8.15 球体体体积

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  Maria has a paperweight that is a glass sphere . The sphere is full of liquid. If the diameter of the paperweight is 6 inches, how much liquid does it contain?
  ::Maria的纸质重量是玻璃球,球体充满液体。如果纸张重量的直径是6英寸,那么它含有多少液体?

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  In this concept, you will learn to calculate the volume of spheres.
  ::在这个概念中,你会学会计算球体的体积。

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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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  is the measure of three-dimensional space a figure takes up. You can also think of it as how much space the figure “holds.”
  ::是一个数字所占用的三维空间的度量。您也可以把它视为数字“值”的度量。

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  To find the volume of spheres, you can use pyramids . Imagine a pyramid with its base on the surface of the sphere and its point as the center of the sphere. The radius of the sphere would be the height of the pyramid.
  ::为了找到球体的体积, 您可以使用金字塔 。 想象一个金字塔, 它的底部是球体表面, 点是球体的中心。 球体的半径是金字塔的高度 。

  

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  The pyramid makes up a portion of the sphere’s volume. If you can fill the whole sphere with pyramids like this, you would know the volume of the sphere. It would be equal to the volumes of all the pyramids put together. How many pyramids would it take to fill a sphere? That depends on the of the sphere.
  ::金字塔占了球体体积的一部分。如果您能够用这样的金字塔填充整个球体,您就会知道球体的体积。它相当于所有金字塔体积的体积。要填充球体需要多少金字塔?这取决于球体。

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  You can combine the surface area of a sphere with the volume formula for a pyramid to calculate the volume of all the pyramids contained within the sphere.
  ::您可以将球的表面面积与金字塔的体积公式结合起来,计算球内所有金字塔的体积。

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  The formula for the volume of a pyramid is $\begin{array}{r}V=\frac{1}{3}Bh\end{array}$ .
  ::金字塔体积的公式是V=13Bh。

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  You know that the height of the pyramid is the radius ( $\begin{array}{r}r\end{array}$ ) of the sphere and the base of the pyramid is the surface area of the sphere ( $\begin{array}{r}SA=4\pi r^2\end{array}$ ). If you substitute this information into the formula for the volume of a pyramid, you would get the following.
  ::您知道金字塔的高度是球体的半径(r),金字塔的底部是球体的表面(SA=4r2)。 如果您用此信息来替代金字塔体积的公式,您将得到以下信息。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{1}{3}Bh\\ V& =& \frac{1}{3}(4\pi r^2)r\\ V& =& \frac{4}{3}\pi r^3\end{array}\end{array}$$

  
  ::V=13BhV=13(4°r2rV=43°r3)

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  The volume of a sphere can be found using the formula
  ::使用公式可以找到一个球体的体积

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  $$\begin{array}{r}V=\frac{4}{3}\pi r^3\end{array}$$

  
  ::V=43°r3

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  Again, all you need to know is the radius of the sphere. You put the value in for $\begin{array}{r}r\end{array}$  in the formula and solve for $\begin{array}{r}V\end{array}$ , the volume.
  ::再说一遍,你需要知道的就是球的半径。您在公式中输入 r 值的值,然后解析 V, 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 sphere below.
  ::查找下方球体的体积 。

  

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  First, substitute what you know into the volume formula.
  ::首先,用数量公式来取代你所知道的东西。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (6{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(6)3

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  Next, use algebra to calculate the volume.
  ::接下来,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (6{)}^3\\ V& =& \frac{4}{3}\pi (216)\\ V& =& 288\pi \\ V& =& 904.78\end{array}\end{array}$$

  
  ::V=43(6)3V=43(216V=288V=904.78)

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  The answer is 904.78.
  ::答案是904.78。

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  The volume of the sphere is 904.78 m ³ .
  ::球体体积为904.78立方米。

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  If you wanted to make your measurement more accurate, you could say that the volume is $\begin{array}{r}288\pi m^3\end{array}$ .
  ::如果你想提高测量的准确度 你可以说体积是288立方米

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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  Maria’s liquid paperweight.
  ::更早之前, 你曾因Maria的液态纸质重量而感到疑惑。

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  Maria has a paperweight with a diameter of 6 inches that is full of water, and she wants to find out how much liquid is inside.
  ::Maria的纸张重量直径为6英寸 满是水 她想弄清楚里面有多少液体

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  First, find the radius of the sphere. Remember the radius is half the diameter.
  ::首先,找到球体的半径。记住半径是直径的一半。

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  $$\begin{array}{r}\begin{array}{rcl}r& =& \frac{d}{2}\\ r& =& \frac{6}{2}\\ r& =& 3\end{array}\end{array}$$

  
  ::= rd2r= 62r= 3

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  Next, substitute what you know into the volume formula.
  ::接下来, 将您知道的替换为音量公式 。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (3{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(3)

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  Then, use algebra to calculate the volume.
  ::然后,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (3{)}^3\\ V& =& \frac{4}{3}\pi (27)\\ V& =& 36\pi \\ V& =& 113.10\end{array}\end{array}$$

  
  ::V=43(3)3V=43(27V=36V=113.10)

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  The answer is 113.10.
  ::答案是113.10。

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  The volume contained in Maria’s paperweight is 113.10 in ³ or $\begin{array}{r}36\pi i{n}^3\end{array}$ .
  ::Maria的纸张重量中包含的体积为113.10英寸3或363英寸3。

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  Example 2
  ::例2

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  Find the volume of the following sphere.
  ::查找下列球体的体积。

  

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  First, substitute what you know into the volume formula.
  ::首先,用数量公式来取代你所知道的东西。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (8{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(8)(8)3

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  Next, use algebra to calculate the volume.
  ::接下来,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (8{)}^3\\ V& =& \frac{4}{3}\pi (512)\\ V& =& 682.67\pi \\ V& =& 2144.66\end{array}\end{array}$$

  
  ::V=43(8)3V=43(512)V=682.67V=2144.66

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  The answer is 2144.66. The volume of the sphere is 2144.66 cm ³ .
  ::答案是2144.66,球体体积为2144.66立方厘米。

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  If you wanted to make your measurement more accurate, you could say that the volume is $\begin{array}{r}682.67\pi c{m}^3\end{array}$ .
  ::如果你想提高测量的准确度 你可以说音量是682.67立方厘米

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  Example 3
  ::例3

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  Find the volume of a sphere with a radius of 4 inches.
  ::查找圆球的体积,半径为4英寸。

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  First, substitute what you know into the volume formula.
  ::首先,用数量公式来取代你所知道的东西。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (4{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(4)3

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  Next, use algebra to calculate the volume.
  ::接下来,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (4{)}^3\\ V& =& \frac{4}{3}\pi (64)\\ V& =& 85.33\pi \\ V& =& 268.08\end{array}\end{array}$$

  
  ::V=43(4)3V=43(64V=85.33V=268.08)

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  The answer is 268.08. 
  ::答案是268.08

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  The volume of the sphere is 268.08 in ³ .
  ::球体体积是268.08英寸3

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  Example 4
  ::例4

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  Find the volume of a sphere with a radius of 5 ft.
  ::查找半径为 5 英尺的球体体体积。

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  First, substitute what you know into the volume formula.
  ::首先,用数量公式来取代你所知道的东西。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (5{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(5)3

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  Next, use algebra to calculate the volume.
  ::接下来,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (5{)}^3\\ V& =& \frac{4}{3}\pi (125)\\ V& =& 166.67\pi \\ V& =& 523.60\end{array}\end{array}$$

  
  ::V=43(5)3V=43(125)V=166.667V=523.60

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  The answer is 523.60.
  ::答案是523.60

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  The volume of the sphere is 523.60 ft ³ .
  ::球体体积为523.60平方英尺。

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  Example 5
  ::例5

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  Find the volume of a sphere with a radius of 3.5 inches.
  ::查找圆球的体积,半径为3.5英寸。

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  First, substitute what you know into the volume formula.
  ::首先,用数量公式来取代你所知道的东西。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi r^3\\ V& =& \frac{4}{3}\pi (3.5{)}^3\end{array}\end{array}$$

  
  ::V=43r3V=43(3.5)3

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  Next, use algebra to calculate the volume.
  ::接下来,使用代数计算音量。

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  $$\begin{array}{r}\begin{array}{rcl}V& =& \frac{4}{3}\pi (3.5{)}^3\\ V& =& \frac{4}{3}\pi (42.875)\\ V& =& 57.17\pi \\ V& =& 179.59\end{array}\end{array}$$

  
  ::V=43-(3.5)3V=43-(42.875)V=57.17-V=179.59

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  The answer is 179.59.
  ::答案是179.59

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  The volume of the sphere is 179.59 in ³ .
  ::球体体积是179.59英寸3

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  Review
  ::回顾

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  Find the volume of each sphere. Use 3.14 to approximate $\begin{array}{r}\pi \end{array}$  and round your answers to the nearest hundredths place.
  ::查找每个球体的体积。 使用 3. 14 来大约 ° , 并绕过您的答案到最近的一百个位置 。

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  1. A sphere with a radius of 3 m.
  ::1. 半径为3米的球体。

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  2. A sphere with a radius of 2.5 m.
  ::2. 半径为2.5米的球体。

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  3. A sphere with a radius of 5 in.
  ::3. 半径为5的球体。

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  4. A sphere with a radius of 6 in.
  ::4. 半径为6的球体。

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  5. A sphere with a radius of 7 ft.
  ::5. 半径为7英尺的球体。

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  6. A sphere with a radius of 4.5 cm.
  ::6. 半径4.5厘米的球体。

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  7. A sphere with a radius of 5.5 m.
  ::7. 半径为5.5米的球体。

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  8. A sphere with a radius of 13 mm.
  ::8. 半径为13毫米的球体。

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  9. A sphere with a diameter of 8 in.
  ::9. 直径为8的球体。

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  10. A sphere with a diameter of 10 ft.
  ::10. 直径为10英尺的球体。

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  11. A sphere with a diameter of 3 m.
  ::11. 直径为3米的球体。

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  12. A sphere with a diameter of 13 m.
  ::12. 直径13米的球体。

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  13. A sphere with a diameter of 22 ft.
  ::13. 直径为22英尺的球体。

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  Use what you have learned to solve each problem.
  ::利用你学到的东西解决每个问题

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  14. A sphere has a diameter of 12 feet. What is its volume?
  ::14. 球体直径为12英尺,体积多少?

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  15. Kelly has a perfume bottle in the shape of a sphere. The diameter of the bottle is 6 inches. How much perfume does Kelly have left if the bottle is only half full?
  ::15. Kelly有一个圆球形的香水瓶,瓶子直径6英寸,如果瓶子只有半满,Kelly还剩多少香水?

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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 。