SOLUTION: A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 14°, and that the a
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Question 1203323: A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 14°, and that the angle of depression to the bottom of the tower is 2°. How far is the person from the monument? (Round your answer to three decimal places.) Answer by math_tutor2020(3817) (Show Source):
Let's start off with a rough sketch of the diagram
The key points are
A = base of the building
B = location of the person's eye
C = base of the monument
D = location on monument at the same height level as the person's eye
E = top of the monument
We have these variables
x = distance from B to D = distance from A to C
y = distance from C to D
Note how segment DE is 200-y feet long so that CE = CD+DE = y + (200-y) = 200
Let's focus on triangle BCD.
tan(angle) = opposite/adjacent
tan(angle DBC) = CD/BD
tan(2) = y/x
y = x*tan(2)
This will be used later in a substitution step.
Now focus on triangle DEB.
tan(angle) = opposite/adjacent
tan(angle DBE) = DE/BD
tan(14) = (200-y)/x
tan(14) = (200-x*tan(2))/x ........ substitution
x*tan(14) = 200-x*tan(2)
x*tan(14)+x*tan(2) = 200
x*( tan(14)+tan(2) ) = 200
x = 200/( tan(14)+tan(2) )
x = 703.609019511759 approximately
x = 703.609
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