11.7 球体

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  球球面

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  Consider a consisting of the set of all points in three-dimensional space that are equidistant from a single point . How could you determine how much two-dimensional and three-dimensional space that figure occupies?
  ::考虑一组三维空间的所有点,它们从一个点的距离相等。 您如何确定该数字占用的二维和三维空间的大小 ?

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  Spheres
  ::球球面

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  A sphere is the set of all points in three-dimensional space that are equidistant from a single point. The radius  of a sphere has one endpoint on the sphere surface and the other endpoint at the center of that sphere. The diameter of a sphere must contain the center.
  ::一个球体是三维空间中所有点的集合,三维空间从一个点起就相等。一个球体的半径在球体表面有一个端点,而在球体中心另一个端点。一个球体的直径必须包含中心。

  

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  A great circle is the largest circular cross-section in a sphere. The of a sphere is the circumference of a great circle. Every great circle divides a sphere into two congruent hemispheres .
  ::一个大圆圈是一个球体中最大的圆环交叉体。一个球体环绕一个大圆圈。每个大圆圈将一个球体分成两个相近的半球。

  

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  Surface Area
  ::地表地区

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  is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit .
  ::是一个二维测量,即所有将固体捆绑在一起的表面的总面积。区域的基本单位是平方单位。

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  Surface Area of a Sphere: $\begin{array}{r}SA=4\pi r^2\end{array}$
  ::球体表面面积:SA=4°r2

  

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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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  To find the of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.
  ::要找到任何固体,你必须弄清楚它占用了多少空间。基本体积单位是立方体单位。

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  Volume of a Sphere: $\begin{array}{r}V=\frac{4}{3}\pi r^3\end{array}$
  ::球体体积:V=43°r3

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  Examples
  ::实例

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  Example 1
  ::例1

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  Find the surface area of the figure below, a hemisphere with a circular base.
  ::查找下图的表面区域, 以圆基为圆基的半球 。

  

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  Use the formula for surface area:
  ::使用表面积公式:

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  $$\begin{array}{rl}SA& =\pi r^2+\frac{1}{2}4\pi r^2\\ & =\pi (6^2)+2\pi (6^2)\\ & =36\pi +72\pi =108\pi c{m}^2\end{array}$$

  
  ::2+1242(62)+2(62)=3672108cm2

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  Example 2
  ::例2

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  A sphere has a volume of $\begin{array}{r}14,137.167f{t}^3\end{array}$ . What is the radius?
  ::球体体积为14,137,167平方英尺。 方圆是多少?

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  Use the formula for volume, plug in the given volume and solve for the radius, $\begin{array}{r}r\end{array}$ :
  ::使用音量的公式, 插入给定音量并解析半径 r:

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  $$\begin{array}{rl}V& =\frac{4}{3}\pi r^3\\ 14,137.167& =\frac{4}{3}\pi r^3\\ \frac{3}{4\pi }\cdot 14,137.167& =r^3\\ 3375\approx r^3\end{array}$$

  
  ::V=43r314,137.167=43r33414,137.167=r33375r3

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  At this point, you will need to take the cubed root of 3375. Your calculator might have a button that looks like $\begin{array}{r}\sqrt[3]{}\end{array}$ , or you can use   $\begin{array}{r}{3375}^{\frac{1}{3}}\end{array}$ .
  ::此时, 您需要选择 3375 的立方根 。 您的计算器可能有一个类似 3 的按钮, 或者您可以使用 337513 的按钮 。

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  $$\begin{array}{r}\sqrt[3]{3375}=15\approx r\end{array}$$

  
  ::33753=15_____________________________________________________________________________________________________________________________________________________________________________________

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  Example 3
  ::例3

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  The circumference of a sphere is $\begin{array}{r}26\pi feet\end{array}$ . What is the radius of the sphere?
  ::球的环绕是26英尺 球的半径是多少?

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  The circumference is referring to the circumference of a great circle.
  ::环绕是指大圆圈的环绕。

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  Use $\begin{array}{r}C=2\pi r\end{array}$ :
  ::使用 C=2°r :

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  $$\begin{array}{rl}2\pi r& =26\pi \\ r& =13ft\end{array}$$

  
  ::2°r=26°r=13英尺

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  Example 4
  ::例4

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  Find the surface area of a sphere with a radius of 14 feet.
  ::找到圆球的表面区域 半径为14英尺

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  Use the formula for surface area:
  ::使用表面积公式:

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  $$\begin{array}{rl}SA& =4\pi (14{)}^2\\ & =784\pi f{t}^2\end{array}$$

  
  ::SA=4______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Example 5
  ::例5

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  Find the volume of a sphere with a radius of 6 m.
  ::查找半径为6米的球体体积。

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  Use the formula for volume:
  ::使用音量公式:

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

  
  ::V=43-1963=43-(216)=288-(立方米)

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  Review
  ::回顾

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  1. Are there any cross-sections of a sphere that are not a circle? Explain your answer.
     ::是否有一个球的交叉截面不是圆形? 请解释您的答复 。
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  2. List all the parts of a sphere that can also be found in a circle.
     ::列出在圆圈中也可以找到的球体的所有部分。
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  3. List any parts of a sphere that a circle does not have.
     ::列出圆没有的球体的任何部分。

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  For 4 - 11, find the surface area and volume of a sphere with the given dimension. Leave your answer in terms of $\begin{array}{r}\pi .\end{array}$
  ::4 - 11, 找到一个具有给定维度的球体的表面面积和体积。 请保留您的答复 。

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  4. a radius of 8 in.
     ::a 半径为 8 英寸。
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  5. a diameter of 18 cm.
     ::a直径为18厘米。
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  6. a radius of 20 ft.
     ::a 半径为20英尺。
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  7. a diameter of 4 m.
     ::a直径为4米。
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  8. a radius of 15 ft.
     ::a 半径15英尺。
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  9. a diameter of 32 in.
     ::a 直径32英寸。
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  10. a circumference of $\begin{array}{r}26\pi cm\end{array}$ .
      ::a 宽度为26厘米。
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  11. a circumference of $\begin{array}{r}50\pi yds\end{array}$ .
      ::a 50 yds 的周长。
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  12. The surface area of a sphere is $\begin{array}{r}121\pi i{n}^2\end{array}$ . What is the radius?
      ::球体的表面面积是121. 英寸 2. 半径是多少?
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  13. The volume of a sphere is $\begin{array}{r}47916\pi m^3\end{array}$ . What is the radius?
      ::球体的体积是47916瓦米3 方圆是多少?
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  14. The surface area of a sphere is $\begin{array}{r}4\pi f{t}^2\end{array}$ . What is the volume?
      ::球体的表面面积是42平方英尺,体积是多少?
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  15. The volume of a sphere is $\begin{array}{r}36\pi m{i}^3\end{array}$ . What is the surface area?
      ::球体的体积是36°MI3,表面积是多少?
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  16. Find the radius of the sphere that has a volume of $\begin{array}{r}335c{m}^3\end{array}$ . Round your answer to the nearest hundredth.
      ::查找球体的半径,其体积为335厘米3。 将答案转至最接近的一百度。
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  17. Find the radius of the sphere that has a surface area $\begin{array}{r}225\pi f{t}^2\end{array}$ . Round your answer to the nearest hundredths place.
      ::查找球体的半径,该球体的表面面积为225- ft2. 将答案四舍五入到最近的一百个位置。

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  Find the surface area and volume of the following shape. Leave your answers in terms of $\begin{array}{r}\pi \end{array}$ .
  ::查找以下形状的表面面积和体积。 请保留您的回答 。

  18. 

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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 。