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Charles lives in a tall apartment building. He looks out his window and spots a car on the road that runs straight toward his building. His angle of depression to the car is 24°.

Charles looks farther down the road and spots a dump truck. He estimates that the car is 800 feet in front of the dump truck. His angle of depression to the dump truck is 10°.

How far above the ground is Charles's line of sight?

approximately 113.9 feet

approximately 138.9 feet

approximately 233.6 feet

approximately 264.6 feet

 Apr 9, 2020
 #1
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Drawing a sketch, placing Charles at point C, the base of the building immediately below Charlie at point Z, the car at point Y, and the truck at point X. 

 

Call the distance from Y to Z 'c', the distance from  to Z 'c + 800', and the height of the building 'h'.

 

Triangle CZY is a right triangle with Z the right angle and angle C = 66o   --->   tan(66)  =  x / h

 

Triangle CZX is a right triangle with angle XCZ = 80o   --->   tan(80)  =  (x + 800) / h

 

We now have:  tan(66)  =  x/h   --->   x  =  h·tan(66)

                         tan(80)  =  (x + 800)/h   --->   x + 800  =  h·tan(80)   --->   x  =  h·tan(80) - 800

 

Combining these two equations:  h·tan(66)  =  h·tan(80) - 800

                                   h·tan(66) -  h·tan(80)  =  -800

                                      h(tan(66) - tan(80))  =  -800

                                                                   h  =  -800 / (tan(66) - tan(80))

                                                                   h  =  233.6'

 Apr 9, 2020
 #2
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tan 10 = h/(x+800)    h = (x+800) tan 10

tan 24 = h/x              h = x tan 24

 

x tan 24 = (x+800) tan 10      solve for   x  = 524.584

 

tan 24 =  h/ 524.584          h = 233.56 ft

 Apr 9, 2020

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