1.13 高高和萧条之角

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  高高和萧条之角

  You have decided to go camping with some friends. While out on a hike, you reach the top of a ridge and look down at the trail behind you. In the distance, you can see your camp. You're thinking about how far you've traveled, and wonder if there is a way to determine it.

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  By using a small device called a clinometer, you're able to measure the angle between your horizontal line of sight and the camp as $\begin{array}{r}{37}^{\circ }\end{array}$ , and you know that the hill you just hiked up has a height of 300 m. Is it possible to find out how far away your camp is using this information? (Assume that the trail you hiked is slanted like the side of a triangle.)
  ::通过使用一个叫做光谱仪的小装置, 你可以测量你水平视线与营地之间的角值, 也就是37英寸, 你知道刚刚爬上的那个山峰高度是300米。 能否找出你的营地使用这个信息有多远? (假设你徒步的足迹像三角形的侧面一样倾斜 )

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  Angles of Elevation and Depression
  ::高高和萧条之角

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  You can use right triangles to find distances, if you know an angle of elevation or an angle of depression.
  ::您可以使用右三角来寻找距离, 如果您知道一个高度角度或萧条角度 。

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  The figure below shows each of these kinds of angles.
  ::下图显示了上述每一种角度。

  

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  The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. For example, if you are standing on the ground looking up at the top of a mountain, you could measure the angle of elevation. The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. For example, if you were standing on top of a hill or a building, looking down at an object, you could measure the angle of depression. You can measure these angles using a clinometer or a theodolite. People tend to use clinometers or theodolites to measure the height of trees and other tall objects. Here we will solve several problems involving these angles and distances.
  ::高度角度是水平视线和向上视线之间的角。 例如, 如果您站在地上仰望山顶, 您可以测量高地角。 低地角度是水平视线和向下视线之间的角。 例如, 如果您站在山顶或建筑物上, 向下看一个物体, 您可以测量低地角。 您可以使用光度计或地平线测量这些角。 人们倾向于使用光度计或神像测量树和其他高物体的高度。 在这里, 我们将解决涉及这些角度和距离的一些问题 。

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  Finding the angle of elevation
  ::查找高度角度

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  You are standing 20 feet away from a tree, and you measure the angle of elevation to be $\begin{array}{r}{38}^{\circ }\end{array}$ . How tall is the tree?
  ::你站在离树20英尺远的地方, 测量高度的角度是38英寸。那棵树有多高?

  

   

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  The solution depends on your height, as you measure the angle of elevation from your line of sight. Assume that you are 5 feet tall.
  ::溶液取决于你的高度, 当您从视线上测量高度角度时。 假设您有5英尺高 。

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  The figure shows us that once we find the value of $\begin{array}{r}T\end{array}$ , we have to add 5 feet to this value to find the total height of the triangle. To find $\begin{array}{r}T\end{array}$ , we should use the tangent value:
  ::该图向我们显示,一旦我们找到T的值,我们就必须在这个值上增加5英尺,才能找到三角形的总高度。要找到T,我们应该使用正切值:

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  $$\begin{array}{rl}\tan {38}^{\circ }& =\frac{opposite}{adjacent}=\frac{T}{20}\\ \tan {38}^{\circ }& =\frac{T}{20}\\ T& =20\tan {38}^{\circ }\approx 15.63\\ \text{Height of tree}& \approx 20.63ft\end{array}$$

  
  ::388838T20T=20tan3815.63820.63英尺

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  You are standing on top of a building, looking at a park in the distance. The angle of depression is $\begin{array}{r}{53}^{\circ }\end{array}$ . If the building you are standing on is 100 feet tall, how far away is the park? Does your height matter?
  ::你站在一栋大楼的顶端, 看着远处的公园。 压抑的角度是53。 如果您站在100英尺高的建筑上, 公园离这有多远? 您的身高重要吗 ?

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  Finding the angle of depression
  ::寻找抑郁的角

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  If we ignore the height of the person, we solve the following triangle:
  ::如果我们忽略一个人的身高, 我们就可以解决以下三角形 :

  

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  Given the angle of depression is $\begin{array}{r}{53}^{\circ }\end{array}$ , $\begin{array}{r}\mathrm{\angle }A\end{array}$ in the figure above is $\begin{array}{r}{37}^{\circ }\end{array}$ . We can use the tangent function to find the distance from the building to the park:
  ::以压抑角度为53°Q, 上图A为37°C。我们可以使用正切函数找到从大楼到公园的距离:

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  $$\begin{array}{rl}\tan {37}^{\circ }& =\frac{opposite}{adjacent}=\frac{d}{100}\\ \tan {37}^{\circ }& =\frac{d}{100}\\ d& =100\tan {37}^{\circ }\approx 75.36ft.\end{array}$$

  
  ::37775.36英尺

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  If we take into account the height if the person, this will change the value of the adjacent side. For example, if the person is 5 feet tall, we have a different triangle:
  ::如果我们考虑到一个人的身高,这将改变相邻侧的值。例如,如果这个人身高5英尺,我们有一个不同的三角形:

  

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  $$\begin{array}{rl}\tan {37}^{\circ }& =\frac{opposite}{adjacent}=\frac{d}{105}\\ \tan {37}^{\circ }& =\frac{d}{105}\\ d& =105\tan {37}^{\circ }\approx 79.12ft.\end{array}$$

  
  ::37779.12英尺 105tan37d105d=105tan3779.12英尺。

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  If you are only looking to estimate a distance, then you can ignore the height of the person taking the measurements. However, the height of the person will matter more in situations where the distances or lengths involved are smaller. For example, the height of the person will influence the result more in the tree height problem than in the building problem, as the tree is closer in height to the person than the building is.
  ::如果您只是想要估计距离, 那么您就可以忽略测量对象的高度。 但是, 在距离或长度较小的情况下, 个人的高度会比较重要 。 例如, 一个人的高度会比建筑问题对树高问题的影响更大, 因为树与人的高度比建筑的高度更近 。

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  Real-World Application: The Horizon 
  ::现实世界应用:地平线

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  You are on a long trip through the desert. In the distance you can see mountains, and a quick measurement tells you that the angle between the mountaintop and the ground is $\begin{array}{r}{13.4}^{\circ }\end{array}$ . From your studies, you know that one way to define a mountain is as a pile of land having a height of at least 2,500 meters. If you assume the mountain is the minimum possible height, how far are you away from the center of the mountain?
  ::你们在沙漠中长途跋涉。在你们可以看到山岳的远处,快速测量显示山顶与地面之间的角是13.4。从你们的研究中,你们知道山的一个定义方式是,山地的高度至少为2,500米。如果你们认为山是可能的最低高度,那么离山中心还有多远?

   

  

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  $$\begin{array}{rl}\tan {13.4}^{\circ }& =\frac{opposite}{adjacent}=\frac{2500}{d}\\ \tan {13.4}^{\circ }& =\frac{2500}{d}\\ d& =\frac{2500}{\tan {13.4}^{\circ }}\approx 10,494meters.\end{array}$$

  
  ::-=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked if it was possible to find out how far away your camp is using the information given.
  ::更早之前,有人问过您是否能够了解您的营地使用所提供的信息离您的营地有多远。

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  Since you know the angle of depression is $\begin{array}{r}{37}^{\circ }\end{array}$ , you can use this information, along with the height of the hill, to create a trigonometric relationship:
  ::既然你知道萧条的角度是37, 你可以使用这个信息, 连同山高, 来建立三角关系:

  

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  Since the unknown side of the triangle is the hypotenuse, and you know the opposite side, you should use the sine relationship to solve the problem:
  ::因为三角形的未知侧面是下限, 你知道相反的侧面, 您应该使用正弦关系来解决问题:

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  $$\begin{array}{r}\sin {37}^{\circ }=\frac{300}{hypotenuse}\\ hypotenuse=\frac{300}{\sin {37}^{\circ }}\\ hypotenuse\approx 498.5\end{array}$$

  
  ::300-hypotenuse hypotenuse = 300sin 37 hypotenuse 498.5 498.5

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  You have traveled approximately 498.5 meters up the hill.
  ::你已登上山丘498.5米

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  Example 2
  ::例2

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  You are six feet tall and measure the angle between the horizontal and a bird in the sky to be $\begin{array}{r}{40}^{\circ }\end{array}$ . You can see that the shadow of the bird is directly beneath the bird, and 200 feet away from you on the ground. How high is the bird in the sky?
  ::你身高六英尺高, 测量水平与天空鸟类之间的角角是40英寸。 你可以看到鸟的影子直接在鸟的下面, 距离地面200英尺。 鸟在天空中的高度是多高?

  

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  We can use the tangent function to find out how high the bird is in the sky:
  ::我们可以用正切函数 找出鸟在天空中的高度

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  $$\begin{array}{r}\tan {40}^{\circ }=\frac{height}{200}\\ height=200\tan {40}^{\circ }\\ height=(200)(.839)\\ height=167.8\end{array}$$

  
  ::{\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? {\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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  We then need to add your height to the solution for the triangle. Since you are six feet tall, the total height of the bird in the sky is 173.8 feet.
  ::那么我们需要将你的身高加到三角形的溶液中。既然你身高6英尺, 天空鸟儿的总身高是173.8英尺。

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  Example 3
  ::例3

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  While out swimming one day you spot a coin at the bottom of the pool. The pool is ten feet deep, and the angle between the top of the water and the coin is $\begin{array}{r}{15}^{\circ }\end{array}$ . How far away is the coin from you along the bottom of the pool?
  ::当有一天你游出去游泳时,你会在游泳池底部发现一枚硬币。游泳池深10英尺,水顶与硬币之间的角是15英寸。硬币离你离游泳池底部有多远?

  

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  Since the distance along the bottom of the pool to the coin is the same as the distance along the top of the pool to the coin, we can use the tangent function to solve for the distance to the coin:
  ::由于池底至硬币的距离与池顶至硬币的距离相同,我们可以使用正切函数解析硬币的距离:

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  $$\begin{array}{r}\tan {15}^{\circ }=\frac{opposite}{adjacent}\\ \tan {15}^{\circ }=\frac{10}{x}\\ x=\frac{10}{tan{15}^{\circ }}\\ x\approx {37.32}^{\circ }\end{array}$$

  
  ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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  Example 4
  ::例4

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  You are hiking and come to a cliff at the edge of a ravine. In the distance you can see your campsite at the base of the cliff, on the other side of the ravine. You know that the distance across the ravine is 500 meters, and the angle between your horizontal line of sight and your campsite is $\begin{array}{r}{25}^{\circ }\end{array}$ . How high is the cliff? (Assume you are five feet tall.)
  ::您正在远足, 来到峡谷边缘的悬崖上。 距离您可以看到您在悬崖底部、 峡谷另一侧的营地。 您知道峡谷的距离是500米, 您的横向视线与阵营之间的角是25英寸。 悬崖有多高? (假设您有5英尺高 ) 。

  

   

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  Using the information given, we can construct a solution:
  ::利用所提供的信息,我们可以找到解决办法:

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  $$\begin{array}{r}\tan {25}^{\circ }=\frac{opposite}{adjacent}\\ \tan {25}^{\circ }=\frac{height}{500}\\ height=500\tan {25}^{\circ }\\ height=(500)(.466)\\ height=233\text{meters}\end{array}$$

  
  ::252525500里拉=500多 25588=500(500)(4668=233米)

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  This is the total height from the bottom of the ravine to your horizontal line of sight. Therefore, to get the height of the ravine, you should take away five feet for your height, which gives an answer of 228 meters.
  ::这是从峡谷底部到水平视线的总高度。 因此, 要到达峡谷的高度, 您应该取走5英尺的高度, 回答为 228米 。

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  Review
  ::回顾

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  Round your answers to the nearest hundredths place for the following questions.
  ::将以下各题的答案回合到最近的一百个地方。

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  1. A 70 foot building casts an 50 foot shadow. What is the angle that the sun hits the building?
     ::70英尺的建筑物投下了50英尺的阴影 太阳击中大楼的角度是什么?
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  2. You are standing 10 feet away from a tree, and you measure the angle of elevation to be $\begin{array}{r}{65}^{\circ }\end{array}$ . How tall is the tree? Assume you are 5 feet tall up to your eyes.
     ::你站立在离树10英尺远的地方, 你测量的高度角是65英寸。那棵树有多高?假设你的眼睛高到5英尺。
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  3. Kaitlyn is swimming in the ocean and notices a coral reef below her. The angle of depression is $\begin{array}{r}{35}^{\circ }\end{array}$ and the depth of the ocean, at that point is 350 feet. How far away is she from the reef?
     ::Kaitlyn在海洋中游泳,发现她下面有珊瑚礁。 低沉的角是35°C,海洋的深度是350英尺。 她离珊瑚礁有多远?
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  4. The angle of depression from the top of a building to the base of a car is $\begin{array}{r}{60}^{\circ }\end{array}$ . If the building is 78 ft tall, how far away is the car?
     ::从楼顶到车底的压抑角度是60英尺。如果大楼高78英尺,车离这有多远?
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  5. The Leaning Tower of Pisa currently “leans” at a $\begin{array}{r}{4}^{\circ }\end{array}$ angle and has a vertical height of 55.86 meters. How tall was the tower when it was originally built?
     ::皮萨的落叶塔目前“利宁塔”,角度为4,垂直高度为55.86米。 该塔最初建造时有多高?
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  6. The angle of depression from the top of an apartment building to the base of a fountain in a nearby park is $\begin{array}{r}{72}^{\circ }\end{array}$ . If the building is 78 ft tall, how far away is the fountain?
     ::从公寓楼顶到附近公园喷泉底部的压抑角度是72英尺。 如果大楼高78英尺,喷泉距离多远?
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  7. You are standing 15 feet away from a tree, and you measure the angle of elevation to be $\begin{array}{r}{35}^{\circ }\end{array}$ . How tall is the tree? Assume you are 5 feet tall up to your eyes.
     ::你站在离树15英尺远的地方, 测量高度角是35英寸。 树有多高? 假设你的眼睛高到5英尺。
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  8. Bill spots a tree directly across the river from where he is standing. He then walks 18 ft upstream and determines that the angle between his previous position and the tree on the other side of the river is $\begin{array}{r}{55}^{\circ }\end{array}$ . How wide is the river?
     ::比尔将一棵树直接指向河对面他所站的地方。然后他沿着18英尺高的上游行走,确定他先前的位置与河另一侧的树之间的角是55英寸。河宽多少?
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  9. A 50 foot building casts an 50 foot shadow. What is the angle that the sun hits the building?
     ::一个50英尺的建筑物投下了50英尺的阴影。太阳击中大楼的角度是什么?
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  10. Eric is flying his kite one afternoon and notices that he has let out the entire 100 ft of string. The angle his string makes with the ground is $\begin{array}{r}{60}^{\circ }\end{array}$ . How high is his kite at this time?
      ::Eric有一个下午在放风筝 并且注意到他已经释放了整个100英尺的绳子。他的绳子与地面的角是60英寸。这个时候他的风筝有多高?
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  11. 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 $\begin{array}{r}{36}^{\circ }\end{array}$ angle with the ground 25 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。树的顶端从树底25英尺处以地面为角,取角为36英寸。树高到最近的脚是多少?
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  12. Upon descent an airplane is 15,000 ft above the ground. The air traffic control tower is 200 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. (Note that 1 mile = 5280 feet)
      ::飞机降落时在地面上方15 000英尺,空中交通管制塔高200英尺,确定从塔顶到飞机的高度角为15英寸。 距离最近的里程,找到从飞机到塔的地面距离。 (注意1英里=5 280英尺)
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  13. Tara is trying to determine the angle at which to aim her sprinkler nozzle to water the top of a 10 ft bush in her yard. Assuming the water takes a straight path and the sprinkler is on the ground 4 ft from the tree, at what angle of inclination should she set it?
      ::Tara正试图确定用喷水器喷嘴向院子里10英尺灌木顶部浇水的角。 假设水走直道,喷水器在树的地面4英尺处,她应该按什么倾角设定倾角?
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  14. Over 3 miles (horizontal), a road rises 1000 feet (vertical). What is the angle of elevation?
      ::3英里以上(横向),一条公路上升1000英尺(垂直)。高处的角是什么?
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  15. Over 4 miles (horizontal), a road rises 1000 feet (vertical). What is the angle of elevation?
      ::4英里以上(横向),一条公路上升1000英尺(垂直)。高处的角是什么?

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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 。