}{rcl}V& =& 16\times 3\\ V& =& 48i{n}^3\end{array}\end{array}$$

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  The volume of this rectangular prism is  $\begin{array}{r}48i{n}^3\end{array}$ .
  ::此矩形棱柱的体积为 48 i n 3 。

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  You can use the same formula for finding the volume of the  triangular prism . Except this time, the area of the base is a  triangle  and not a  rectangle .
  ::您可以使用相同的公式来查找三角棱晶的体积。除了这次外,基础区域是一个三角区域,而不是矩形。

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  Take a look at the triangular prism below.
  ::看看下面的三角棱镜。

  

  [Figure 4]

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  To find the volume of a triangular prism, multiply the area of the base ( $\begin{array}{r}B\end{array}$ ) with the height of the prism.
  ::要找到三角棱镜的体积, 请将基点( B) 的面积乘以棱镜的高度 。

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, find the area of the triangular base using the formula for area of a triangle.
  ::首先,使用三角形区域的公式查找三角基点区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(15\times 6)\\ A& =& \frac{1}{2}(90)\\ A& =& 45\text{ sq. units}\end{array}\end{array}$$

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  Next, multiply this by the height.
  ::下一个,乘以这个乘以高度。

  $$\begin{array}{r}\begin{array}{rcl}V& =& Bh\\ V& =& (45)h\\ V& =& 45(2)\\ V& =& 90\text{cubic centimeters or}c{m}^3\end{array}\end{array}$$

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  The volume of the prism is  $\begin{array}{r}90c{m}^3\end{array}$ .
  ::棱晶的体积是90厘米3

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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 Lance and Karen’s sandbox.
  ::更早之前, 兰斯和凯伦的沙箱问题就发生在你身上。

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  The sandbox is 6 feet wide, 8 feet long, and 1 feet deep. Lance needs to know the volume.
  ::沙箱宽6英尺,长8英尺,深1英尺

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  To find the volume of the sandbox, which is a prism with one of its bases removed, use the following formula.
  ::要找到沙箱的体积, 沙箱是一个棱镜, 其基地之一被拆除, 使用以下公式 。

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, substitute in the given values. Remember,  $\begin{array}{r}B\end{array}$  is length times width.
  ::首先, 替换给定值。 记住, B 是长度乘以宽度 。

  $$\begin{array}{r}\begin{array}{rcl}V& =& (8\times 6)(1)\\ V& =& 48f{t}^3\end{array}\end{array}$$

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  This is the total volume.
  ::这是总量。

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  Next, to find out how much sand he needs to fill the sandbox halfway, divide the total volume by 2.
  ::接下来,为了找出他需要多少沙子来填满沙箱的中途, 将总量除以2。

  $$\begin{array}{r}48÷2=24\end{array}$$

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  Lance needs 24 cubic feet of sand to fill the sandbox halfway.
  ::兰斯需要24立方英尺的沙子 填满沙箱中途

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  Example 2
  ::例2

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  Find the volume of the prism.
  ::寻找棱镜的体积 。

  

  [Figure 5]

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  To find the volume of a prism, use the following formula.
  ::要找到棱晶的体积,请使用以下公式。

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, substitute in the given values. Remember  $\begin{array}{r}B\end{array}$  is length times width.
  ::首先, 替换给定值。 记住 B 是长度乘以宽度 。

  $$\begin{array}{r}\begin{array}{rcl}V& =& (16\times 9)(4)\\ V& =& 576c{m}^3\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}576c{m}^3\end{array}$ .
  ::答案是576米3

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  Example 3
  ::例3

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  Find the volume of the prism.
  ::寻找棱镜的体积 。

  

  [Figure 6]

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  First, to find the volume of a prism, use the following formula.
  ::首先,要找到棱晶的体积,请使用以下公式。

  $$\begin{array}{r}V=Bh\end{array}$$

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  Next, substitute in the given values. Remember  $\begin{array}{r}B\end{array}$  is length times width. This is a  square  cube, so the length, width, and height are the same.
  ::接下来,在给定值中替换。 记住 B 是长度乘以宽度。 这是一个方形立方体, 长度、 宽度和高度相同 。

  $$\begin{array}{r}\begin{array}{rcl}V& =& (5\times 5)(5)\\ V& =& 125i{n}^3\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}125i{n}^3\end{array}$ .
  ::答案是125 i n 3。

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  Example 4
  ::例4

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  Find the volume of the prism.
  ::寻找棱镜的体积 。

  

  [Figure 7]

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  First, to find the volume of a prism, use the following formula.
  ::首先,要找到棱晶的体积,请使用以下公式。

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, substitute in the given values. Remember,  $\begin{array}{r}B\end{array}$  is length times width.
  ::首先, 替换给定值。 记住, B 是长度乘以宽度 。

  $$\begin{array}{r}\begin{array}{rcl}V& =& (30\times 5)(3)\\ V& =& 450i{n}^3\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}450i{n}^3\end{array}$ .
  ::答案是450 i n 3。

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  Example 5
  ::例5

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  Find the volume of the prism.
  ::寻找棱镜的体积 。

  

  [Figure 8]

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  To find the volume of a triangular prism, multiply the area of the base ( $\begin{array}{r}B\end{array}$ ) with the height of the prism.
  ::要找到三角棱镜的体积, 请将基点( B) 的面积乘以棱镜的高度 。

  $$\begin{array}{r}V=Bh\end{array}$$

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  First, find the area of the triangular base using the formula for area of a triangle.
  ::首先,使用三角形区域的公式查找三角基点区域。

  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{1}{2}bh\\ A& =& \frac{1}{2}(5\times 7)\\ A& =& \frac{1}{2}(35)\\ A& =& 17.5sq.m\end{array}\end{array}$$

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  Next, multiply this by the height.
  ::下一个,乘以这个乘以高度。

  $$\begin{array}{r}\begin{array}{rcl}V& =& Bh\\ V& =& (17.5)1\\ V& =& 17.5c{m}^3\end{array}\end{array}$$

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  The volume of the prism is  $\begin{array}{r}17.5c{m}^3\end{array}$ .
  
  ::棱晶体积为17.5厘米3米。

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  Review
  ::回顾

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  Find the volume of each rectangular prism. Remember to label your answer in cubic units.
  ::查找每个矩形棱柱的体积。 记住要将答案标签为立方单位 。

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  1. Length = 5 in, width = 3 in, height = 4 in
     ::长度 = 5 英寸,宽 = 3 英寸,身高 = 4 英寸
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  2. Length = 7 m, width = 6 m, height = 5 m
     ::长度=7米,宽度=6米,高度=5米
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  3. Length = 8 cm, width = 4 cm, height = 9 cm
     ::长度 = 8 厘米,宽 = 4 厘米,高度 = 9 厘米
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  4. Length = 8 cm, width = 4 cm, height = 12 cm
     ::长度 = 8 厘米,宽 = 4 厘米,高度 = 12 厘米
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  5. Length = 10 ft, width = 5 ft, height = 6 ft
     ::长度 = 10英尺,宽 = 5英尺,高度 = 6英尺
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  6. Length = 9 m, width = 8 m, height = 11 m
     ::长度=9米,宽度=8米,高度=11米
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  7. Length = 5.5 in, width = 3 in, height = 5 in
     ::长度=5.5英寸,宽度=3英寸,身高=5英寸
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  8. Length = 6.6 cm, width = 5 cm, height = 7 cm
     ::长度=6.6厘米,宽度=5厘米,高度=7厘米
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  9. Length = 7 ft, width = 4 ft, height = 6 ft
     ::长度 = 7 英尺,宽 = 4 英尺,高度 = 6 英尺
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  10. Length = 15 m, width = 8 m, height = 10 m
      ::长度=15米,宽度=8米,高度=10米

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  Find the volume of each triangular prism. Remember that  $\begin{array}{r}h\end{array}$  means the height of the triangular base and  $\begin{array}{r}H\end{array}$  means the height of the whole prism.
  ::查找每个三角棱柱的体积。 记住, h 表示三角基点的高度, H 表示整个棱柱的高度。

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  11. $\begin{array}{r}b=6in,h=4in,H=5in\end{array}$  
      ::b = 6 i n, h = 4 i n, H = 5 i n
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  12. $\begin{array}{r}b=7in,h=5in,H=9in\end{array}$  
      ::b = 7 i n, h = 5 i n, H = 9 i n
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  13. $\begin{array}{r}b=10m,h=8m,H=9m\end{array}$  
      ::b = 10米,h = 8米,H = 9米
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  14. $\begin{array}{r}b=12m,h=10m,H=13m\end{array}$  
      ::b = 12米,h = 10米,H = 13米
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  15. $\begin{array}{r}b=8cm,h=6cm,H=9cm\end{array}$  
      ::b = 8厘米,h = 6厘米,H = 9厘米

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  Answer true or false for each of the following questions.
  ::回答下列每个问题是否真实或虚假。

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  16. Volume is the amount of space that a figure can hold inside it.
      ::体积是数字中可以容纳的空间数量。
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  17. The volume of a rectangular prism is always greater than the volume of a cube.
      ::矩形棱柱的体积总是大于立方体的体积。
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  18. The volume of a triangular prism is less than a rectangular prism with the same size base.
      ::三角棱镜的体积小于大小相同的矩形棱镜。
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  19. A painter would need to know the surface area of a house to do his/her job correctly.
      ::画家需要知道房子的表面 才能正确地完成他/她的工作
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  20. If Marcus is covering his book with a book cover, Marcus is covering the surface area of the book.
      ::如果马库斯用书封面遮盖他的书, 马库斯就覆盖书的表面面积。