A 100 ft cell tower on the top of a hill. From the same point near the hill but on level ground, the angle of elevation to the top of the tower is 72° ,

Question

and the angle of elevation to the top of the hill is 70°. How tall is the hill?

Answer 1

Draw and label a diagram.

h represents the height of the hill

d represents the distance from the hill to the observation point

tan 72° = (100 + h) / d

d tan 72° = 100 + h

d = (100 + h) / tan 72°

tan 70° = h / d

d tan 70° = h

d = h / tan 70°

(100 + h) / tan 72° = h / tan 70°

Can you solve for h and answer?

Answer 2

We can use the sine law to get the distance to the top of the hill d:

100/sin(2) = d/sin(18)

d = 100*sin(18)/sin(2)

Sine law can be used again to get the height of the hill h:

h/sin(70) = d/sin(90)

h = d*sin(70)

h = 100*sin(18)*sin(70)/sin(2)

h ~ 832 ft