Question

While visiting Yosemite National Forrest, Joe approximated the angle of elevation to the top of a hill to be 40 degrees. After walking 450 ft closer, he guessed that the angle of elevation had increased by 18 degrees. Approximately how tall is the hill?

Answer 1

Draw a diagram to illustrate the problem as shown in the figure below.

Let h = the height of the hill.
At position A, the angle of elevation is 40°, and the horizontal distance to the foot of the hill is x.
By definition,
tan(40°) = h/x
h = x tan40 = 0.8391x             (1)

At position B, Joe is (x - 450) ft from the foot of the hill. His angle of elevation is 40 + 18 = 58°.
By definition,
tan(58°) = h/(x - 450)
h = (x - 450) tan(58°) = 1.6003(x-450)
h = 1.6003x - 720.135          (2)

Equate (1) and (2).
1.6003x - 720.135 = 0.8391x
0.7612x = 720.135
x = 946.0523

From (1), obtain
h = 0.8391*946.0523 = 793.8 ft

Answer:  The height of the hill is approximately 794 ft (nearest integer)