13.4 应用问题

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  应用问题

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  A 40-foot-tall tree casts a shadow of 80 feet. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground?
  ::一棵40英尺长的树投下80英尺长的阴影,从阴影的尽头高到树的顶端的地面,其角度是什么?

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  Application Problems
  ::应用问题

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  When solving word problems, it is important to understand the terminology used to describe angles. In trigonometric problems, the terms angle of elevation and angle of depression are commonly used. Both of these angles are always measured from a horizontal line as shown in the diagrams below.
  ::在解决字词问题时,必须理解用来描述角度的术语。在三角问题中,通常使用高高角度和低压角度这两个术语。这两个角度总是从水平线测量,如下图所示。

  

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  Let's solve the following problems.
  ::让我们解决以下的问题。

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  1. An airplane approaching an airport spots the runway at an angle of depression of $\begin{array}{r}{25}^{\circ }\end{array}$ . If the airplane is 15,000 ft above the ground, how far (ground distance) is the plane from the runway? Give your answer to the nearest 100 ft.
     ::接近机场的一架飞机将跑道置于25英尺的压抑度角。如果飞机在地面上方15000英尺,则飞机离跑道有多远(地面距离)?请回答最近的100英尺。

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  Make a diagram to illustrate the situation described and then use a trigonometric ratio to solve. Keep in mind that an angle of depression is down from a horizontal line of sight-in this case a horizontal line from the pilot of the plane parallel to the ground.
  ::绘制一个图表来说明描述的情况, 然后使用三角比来解决问题。 请注意, 压抑的角度是从水平的视线向下, 在此情况下, 从平面的飞行员到地面的平行水平线 。

  

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  Note that the angle of depression and the alternate interior angle will be congruent, so the angle in the triangle is also $\begin{array}{r}{25}^{\circ }\end{array}$ .
  ::请注意,低音角和内部其他角度是相同的, 所以三角形中的角度也是 25 。

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  From the picture, we can see that we should use the tangent ratio to find the ground distance.
  ::从画面中,我们可以看到,我们应该使用正切比来寻找地面距离。

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  $$\begin{array}{rl}\tan {25}^{\circ }& =\frac{15000}{d}\\ d& =\frac{15000}{\tan {25}^{\circ }}\approx 32,200ft\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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  2. Rachel spots a bird in a tree at an angle of elevation of $\begin{array}{r}{30}^{\circ }\end{array}$ . If Rachel is 20 ft from the base of the tree, how high up in the tree is the bird? Give your answer to the nearest tenth of a foot.
     ::瑞秋在树上标出一只鸟在高度30英尺角的树上。如果瑞秋离树底20英尺,那鸟在树上有多高?请回答最近的十英尺。

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  Make a diagram to illustrate the situation. Keep in mind that there will be a right triangle and that the right angle is formed by the ground and the trunk of the tree.
  ::绘制图表以说明情况。 记住将有一个右三角形, 右角由树的地面和树干组成 。

  

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  Here we can use the tangent ratio to solve for the height of the bird
  ::在这里,我们可以用正切比来解决鸟的高度

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  $$\begin{array}{rl}\tan {30}^{\circ }& =\frac{h}{20}\\ h& =20\tan {30}^{\circ }\approx 11.5ft\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么?

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  3. A 12 ft ladder is leaning against a house and reaches 10 ft up the side of the house. To the nearest degree , what angle does the ladder make with the ground?
     ::12英尺高的梯子靠着一栋房子,在房子的侧面达到10英尺高。

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  In this problem, we will need to find an angle. By making a sketch of the triangle we can see which inverse trigonometric ratio to use.
  ::在这个问题上,我们需要找到一个角度。通过绘制三角形的草图,我们可以看到哪个反三角比可以使用。

  

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  $$\begin{array}{rl}\sin x^{\circ }& =\frac{10}{12}\\ {\sin }^{-1}\left(\frac{10}{12}\right)& \approx {56}^{\circ }\end{array}$$

  
  ::-=YTET -伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=- 翻译:

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find  the angle of elevation from the end of the shadow to the top of the tree with respect to the ground.
  ::早些时候,有人要求你找到从阴影的尽头到树顶的高度角,从地面到地面。

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  If you draw this situation, you will see that we are dealing with a right triangle. The side opposite the angle of elevation is 40. The side adjacent to the angle is 80. Therefore, we can use the tangent to find the angle of elevation.
  ::如果你画出这种情况, 你会看到我们正在处理一个右三角形。 向右的角是40。 向右的角是40。 向右的角是80。 因此, 我们可以用正切的角来找到高的角 。

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  $$\begin{array}{rl}\tan x^{\circ }& =\frac{40}{80}=\frac{1}{2}\\ {\tan }^{-1}(\frac{1}{2})=\approx {26.57}^{\circ }\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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  Example 2
  ::例2

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  A ramp makes a $\begin{array}{r}{20}^{\circ }\end{array}$ angle with the ground. If door the ramp leads to is 2 ft above the ground, how long is the ramp? Give your answer to the nearest tenth of a foot.
  ::斜坡以地面为角度 20 ° 。 如果斜坡通往的门在地面上方两英尺, 斜坡长度是多少? 请将答案给予最近的十分之一英尺 。

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  $$\begin{array}{rl}\sin {20}^{\circ }& =\frac{2}{x}\\ x& =\frac{2}{\sin {20}^{\circ }}\approx 5.8ft\end{array}$$

  
  ::202xx=2sin 205.8英尺

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  Example 3
  ::例3

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  Charlie lets out 90 ft of kite string. If the angle of elevation of the string is $\begin{array}{r}{70}^{\circ }\end{array}$ , approximately how high is the kite? Give your answer to the nearest foot.
  ::Charlie 释放90英尺风筝弦。 如果弦的高度角度是 70 英寸, 风筝的高度大约是多少? 请回答最近的脚 。

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  $$\begin{array}{rl}\sin {70}^{\circ }& =\frac{x}{90}\\ x& =90\sin {70}^{\circ }\approx 85ft\end{array}$$

  
  ::70x90x=90sin7085英尺

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  Example 4
  ::例4

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  A ship’s sonar spots a wreckage at an angle of depression of $\begin{array}{r}{32}^{\circ }\end{array}$ . If the depth of the ocean is about 250 ft, how far is the wreckage (measured along the surface of the water) from the ship, to the nearest foot.
  ::一艘船舶的声纳在32°Z的低沉降角发现残骸。 如果海洋的深度约为250英尺,那么从船舶到最近的脚的残骸(测量水面)有多远。

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  $$\begin{array}{rl}\tan {32}^{\circ }& =\frac{250}{x}\\ x& =\frac{250}{\tan {32}^{\circ }}\approx 400ft\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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  Review
  ::回顾

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  Use the Pythagorean Theorem and/or trigonometry to solve the following word problems.
  ::使用毕达哥里安定理和(或)三角测量法解决以下字词问题。

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  1. A square has sides of length 8 inches. To the nearest tenth of an inch, what is the length of its diagonal?
     ::方形有8英寸长的两边。至最近的十分之一英寸,其对角长度是多少?
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  2. Layne spots a sailboat from her fifth floor balcony, about 25 m above the beach, at an angle of depression of $\begin{array}{r}{3}^{\circ }\end{array}$ . To the nearest meter, how far out is the boat?
     ::Layne从她的五楼阳台,大约在海滩上25米处发现一艘帆船,方向是3°C的压抑度角。
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  3. A zip line takes passengers on a 200 m ride from high up in the trees to a ground level platform. If the angle of elevation of the zip line is $\begin{array}{r}{10}^{\circ }\end{array}$ , how high above ground is the tree top start platform? Give your answer to the nearest meter.
     ::拉链线将乘坐200米长的乘车乘客从树上到地面平台。如果拉链的高度角是 10 ,那么树顶起站平台的高度是 10 ,那么树顶起站平台的高度是 多少? 请将答案给最近的计数器 。
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  4. The angle of depression from the top of an apartment building to the base of a fountain in a nearby park is $\begin{array}{r}{57}^{\circ }\end{array}$ . If the building is 150 ft tall, how far away, to the nearest foot, is the fountain?
     ::从公寓楼顶到附近公园喷泉底部的压抑角度是57英寸。 如果该建筑物高150英尺,离最近的脚还有多远? 喷泉离最近的脚有多远?
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  5. A playground slide platform is 6 ft above ground. If the slide is 8 ft long and the end of the slide is 1 ft above ground, what angle does the slide make with the ground? Give your answer to the nearest degree.
     ::游乐场幻灯片平台在地面上方为 6 英尺。 如果幻灯片长8 英尺, 幻灯片结尾处在地面上方为 1 英尺, 幻灯片用地面产生什么角度 ? 请将您的答案给予最近的程度 。
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  6. Benjamin spots a tree directly across the river from where he is standing. He then walks 27 ft upstream and determines that the angle between his previous position and the tree on the other side of the river is $\begin{array}{r}{73}^{\circ }\end{array}$ . How wide, to the nearest foot, is the river?
     ::Benjamin将一棵树直接指向河对面他所站的地方。然后他沿着27英尺高的上游行走,确定他原先的位置与河另一边的树之间的角是73英尺。河的宽度是多少?
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  7. A rectangle has sides of length 6 in and 10 in. To the nearest degree, what angle does the diagonal make with the longer side?
     ::矩形的边距为 6 和 10 。 最接近的边距, 斜角与长边的角是多少 ?
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  8. Tommy is flying his kite one afternoon and notices that he has let out the entire 130 ft of string. The angle his string makes with the ground is $\begin{array}{r}{48}^{\circ }\end{array}$ . How high, to the nearest foot, is his kite at this time?
     ::汤米有一天下午飞着风筝 发现他已经释放了整个130英尺的绳子。他的绳子与地面的角是48英寸。最近的脚有多高,现在他的风筝是多少?
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  9. A tree struck by lightning in a storm breaks and falls over to form a triangle with the ground. The tip of the tree makes a $\begin{array}{r}{18}^{\circ }\end{array}$ angle with the ground 21 ft from the base of the tree. What was the height of the tree to the nearest foot?
     ::一棵树在暴风雨中被闪电击中,倒下来形成一个与地面的三角形。树的顶端以树底21英尺的地面为角,以18英寸角。树高到最近的脚是多少?
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  10. Upon descent an airplane is 19,000 ft above the ground. The air traffic control tower is 190 ft tall. It is determined that the angle of elevation from the top of the tower to the plane is $\begin{array}{r}{15}^{\circ }\end{array}$ . To the nearest mile, find the ground distance from the airplane to the tower.
      ::空中控制塔高190英尺,确定从塔顶到飞机的高度角为15英尺。 距离最近的里程,找到从飞机到塔的地面距离。
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  11. Why will the sine and cosine ratios always be less than 1?
      ::为什么正弦和共弦比率总是低于1?

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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 。