10.6 气瓶表面面积

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  圆柱体表面面积

  

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  Grandma Marion is knitting toilet paper covers as presents for the upcoming holiday season.  She needs to figure out how many balls of yarn she will need for each cover.  In order to do this, she needs to know the total outside area of the toilet paper roll. The radius of the top, and base, of the toilet paper roll measures 2 inches, and the height of the roll is 5 inches.  If one of Grandma Marion's balls of yarn can knit a piece that is 100 square inches, will one ball be enough to make a toilet paper cover?
  ::玛格丽娜·马里恩正在编织马桶纸的封面,作为即将到来的节假日的礼物。 她需要弄清楚她需要多少个毛线。 为了做到这一点,她需要知道马桶纸卷的外部区域。 顶部和底部的半径是2英寸,卷的高度是5英寸。 如果玛格丽娜·玛莉恩的毛线球之一可以织上一块100平方英寸的毛线,那么一个球是否足以做马桶纸封面?

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  In this concept, you will learn how to find the of a cylinder . 
  ::在这个概念中,你会学会如何找到圆柱体。

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  Finding the Surface Area of a Cylinder
  ::寻找圆柱体的表面区域

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  A cylinder is a that has two circular faces at each end.  The side of a cylinder is not called a face because it is curved. The surface area of a cylinder is the total of area of each circular face and the side of the cylinder. Imagine a can of soup. The top, bottom, and label around the can would make up the surface area of the can. To find the surface area, you must be able to calculate the area of each face and the side, and then add these areas together.
  ::圆柱形是一个圆形圆形圆形的圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形的圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形的圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形

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  One way to calculate the surface area of a cylinder is to use a .  Imagine you could unroll the soup can so that it is completely flat. You would have something that looks like this.
  ::计算圆柱体表面面积的一个方法就是使用 。 想象一下, 您可以打开汤罐, 这样它就完全平了 。 你会有类似这样的东西 。

  

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  The circles show the top and bottom faces of the cylinder, and the rectangle shows the side, as if it were unrolled.  With the net, you can see each face of the cylinder more clearly.
  ::圆圈显示圆柱体的顶部和底部面,矩形显示侧面,好像是开动的。用网,你可以更清楚地看到圆柱体的每个面。

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  To find the surface area, you need to calculate the area for each circle in the net.  You use the formula $\begin{array}{r}A=\pi r^2\end{array}$ to find the area of a circle, where A = area, r = radius, and $\begin{array}{r}\pi \end{array}$  is a constant that, when rounded, equals 3.14. The radius is the distance from the center point of a circle to any point on the , or perimeter , of the circle.  If you know the radius, you can calculate its area. Look closely again at the cylinder above. The two circular faces are congruent , so they must have the same radius and diameter . Let’s calculate the area for each face.
  ::要找到表面区域, 您需要计算网中每个圆的面积。 您需要使用公式 Ar2 来找到圆的面积, 其中 A = 区域, r = 半径, 和 是一个常数, 当四舍五入时, 等于 3. 14 。 半径是圆的中心点到圆圈或圆圈周围任何一点的距离。 如果您知道圆圈的半径, 您可以计算其面积 。 仔细查看上面的圆柱体 。 两个圆形是相近的, 所以它们必须具有相同的半径和直径 。 lets 计算每个圆形的面积 。

  

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  $$\begin{array}{rlrl}& \text{bottom face}& & \text{top face}\\ A& =\pi r^2& & A=\pi r^2\\ A& =\pi (4{)}^2& & A=\pi (4{)}^2\\ A& =16\pi & & A=16\pi \\ A& =16\times 3.14& & A=16\times 3.14\\ A& =50.24c{m}^2& & A=50.24c{m}^2\end{array}$$

  
  ::=16A=16A=16A=16A×3.14A=16x3.14A=16x3.14A=50.24cm2A=5024cm24cm2A=50.24cm2A=50.24cm2

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  The area of each circular face is 50.24 square centimeters.
  ::每张圆形脸的面积为50.24平方厘米。

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  Now you need to find the area of the side. The net shows that when you “unroll” the cylinder, the side is actually a rectangle. Recall that the formula used to find the area of a rectangle is $\begin{array}{r}A=lw\end{array}$ . For cylinders, the width of the rectangle is the same as the height of the cylinder. In this case, the height of the cylinder is 8 centimeters. The length is actually the same as the circumference of the circle. When you “roll” up the side, it fits exactly once around the circle. Find the circumference of a circle with the formula $\begin{array}{r}C=2\pi r\end{array}$ .
  ::现在您需要找到侧边的区域。 网形显示, 当您“ 滚动” 圆柱体时, 侧面实际上是一个矩形。 回顾用来查找矩形区域的公式是 A=lw 。 对于圆柱体, 矩形的宽度与圆柱体的高度相同。 在这种情况下, 圆柱体的高度是 8 厘米。 长度实际上与圆圈的环绕相同。 当您“ 滚动” 时, 侧面正好适合圆圈周围的一环。 找到一个圆形的环形, 与公式 C= 2Qr 相同 。

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  $$\begin{array}{rl}C& =2\pi r\\ C& =2\pi 4\\ C& =8\pi \\ C& =8\times 3.14=25.12cm\end{array}$$

  
  ::C=2rC=24C=8C=8×3.14=25.12厘米

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  The circumference of the top and bottom of the cylinder, or the length of the side, is 25.12 centimeters.  
  ::圆柱体顶部和底部或侧面长度的周长为25.12厘米。

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  To find the area of the cylinder's side, multiply the circumference of the circle, or length of the cylinder, by the height, or width, of the cylinder.
  ::要找到圆柱体的侧面区域,将圆圆或圆柱体的长度乘以圆柱体的高度或宽度。

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  $$\begin{array}{rl}A& =Ch\\ A& =25.12\times 8\\ A& =200.96{cm}^2\end{array}$$

   
  
  ::A=CA=25.12x8A=200.96平方厘米2

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  Now that you know the area of both circular faces and the side, add them together to find the surface area of the cylinder.
  ::现在你知道圆形面和侧面的面积了 把它们加在一起 找到圆柱体的表面面积

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  $$\begin{array}{rl}& \text{bottom face}\text{top face}\text{side}\text{surface area}\\ & 50.24c{m}^2+50.24c{m}^2+200.96c{m}^2=301.44c{m}^2\end{array}$$

  
  ::50.24厘米2+50.24厘米2+200.96厘米2=301.44厘米2

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  The answer is the total surface area of the cylinder is 301.44 square centimeters.
  ::答案是圆柱体的总面积为301.44平方厘米。

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  You can also use one formula to represent the surface area of the faces and side of a cylinder.
  ::您也可以使用一种公式来表示圆柱体表面和侧面的表面面积。

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  You may have noticed in the previous section that the two circular faces had the same area. This is because they are congruent , and thus have the same radius. You can therefore calculate the area of the pair of circular faces at one time. Simply double the area formula, which gives you  $\begin{array}{r}2\pi r^2\end{array}$ .
  ::您可能在前一节中注意到,两个圆形面有相同的区域。 这是因为两个圆形面是相同的, 并且具有相同的半径。 因此您可以同时计算圆形面对的面积。 只需将区域公式翻一番, 给您 2Qr2 。

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  You can also combine the measurements for the side into a simpler equation. You need to find the circumference by using the formula $\begin{array}{r}2\pi r\end{array}$ , and then multiply this by the height of the cylinder  $\begin{array}{r}h\end{array}$ .  The formula for the side of the cylinder then becomes  $\begin{array}{r}2\pi rh\end{array}$ .
  ::您也可以将侧面的测量结果合并成一个简单的方程。您需要使用公式 2°r 来找到环形,然后乘以圆柱的高度。圆柱的侧面的公式随后变为 2°rh 。

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  When you combine the formulas for the faces and the side, you get this formula:
  ::当组合面和侧面的公式时,你会得到这个公式:

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  $\begin{array}{r}SA=2\pi r^2+2\pi rh\end{array}$
  ::SA=2r2+2rh

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  Now let’s apply this formula to the example from the previous section.
  ::现在让我们将这个公式应用到上一节的示例中。

  

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  In this cylinder, $\begin{array}{r}r=4\end{array}$ inches and $\begin{array}{r}h=8\end{array}$ inches. Simply put these numbers into the formula and solve for surface area. 
  ::在这个气瓶中, r=4英寸和h=8英寸。 简单地把这些数字放在公式中, 并解决表面区域 。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (4{)}^2+2\pi (4)(8)\\ SA& =2\pi (16)+2\pi (32)\\ SA& =32\times 3.14+64\times 3.14\\ SA& =100.48+200.96\\ SA& =301.44cm{.}^2\end{array}$$

  
  ::SA=2r2+2rhSA=2(4)2+2(4)(8)SA=2(16)+2(32)SA=32×3.14+64×3.14SA=100.48+200.96SA=301.44厘米。

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  Again, the answer is the surface area of this cylinder is 301.44 square centimeters.
  ::同样,答案是这个气瓶的表面面积是301.44平方厘米。

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  This formula just saves a little time. Let’s look at another example.
  ::这个公式只是节省了一点时间。让我们看看另一个例子。

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  What is the surface area of the figure below?
  ::下图的表面积是多少?

  

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  You have all of the measurements you need. Let’s put them into the formula and solve for surface area, $\begin{array}{r}SA\end{array}$ .
  ::我们把它们放到公式中,解决地表面积问题,SA。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (3.5{)}^2+2\pi (3.5)(28)\\ SA& =2\pi (12.25)+2\pi (98)\\ SA& =24.5\pi +196\pi \\ SA& =220.5\times 3.14\\ SA& =692.37c{m}^2\end{array}$$

  
  ::SA=2r2+2rhSA=2(3.5)2+2(3.5)(28)SA=2(12.25)+2(98)SA=24.5(196)SA=220.5×3.14SA=692.37cm2

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  This cylinder has a surface area of 692.37 square centimeters.
  ::该气瓶的表面面积为692.37平方米。

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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 Grandma Marion's toilet paper covers.
  ::之前,有人给你一个问题 关于玛莉恩奶奶的卫生纸封面。

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  She wants to know how many balls of yarn are required for one cover.  The dimensions of the toilet paper are height is 5 inches and radius is 2 inches.  One ball of yarn will create a knitted piece of 100 square inches.
  ::她想知道一个封面需要多少个毛线球。 卫生纸的尺寸是5英寸高, 半径是2英寸。 一个毛线球将产生一个100平方英寸的编织片。

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  First, plug in the values of the radius and height into the surface area formula and multiply.
  ::首先,将半径和高度的值插入到表面积公式和乘以中。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (2{)}^2+2\pi (2)(5)\\ SA& =2\pi (4)+2\pi (10)\end{array}$$

  
  ::SA=2r2+2rhSA=2(2)2+2(2)(5)SA=2(4)+2(10)

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  $$\begin{array}{rl}SA& =2\pi (4)+2\pi (10)\\ SA& =8\pi +20\pi \end{array}$$

  
  ::SA=2_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Next, replace the value for pi and multiply.
  ::下一步,替换 pi 和乘法的值。

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  $$\begin{array}{rl}SA& =8\times 3.14+20\times 3.14\\ SA& =25.12+62.8\end{array}$$

  
  ::SA=8x3.14+20x3.14SA=25.12+62.8

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  Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将两个数值加在一起作为答案,确保包括适当的计量单位。

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  $$\begin{array}{rl}SA& =25.12+62.8\\ SA& =87.92{in}^2\end{array}$$

  
  ::SA=25.12+62.8SA=87.92英寸2

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  The answer is the surface area of the toilet paper roll is 87.92 square inches.  Therefore, one ball of yarn is enough to knit a toilet paper cover.
  ::答案是马桶纸卷的表面面积是87.92平方英寸,因此,一团毛线足以织上马桶纸盖。

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  Example 2
  ::例2

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  What is the surface area of the cylinder below using the net of the figure?
  ::利用数字网,下面圆柱体的表面面积是多少?

  

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  First, draw the net. This is done by drawing the bottom and top faces, each with a radius of 7 inches, and the side rectangle, which is 14 inches in length, between the circular faces.  
  ::首先,绘制网。这是通过绘制下方和顶面的图画完成的,每个图画半径为7英寸,以及圆形面之间14英寸的侧矩形。

  

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  Next, calculate the areas of the circles representing the top and bottom of the cylinder, and the rectangle representing the side of the cylinder.
  ::接下来,计算圆圈的面积,代表圆柱体的顶部和底部,矩形代表圆柱体的侧面。

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  Areas of circles:
  ::圆圈区域:

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  $$\begin{array}{rlrl}& \text{bottom face}& & \text{top face}\\ A& =\pi r^2& & A=\pi r^2\\ A& =\pi (7{)}^2& & A=\pi (7{)}^2\\ A& =49\times 3.14& & A=49\times 3.14\\ A& =153.86in{.}^2& & A=153.86in{.}^2\end{array}$$

  
  ::下方面部Ar2A(7)2A(7)2A(7)2A=49×3.14A=49×3.14A=49×3.14A=153.86在2A=153.86英寸

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  Area of rectangle:
  ::矩形区域 :

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  $$\begin{array}{rl}& \text{side}\\ C& =2\pi r\\ C& =2\pi (7)\\ C& =14\times 3.14\\ C& =43.96in{.}^2\\ \\ Ch& =43.96\times 14=615.44in{.}^2\end{array}$$

  
  ::外侧C=2rC=2(7)(C)=14×3.14C=43.96英寸2.Ch=43.96×14=615.44英寸2.

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  Then, add these areas together to find the surface area of the cylinder, making sure to include the appropriate unit of measurement.
  ::然后将这些区域加在一起,以找到气瓶的表面积,确保包括适当的测量单位。

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  $$\begin{array}{r}153.86+153.86+615.44=923.16in{.}^2\end{array}$$

  
  ::153.86+153.86+615.44=923.16英寸。

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  The answer is the surface area of the cylinder pictured above is 923.16 square inches.
  ::答案是上面的圆筒表面面积是923.16平方英寸。

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  Calculate the surface area of a cylinder with a radius of 7 in and a height of 12 in.
  ::计算圆柱体的表面面积,圆柱体半径为7,高度为12。

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  First, plug in the values of the radius and height into the surface area formula, then multiply.
  ::首先,将半径和高度值插入表面积公式,然后乘以。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (7{)}^2+2\pi (7)(12)\\ SA& =2\pi (49)+2\pi (84)\end{array}$$

  
  ::SA=2r2+2rhSA=2(7)2+2(7)(12)SA=2(49)+2(84)

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  $$\begin{array}{rl}SA& =2\pi (49)+2\pi (84)\\ SA& =98\pi +168\pi \end{array}$$

  
  ::SA=2______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Next, replace the value for pi and multiply.
  ::下一步,替换 pi 和乘法的值。

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  $$\begin{array}{rl}SA& =98\times 3.14+168\times 3.14\\ SA& =307.72+527.52\end{array}$$

  
  ::SA=98x 3.14+168x 3.14SA=307.72+527.52

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  Then, add the two values together for the surface area, making sure to include the unit of measurement.
  ::然后将表面积的两个数值加在一起,确保包括测量单位。

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  $$\begin{array}{rl}SA& =307.72+527.52\\ SA& =835.24{in}^2\end{array}$$

  
  ::SA=307.72+527.52SA=835.24英寸2

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  The answer is the surface area of the cylinder is   $\begin{array}{r}835.24\end{array}$   square inches.
  ::答案是圆柱体的表面面积是835.24平方英寸。

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  Example 3
  ::例3

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  Calculate the surface area of a cylinder with a radius of 5 ft and a height of 10 ft.
  ::计算圆柱体的表面面积,圆柱体半径为5英尺,高度为10英尺。

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  First, plug in the values of the radius and height into the surface area formula and multiply.
  ::首先,将半径和高度的值插入到表面积公式和乘以中。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (5{)}^2+2\pi (5)(10)\\ SA& =2\pi (25)+2\pi (50)\end{array}$$

  
  ::SA=2r2+2rhSA=2(5)2+2(5)(10)SA=2(25)+2(50)

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  $$\begin{array}{rl}SA& =2\pi (25)+2\pi (50)\\ SA& =50\pi +100\pi \end{array}$$

  Next, replace the value for pi and multiply.
  ::SA=2(25)+2(50)SA=50100(Next),替换 pi 和 乘法的值。

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  $$\begin{array}{rl}SA& =50\times 3.14+100\times 3.14\\ SA& =157+314\end{array}$$

  
  ::SA=50x3.14+100x3.14SA=157+314

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  Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将两个数值加在一起作为答案,确保包括适当的计量单位。

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  $$\begin{array}{rl}SA& =157+314\\ SA& =471{ft}^2\end{array}$$

   
  ::SA=157+314SA=471平方英尺

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  The answer is the surface area of the cylinder is   $\begin{array}{r}471\end{array}$ square feet.
  ::答案是圆柱体的表面面积是471平方英尺。

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  Example 4
  ::例4

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  Calculate the surface area of a cylinder with a radius of 7 in. and a height of 12 in.
  ::计算圆柱体表面面积,半径为 7 英寸,高度为 12 英寸。

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  First, plug in the values of the radius and height into the surface area formula and multiply.
  ::首先,将半径和高度的值插入到表面积公式和乘以中。

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  $$\begin{array}{rl}SA& =2\pi r^2+2\pi rh\\ SA& =2\pi (7{)}^2+2\pi (7)(12)\\ SA& =2\pi (49)+2\pi (84)\end{array}$$

  
  ::SA=2r2+2rhSA=2(7)2+2(7)(12)SA=2(49)+2(84)

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  $$\begin{array}{rl}SA& =2\pi (49)+2\pi (84)\\ SA& =98\pi +168\pi \end{array}$$

  
  ::SA=2______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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   Next, replace the value for pi and multiply.
  ::下一步,替换 pi 和乘法的值。

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  $$\begin{array}{rl}SA& =98\times 3.14+168\times 3.14\\ SA& =307.72+527.52\end{array}$$

  
  ::SA=98x 3.14+168x 3.14SA=307.72+527.52

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  Then, add the two values together for the answer, making sure to include the appropriate unit of measurement.
  ::然后,将两个数值加在一起作为答案,确保包括适当的计量单位。

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  $$\begin{array}{rl}SA& =307.72+527.52\\ SA& =835.24{in}^2\end{array}$$

   
  ::SA=307.72+527.52SA=835.24英寸2

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  The answer is the surface area of the cylinder is   $\begin{array}{r}835.24\end{array}$ square inches.
  ::答案是圆柱体的表面面积是835.24平方英寸。

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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 surface area of each cylinder given its height and radius.
  ::根据每个圆柱体的高度和半径,查找每个圆柱体的表面面积。

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  1. $\begin{array}{r}r=1m,h=3m\end{array}$
     ::r=1米,h=3米
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  2. $\begin{array}{r}r=2cm,h=4cm\end{array}$
     ::r=2厘米, h=4厘米
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  3. $\begin{array}{r}r=6in,h=10in\end{array}$
     ::r=6 英寸, h=10 英寸
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  4. $\begin{array}{r}r=4in,h=6in\end{array}$
     ::r=4 英寸, h=6 英寸
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  5. $\begin{array}{r}r=5in,h=10in\end{array}$
     ::r=5 英寸, h=10 英寸
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  6. $\begin{array}{r}r=8ft,h=6ft\end{array}$
     ::r=8英尺,h=6英尺
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  7. $\begin{array}{r}r=10m,h=15m\end{array}$
     ::r=10米,h=15米
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  8. $\begin{array}{r}r=9cm,h=12cm\end{array}$
     ::r=9厘米,h=12厘米
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  9. $\begin{array}{r}r=6m,h=8m\end{array}$
     ::r=6米,h=8米
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  10. $\begin{array}{r}r=2cm,h=3cm\end{array}$
      ::r=2厘米,h=3厘米

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  Find the surface area of each cylinder given its height and diameter.
  ::根据每个圆柱体的高度和直径,寻找每个圆柱体的表面面积。

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  11. $\begin{array}{r}d=8m,h=11m\end{array}$
      ::d=8米,h=11米
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  12. $\begin{array}{r}d=10in,h=14in\end{array}$
      ::d=10 英寸, h=14 英寸
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  13. $\begin{array}{r}d=8cm,h=10cm\end{array}$
      ::d=8厘米,h=10厘米
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  14. $\begin{array}{r}d=12m,h=15m\end{array}$
      ::d=12米,h=15米
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  15. $\begin{array}{r}d=15in,h=20in\end{array}$
      ::d=15 英寸, h=20 英寸

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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 。

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  Resources
  ::资源