Right Triangle Trig

Question

A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is . From a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is .How high (in feet) is the mountain?

Answer 1

Draw and label a diagram

h represents the height of the mountain

d represents the distance from the second observation to the base of the mountain

h / d = tan 32°

h = d tan 32°

h / tan 32° = d

h / (1500 + d) = tan 30°

h = (1500 + d) tan 30°

h = 1500 tan 30° + d tan 30°

h - 1500 tan 30° = d tan 30°

(h - 1500 tan 30°) / tan 30° = d

(h - 1500 tan 30°) / tan 30° = h / tan 32°

Can you solve for h and answer?

Answer 2

T - top of mountain

B - bottom of mountain

A - the point with the the angle of elevation of 30°

D - the point, 1500-feet closer to the bottom of mountain and it has the angle of elevation of 32°

TB - the height of the mountain: let TB be "h"

BD - distance to the closer point of observation: let BD be "x"

BA - distance to the farther point of observation: it is "x + 1500"

Writing the system of equations:

tan(32°) = TB/BD = h/x

tan(30°) = TB/BA = h/x + 1500

x = (tan(30°) • 1500)/(tan(32°) - tan(30°)) = 18,224.79 (feet)

Then

h = x • tan(32°) = 11,338.114 ≅ 11,338 (feet)