A woman 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