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?

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)