Awoman standing on a hill sees a flagpole that she knows is 30 ft tall. the angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. find her distance x from the pole. (round your answer to one decimal place.)

Draw a diagram:
from the top of the hill draw a line 18° above the horizontal to the top of the 30 ft pole.
From the top of the hill we draw a line 14° below horizontal to the bottom of the 30 ft pole.
We now have 2 right triangles sharing a common horizontal line,x, to the pole. 
Let y=part of pole above horizontal line
30-y=part of pole below horizontal line.

We shall have the following equations:
y/x=tan 18
30-y/x=tan14

x=y/tan 18
x=(30-y)/tan 14

y/tan 18=(30-y)/tan 14
y tan 14=tan 18(30-y)
ytan 14=30tan 18-ytan18
y tan 14+y tan 18=30tan 18
y(tan 14+tan 18)=30*tan 18
y=(30*tan 18)/(tan 14+tan 18)
x=y/tan 18=30/(tan 14+tan10)=52.2423 ft
thus her distance x from the pole=52.2 ft

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