10.15 锥体体体积

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  锥体量

  

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  Francine is in charge of pricing the desserts at a local ice cream parlor.  She has ordered new serving cups in the shape of a cone for fancy sundaes.  She wants to know how much ice cream it takes to fill the cup to make sure she is pricing the sundae fairly.  The height of the cone-shaped cup is 6 inches and the diameter at the top is 3 inches.  What is the of this cone-shaped cup? 
  ::弗朗辛负责当地一家冰淇淋店的甜点定价。 她已经为花哨圣代订购了新杯子, 以甜筒为形状。 她想知道要填满杯子需要多少冰淇淋才能确保圣代价格公平。 锥形杯的高度是6英寸, 顶部的直径是3英寸。 这个锥形杯是什么?

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  In this concept, you will learn how to calculate the volume of a cone.
  ::在这个概念中,你会学会如何计算锥体的体积。

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  Finding the Volume of a Cone
  ::查找一个锥体的音量

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  To figure out the volume of a cone, let’s first look at how you can compare it with another . The closest solid figure to a cone is a cylinder . Both cones and cylinders have circular bases. The difference between the two is while a cylinder has two circular bases, a cone only has one.
  ::要想弄清楚锥体的体积,让我们首先看看如何与锥体进行比较。 锥体最接近的固态数字是圆柱体。 锥体和圆锥体都有圆基。 两者的区别在于圆锥体有两个圆基,锥体只有一个圆基。

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  Here is a cylinder and the formula for finding the volume of a cylinder.
  ::这里有一个气瓶和找到气瓶体积的公式。

  

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  $\begin{array}{r}V=\pi r^{2}h\end{array}$
  ::Vr2h

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  In the picture below, you can see how a cylinder and a cone are related:
  ::在下图中,您可以看到圆筒和圆锥是如何关联的:

  

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  To find the volume of the cone, you use a formula similar to that of a pyramid , except that you are going to take into consideration that the base of a cone is a circle . Therefore, you will need to find the area of the circle to find the volume of the cone.  The formula for the area of a circle is  $\begin{array}{r}\pi r^2\end{array}$ .  Take that measurement and multiply it by the height of the cone. Next, because the cone fills a portion of the cylinder, multiply the product by one-third.
  ::要找到锥体的体积,你使用类似于金字塔的公式,但要考虑锥体的底部是一个圆。因此,你需要找到圆的面积才能找到锥体的体积。圆的公式是°r2。用这个公式测量并乘以锥体的高度。接下来,由于圆锥体填充了圆柱体的一部分,将产品乘以1/3。

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  $\begin{array}{r}V=\frac{1}{3}\pi r^{2}h\end{array}$
  ::V=13°r2h

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  Let's look at an example.  What is the volume of the cone below?
  ::让我们来举个例子。下面的锥形体体积是多少?

  

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  First, substitute the given values into the formula.
  ::首先,将给定值替换为公式。

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  $$\begin{array}{rl}V& =\frac{1}{3}\pi r^{2}h\\ V& =\frac{1}{3}(3.14)(5^2)(7)\\ V& =\frac{1}{3}(3.14)(25)(7)\\ V& =\frac{1}{3}(3.14)(175)\\ V& =\frac{549.5}{3}\\ V& =183.16i{n}^3\end{array}$$

      
  ::V=13r2hV=13(3.14)(52)(7)(7)V=13(3.14)(25)(7)V=13(3.14)(175)V=549.53V=183.16英寸

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  The answer is the volume of the cone is 183.16 cubic inches.
  ::答案是锥体的体积是183.16立方英寸

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  Let's look at another example.  Find the volume of the following cone.
  ::让我们再看看另一个例子。 找到以下锥体的体积 。

  

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  First, substitute the given values into the formula.
  ::首先,将给定值替换为公式。

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(3^2)(12)\\ V& =\frac{1}{3}(3.14)(9)(12)\\ V& =\frac{1}{3}(3.14)(108)\\ V& =\frac{339.12}{3}\\ V& =113.04{cm}^3\end{array}$$

  
  ::V=13(3.14)(32)(12)V=13(3.14)(9)9(12)V=13(3.14)(108)V=339.123V=113.04cm3

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  The volume of the cone is $\begin{array}{r}113.04c{m}^3\end{array}$ .
  ::锥体的体积是113.04厘米。

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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 Francine's new dessert cups.  
  ::之前,你得到一个问题 关于弗朗辛的新甜点杯。

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  She needs to figure out the volume of ice cream that will fit into one dessert cup.   The diameter of the cup is 3 inches and the height is 6 inches.
  ::她需要弄清楚冰淇淋的体积 将适合一个甜点杯。 杯的直径是3英寸,身高是6英寸。

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  First, find the value of the radius by dividing the diameter by 2, and substitute the values for pi , the radius, and the height of the cone into the formula for volume of a cone.
  ::首先,通过将直径除以 2 来找到半径的值,并将 pi、 半径和锥体高度的值替换成锥体体体积的公式。

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  $$\begin{array}{rl}r& =3÷2\\ r& =1.5\end{array}$$

  
  ::r=32r=1.5

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  $$\begin{array}{rl}V& =\frac{1}{3}\pi r^{2}h\\ V& =\frac{1}{3}(3.14)({1.5}^2)(6)\end{array}$$

  
  ::V=13r2hV=13(3.14)(1.52)(6)

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  Next, square the radius and multiply the values together.
  ::下一步,将半径平方,然后将值乘在一起。

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)({1.5}^2)(6)\\ V& =\frac{1}{3}(3.14)(2.25)(6)\end{array}$$

  
  ::V13(3.14)(1.52)(6)(6)V=13(3.14)(2.25)(6)

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(13.5)\\ V& =\frac{42.39}{3}\end{array}$$

  
  ::V=13(3.14)(13.5)V=42.393

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  Then, divide by 3 and record the answer, remembering to include the appropriate unit of measurement.
  ::然后,除以3,记录答案,记住包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{42.39}{3}\\ V& =14.13i{n}^3\end{array}$$

  
  ::V=42.393V=14.13 3英寸

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  The answer is the volume of ice cream that will fill Francine's cone-shaped desert cup is 16.75 cubic inches.
  ::答案是冰淇淋的数量 将填充弗朗辛的锥形沙漠杯 是16.75立方英寸

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  Example 2
  ::例2

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  Find the volume of a cone with a radius of 2 inches and a height of 4 inches. 
  ::找到圆锥体的体积,半径为2英寸,高度为4英寸。

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  First, substitute the values for pi, the radius, and the height of the cone into the formula for volume of a cone.
  ::首先,将 pi、 半径和锥体高度的值替换为锥体体体积的公式。

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  $$\begin{array}{rl}V& =\frac{1}{3}\pi r^{2}h\\ V& =\frac{1}{3}(3.14)(2^2)(4)\end{array}$$

  
  ::V=13r2hV=13(3.14)(22)(4)

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  Next, square the radius and multiply the values together.
  ::下一步,将半径平方,然后将值乘在一起。

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(2^2)(4)\\ V& =\frac{1}{3}(3.14)(4)(4)\end{array}$$

  
  ::V13(3.14)(22)(4)(4)V=13(3.14)(4)(4)

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(16)\\ V& =\frac{50.24}{3}\end{array}$$

  
  ::V=13(3.14)(16)V=50.243

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  Then, divide by 3 and record the answer, remembering to include the appropriate unit of measurement.
  ::然后,除以3,记录答案,记住包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{50.24}{3}\\ V& =16.75i{n}^3\end{array}$$

  
  ::V=50243V=16.75英寸3英寸

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  The answer is the volume of the cone is 16.75 cubic inches.
  ::答案是锥体的体积是16.75立方英寸

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  Example 3
  ::例3

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  Find the volume of a cone with a radius of 5 ft and a height of 8 ft.
  ::查找圆锥体的体积,半径为5英尺,高度为8英尺。

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  First, substitute the values for pi, the radius, and the height of the cone into the formula for volume of a cone.
  ::首先,将 pi、 半径和锥体高度的值替换为锥体体体积的公式。

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  $$\begin{array}{rl}V& =\frac{1}{3}\pi r^{2}h\\ V& =\frac{1}{3}(3.14)(5^2)(8)\end{array}$$

  
  ::V=13r2hV=13(3.14)(52)(8)

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  Next, square the radius and multiply the values together.
  ::下一步,将半径平方,然后将值乘在一起。

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(5^2)(8)\\ V& =\frac{1}{3}(3.14)(25)(8)\end{array}$$

  
  ::V13(3.14)(52)(8)V=13(3.14)(25)(8)

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(200)\\ V& =\frac{628}{3}\end{array}$$

  
  ::V=13(3.14)(200)V=6283

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  Then, divide by 3 and record the answer, remembering to include the appropriate unit of measurement.
  ::然后,除以3,记录答案,记住包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{628}{3}\\ V& =209.33i{n}^3\end{array}$$

  
  ::V=6283V=209.33英寸3

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  The answer is the volume of the cone is 209.33 cubic inches.
  ::答案是锥体的体积是209.33立方英寸

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  Example 4
  ::例4

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  A cone with a radius of 4 m and a height of 9 m.
  ::圆锥形半径为4米,高度为9米。

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  First, substitute the values for pi, the radius, and the height of the cone into the formula for volume of a cone.
  ::首先,将 pi、 半径和锥体高度的值替换为锥体体体积的公式。

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  $$\begin{array}{rl}V& =\frac{1}{3}\pi r^{2}h\\ V& =\frac{1}{3}(3.14)(4^2)(9)\end{array}$$

  
  ::V=13r2hV=13(3.14)(42)(9)

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  Next, square the radius and multiply the values together.

  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(4^2)(9)\\ V& =\frac{1}{3}(3.14)(16)(9)\end{array}$$

  
  ::下一步,将半径平方,并将数值乘以一。 V=13(3.14)(42)(9)V=13(3.14)(16)(9)

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  $$\begin{array}{rl}V& =\frac{1}{3}(3.14)(144)\\ V& =\frac{452.16}{3}\end{array}$$

  
  ::V=13(3.14)(144)V=452.163

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  Then, divide by 3 and record the answer, remembering to include the appropriate unit of measurement.
  ::然后,除以3,记录答案,记住包括适当的计量单位。

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  $$\begin{array}{rl}V& =\frac{452.16}{3}\\ V& =150.72m{}^3\end{array}$$

  The answer is the volume of the cone is 150.72 cubic meters.
  ::V=452.163V=150.72米 答案是锥体的体积为150.72立方米。

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  Review
  ::回顾

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  Use 3.14 for the value $\begin{array}{r}\pi \end{array}$  and round your answer to the nearest hundredths place where needed.
  ::以 3. 14 表示 值,然后按需要将您的回答绕到最接近的百分百位 。

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  Find the volume of each cone.
  ::查找每个锥体的体积。

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  1. A cone with a radius of 3 inches and a height of 7 inches.
     ::圆锥形半径为3英寸 高度为7英寸
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  2. A cone with a radius of 5 ft and a height of 9 ft.
     ::圆锥形半径为5英尺,高度为9英尺。
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  3. A cone with a radius of 6 meters and a height of 10 meters.
     ::圆锥形半径为6米,高度为10米。
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  4. A cone with a radius of 10 inches and a height of 12 inches.
     ::圆锥体半径为10英寸 高度为12英寸
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  5. A cone with a radius of 12 mm and a height of 14 mm.
     ::圆锥体半径为12毫米,高度为14毫米。
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  6. A cone with a radius of 5 ft and a height of 12 feet.
     ::圆锥形半径为5英尺,高度为12英尺。
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  7. A cone with a radius of 4.5 inches and a height of 7 inches.
     ::圆锥形半径4.5英寸 高度7英寸
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  8. A cone with a radius of 3.5 inches and a height of 5.5 inches.
     ::圆锥形半径3.5英寸,高5.5英寸。
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  9. A cone with a radius of 7 cm and a height of 13 cm.
     ::圆锥体半径为7厘米,高度为13厘米。
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  10. A cone with a radius of 8 cm and a height of 11 cm.
      ::半径为8厘米、高度为11厘米的锥形。
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  11. A cone with a radius of 7.5 inches and a height of 11 inches.
      ::圆锥体半径7.5英寸,高度11英寸。
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  12. A cone with a radius of 11.5 inches and a height of 15 inches.
      ::圆锥形半径为11.5英寸,高度为15英寸。
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  13. A cone with a radius of 12.5 feet and a height of 15 feet.
      ::圆锥体半径12.5英尺,高度15英尺。

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  Solve each word problem.
  ::解决每个字的问题。

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  14. A cone has a radius of 6 meters and a height of 14 meters. What is its volume?
      ::锥体半径为6米,高度为14米。
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  15. The containers of icing for Tina’s cake decorator are cones. Each container has a radius of 2.4 inches and a height of 7 inches. If Tina buys containers of red, yellow, and blue icing, how much icing will she buy?
      ::蒂娜蛋糕设计器的冰柜是锥形的。 每个容器的半径为2.4英寸,高度为7英寸。 如果蒂娜购买红色、黄色和蓝色的冰柜,她会买多少?

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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 。