Question

A woman standing on a hill sees a flagpole that she knows is 35 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.)

Answer 1

The distance of the woman from the pole is 60.9 ft.

Explanation:

Please, see the attached figure for a graphical description of the problem.

Using trigonometry of right triangles, we can solve this problem.

We know, by trigonometry, that the tangent of an angle is equal to the quotient between the length of the side opposite to the angle and the length of the side adjacent to the angle. For the angle α:

tan α = opposite / adjacent

Then, in this case, we have two equations (see the figure for a better understanding):

tan 14° = x /d

tan 18° = (35 -x) /d

We have a system of 2 equations with 2 unknowns, let´s solve it!

Solving the first equation for x:

tan 14° = x /d

d · tan 14° = x

Replacing x in the second equation:

tan 18° = (35 -x) /d

tan 18° = (35 - d · tan 14°)/d

d · tan 18° = 35 - d · tan 14°

d · tan 18° + d · tan 14° = 35

d (tan 18° + tan 14°) = 35

d = 35/(tan 18° + tan 14°)

d = 60.9 ft

Have a nice day!