5.5 一个部门面积

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  部门面积

  播放段落

  Introduction
  ::导言

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  Very often, a grower waters their property at varying rates. For example, some days the irrigation pipe is only turned through an angle of  $\begin{array}{r}{54}^{\circ }\end{array}$ . The grower needs to know the area watered that day. The grower found a formula on the internet to calculate the irrigated area, but it did not make sense. The formula is  $\begin{array}{r}A=0.0087r^2\theta \end{array}$ . The questions from the grower are: "How is this formula created? Why does the value .0087 appear in this formula?"
  ::通常情况下, 种植者会以不同的速率饮用他们的地产。 例如, 有几天灌溉管道只是经过54°Q的角。 种植者需要知道当日的水源面积。 种植者在互联网上找到了计算灌溉面积的公式, 但它说不通。 公式是 A=0.0087r2。 种植者提出的问题是 : “ 这个公式是如何创建的? 为什么这个公式中会出现 . 0087 值? ”

  

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  Area of Sector Formula
  ::部门公式区区

  播放段落

  One of the most common geometric formulas is the area of a disk :
  ::最常用的几何公式之一是磁盘面积:

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  $$\begin{array}{r}A=\pi r^2.\end{array}$$

  
  ::2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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  From this, we can derive the area of the sector with central angle,  $\begin{array}{r}\theta \end{array}$ radians. Comparing the measure of the central angle with the full circle’s angle of $\begin{array}{r}2\pi \end{array}$ , t he sector of the area is proportional to the area of the disk :
  ::我们可以从中角的弧度中取出扇区面积。 将中央角的度量与整个圆角的角值 2作比较, 扇区与磁盘的面积成正比 :

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  $$\begin{array}{rl}{A}_s& =\frac{\theta }{2\pi }\pi r^2\\ A_s& =\frac{1}{2}{r}^2\theta ,\end{array}$$

  
  ::作为2r2As=12r2,

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  where $\begin{array}{r}r\end{array}$  is the radius. 
  ::r 是半径。

   

  
   

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  Examples
  ::实例

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  Example 1
  ::例1

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  Crops are often grown using a technique called center pivot irrigation, which results in circular- shaped fields.
  ::作物往往使用称为中央支流灌溉的技术种植,从而形成圆形田地。

  

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  Here is a satellite image taken over fields in Kansas that use this type of irrigation system:
  ::使用这种灌溉系统的卫星图象,

  

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  If the irrigation pipe is 450 m in length, what is the area that can be irrigated after a rotation of $\begin{array}{r}\frac{2\pi }{3}\end{array}$ radians?
  ::如果灌溉管道长450米,那么在2+3弧度旋转后可以灌溉的面积是多少?

  

  
  

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  Solution:
  ::解决方案 :

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  Use the formula
  ::使用公式

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(450\text{ m}{)}^2\left(\frac{2\pi }{3}\right).\end{array}$$

  
  ::A=12r2A=12(450米)2(23)。

  
  

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  The area is approximately 212,058 square meters.
  ::该地区面积约为212,058平方米。

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  Example 2
  ::例2

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  A drawing of a doughnut has a hole in the middle with a radius of 1 cm, and the distance from the center of the hole to the outer edge of the doughnut is 3 cm. What is the area of a sector of $\begin{array}{r}\frac{1}{4}\end{array}$ of the doughnut?
  ::甜甜圈的图画中间有一个圆洞,半径为1厘米,而从圆洞中心到甜甜圈外缘的距离是3厘米。甜甜圈14个区块的面积是多少?

  

  
  

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  Solution:
  ::解决方案 :

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  The formula for the area of a sector is

  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta .\end{array}$$

  Use this formula to find the area of the sector from the center outward:
  ::扇区面积的公式为 A= 12r2。 使用此公式从中间向外查找扇区面积 :

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}{3}^2\frac{\pi }{2}\\ A& =\frac{9\pi }{4}\end{array}$$

  
  ::A=12r2A=12322A=94

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  Now subtract the area of the sector that is part of the hole, and therefore not part of the doughnut:
  ::现在去掉作为洞的一部分的部门面积, 而不是甜甜圈的一部分:

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(1{)}^2\frac{\pi }{2}\\ A& =\frac{\pi }{4}\end{array}$$

  
  ::A=12r2A=12(1)(1)22A4

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  Area of the sector of doughnut:

  $$\begin{array}{rl}A& =\frac{9\pi }{4}-\frac{\pi }{4}=\frac{8\pi }{4}=2\pi {\text{ cm}}^2\end{array}$$

  
  ::甜甜圈部门面积:A=944=84=22cm2

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  Example 3
  ::例3

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  A driver is traveling around a circular track that has radius of 70 meters. If the angle from the starting line to her current position is $\begin{array}{r}\frac{\pi }{3}\end{array}$ radians, what is the area of the sector traced out by her car?
  ::驾驶员绕着圆形轨道行驶,圆形轨道半径为70米。如果从起点线到她目前位置的角是3弧度,那么她的车在哪一区?

  

  
  

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  Solution:
  ::解决方案 :

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  The area of a sector is:

  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \end{array}$$

  This leads us to:
  ::A=12r2* 导致我们:

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(70\text{ m}{)}^2\frac{\pi }{3}\\ A& \approx 2565.6{\text{ m}}^2\end{array}$$

  
  ::A=12r2A=12(70米)23A=2565.6平方米

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  Example 4
  ::例4

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  If the radius of a sector is 5 feet and the sector has  a central  angle of $\begin{array}{r}{43}^{\circ }\end{array}$ , find the area of the sector.
  ::如果一个部门的半径是5英尺,而该部门的中心角是43,请找到该部门的面积。

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  Solution:
  ::解决方案 :

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  First, convert the angle from degrees to radians:
  ::首先,将角度从度转换为弧度:

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  $$\begin{array}{r}{43}^{\circ }\cdot \frac{\pi }{{180}^{\circ }}\approx 0.75\text{ radians}.\end{array}$$

  
  ::43180+0.75弧度

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  Use  $\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$  to solve for the area:
  ::使用 A= 12r2 来解析区域 :

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  $$\begin{array}{rl}A& =\frac{1}{2}(5\text{ ft}{)}^2(.75)\\ A& =9.38{\text{ ft}}^2.\end{array}$$

  
  ::A=12(5英尺)(2.75)A=9.38英尺。

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  Example 5
  ::例5

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  If a pie wedge has an area of 15 square inches and the pie has a radius of 9 inches, find the  central angle.
  ::如果馅饼的面积是15平方英寸 而馅饼的半径是9英寸 找到中心角

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  Solution:
  ::解决方案 :

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  Since you know that $\begin{array}{r}A=\frac{1}{2}{r}^2\theta \end{array}$ , you can solve for the angle swept out by the sector:
  ::由于您知道 A=12r2, 您可以解开扇区扫出的角 :

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  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta \\ 15=\frac{1}{2}(81)(\theta )\\ \theta =\frac{(2)(15)}{81}\\ \theta \approx .37\text{ radians}\end{array}$$

  
  ::A=12r215=12(81)() (2)(15)813.37弧度

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  Example 6
  ::例6

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  Returning to the grower's question, if the angle of rotation is in degrees, it must be converted to radians. That conversion factor is  $\begin{array}{r}\pi /180\approx 0.0175\end{array}$  . The area formula is 

  $$\begin{array}{r}A=\frac{1}{2}{r}^2\theta =\frac{1}{2}{r}^2{\theta }_d\frac{\pi }{180}\approx 0.0087r^2{\theta }_d,\end{array}$$

  
  ::回到种植者的问题,如果旋转角度为度,则必须转换成弧度。转换系数为 /1800.0175。区域公式为 A=12r212r222d1800.0087r2d,

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  where  $\begin{array}{r}\theta \end{array}$  is the angle in radians, and  $\begin{array}{r}{\theta }_d\end{array}$  is the angle in degrees.
  ::其中 是弧度的角度, d 是度的角度。

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  Since the angle of rotation is  $\begin{array}{r}{54}^{\circ }\end{array}$  and the length of the irrigation pipe is 400 feet:
  ::由于旋转角度是54英寸,灌溉管道长度为400英尺:

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  $$\begin{array}{rl}A& =\frac{1}{2}{r}^2\theta \\ A& =\frac{1}{2}(400\text{ ft}{)}^2(54)\frac{\pi }{180}\\ A& \approx 75,398{\text{ ft}}^2\end{array}$$

  
  ::A=12r2A=12(400英尺)(54(8)180A75,398英尺)

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  The area irrigated is approximately 75,398 ft ² .
  ::灌溉面积约为75 398平方英尺。

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  Summary
  ::摘要

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  • The area of a sector is  $\begin{array}{r}A=\frac{1}{2}{r}^2\theta ,\end{array}$  where  $\begin{array}{r}\theta \end{array}$  is in radians.
    ::一个扇区的区域为A=12r2,其中为弧度。
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  • Use the fact that $\begin{array}{r}\pi \end{array}$  is equal to $\begin{array}{r}{180}^{\circ }\end{array}$  to convert between degrees and radians.
    ::使用 = 等于 180 = 来转换度和弧度。

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  Review
  ::回顾

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  1. If the radius of a sector is 8 inches and the central angle of the sector is $\begin{array}{r}{40}^{\circ }\end{array}$ , find the area of the sector.
     ::如果一个部门的半径为8英寸,而这个部门的中心角为40英寸,请找到这个部门的面积。
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  2. If the radius of a sector is 12 inches and the central angle of the sector is $\begin{array}{r}\frac{\pi }{6}\end{array}$ radians, find the area of the sector.
     ::如果一个部门的半径是12英寸,而该部门的中心角是 6 弧度,请找到这一部门的面积。
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  3. If the radius of a sector is 6 inches and the central angle of the sector is $\begin{array}{r}{140}^{\circ }\end{array}$ , find the area of the sector.
     ::如果一个部门的半径是6英寸,而这个部门的中心角是140英寸,请找到这个部门的面积。
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  4. If the radius of a sector is 5 inches and the central angle of the sector is $\begin{array}{r}\frac{5\pi }{3}\end{array}$ radians, find the area of the sector.
     ::如果一个部门的半径是5英寸,而该部门的中心角是5-3弧度,请找到这一部门的面积。
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  5. If the radius of a sector is 10 inches and the central angle of the sector is $\begin{array}{r}{100}^{\circ }\end{array}$ , find the area of the sector.  

     

     
     ::如果一个部门的半径是10英寸,而这个部门的中心角是100英寸,请找到这个部门的面积。
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  6. If a pie wedge has an area of 10 square inches and the pie has a radius of 6 inches, find the angle swept out by the sector.
     ::如果馅饼的面积为10平方英寸,而馅饼的半径为6英寸, 找到被这个部门冲走的角。
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  7. If a pie wedge has an area of 15 square inches and the pie has a radius of 4 inches, find the angle swept out by the sector.
     ::如果馅饼的面积为15平方英寸,而馅饼的半径为4英寸, 就能发现这一区段的倾角。
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  8. If a pie wedge has an area of 12 square inches and the pie has a radius of 3 inches, find the angle swept out by the sector.
     ::如果馅饼的面积为12平方英寸,而馅饼的半径为3英寸, 就能发现这一区段的倾角。
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  9. If you have a piece of round cake that has an area of 20 square inches and you know the piece sweeps out an angle of $\begin{array}{r}\frac{\pi }{3}\end{array}$ radians, find the radius of the cake.
     ::如果你有一块圆饼, 面积为20平方英寸, 你知道这块蛋糕扫出 3 弧度角, 找到蛋糕的半径 。
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  10. If you have a piece of round cake that has an area of 100 square inches and you know the piece sweeps out an angle of $\begin{array}{r}{50}^{\circ }\end{array}$ , find the radius of the cake.
      ::如果你有一块圆饼, 面积100平方英寸, 你知道那块蛋糕扫出50角角, 找到蛋糕的半径。
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  11. If you have a piece of round cake that has an area of 35 square inches and you know the piece sweeps out an angle of $\begin{array}{r}\frac{2\pi }{5}\end{array}$ radians, find the radius of the cake.
      ::如果你有一块圆饼, 面积35平方英寸, 你知道那块蛋糕扫出角角 2+5 弧度, 找到蛋糕的半径 。
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  12. If you have a piece of round cake that has an area of 20 square inches and you know the piece sweeps out an angle of $\begin{array}{r}{30}^{\circ }\end{array}$ , find the radius of the cake.
      ::如果你有一块圆饼, 面积为20平方英寸, 你知道那块蛋糕扫出角角30英寸, 找到蛋糕的半径。

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  A pizza has a radius of 10 inches. Use this information to answer questions 13-14. 
  ::披萨半径10英寸 用这个信息回答问题13-14

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  13. A slice is removed. The length of the crust of the missing slice is 3 inches. What is the area of the missing slice?
      ::切片被切除。 遗漏切片的地壳长度为 3 英寸。 遗漏切片的面积是 多少 ?
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  14. You eat 3 pieces with a combined central angle of $\begin{array}{r}\frac{4\pi }{5}\end{array}$ . What is the area of the pizza you ate?
      ::你吃了三片 中央角度是4 5,你吃披萨的面积是多少?
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  15. A large pizza has a radius of 12 inches. What is the area of half of the large pizza?
      ::大比萨饼半径12英寸 大比萨饼的一半面积是多少?

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  Review ( Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。