Challenge Problem 5.
A man is standing on the top of a hill and sees a flagpole he knows is 45 feet high. The angle of
depression to the bottom of the pole is 12 degrees, and the angle of elevation to the top of the
pole is 16 degrees. Find his distance from the pole.

his distance from the pole is 90.16 ft

Explanation:

based on the diagram of the question i will upload along this answer;

45 - d is equal to part of the pole below the horizontal line

d/x = tan(16) ; x = d/tan(16)............equ1

(45-d) / x = tan(12); x = (45-d) / tan(12)-----------equ2

∴ d/tan(16) = (45-d) / tan(12)

d.tan(12) = (45-d).tan(16)  

d.tan12 = (45×tan16) - d.tan16

0.2125d = 12.9034 - 0.2867d

0.2125d + 0.2867d = 12.9034

0.4992d = 12.9034

d = 12.9034/0.4992

d = 25.85 ft

now substitute value of d into any of our previous equation, lets take equation 1

x = 25.85 / tan(16)

x = 25.85 / 0.2867

x = 90.16 ft

Therefore his distance from the pole is 90.16 ft

come closer and become far apart and become steam/gas