Height of Cloud Cover

Question

To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75 degrees from the horizontal. An observer 600 m away measures the angle of elevation to the spot of the light to be 45 degrees. Find the height h of the cloud cover.

Answer 1

draw the triangle

the left base angle is 75º and the right base angle is 45º

draw the height and call it h

the base is 600 m

the height divides the triangle into two right triangles

the one on the right is an isosceles right triangle (45-45-90)

the base of the triangle on the right is the same as the height, h because it is a 45-45-90 triangle

the base of the right triangle on the left is 600-h

use the tangent function

tan=opposite/adjacent

tan75º=h/(600-h)

tan75º=3.73205

3.73205=h/(600-h)

3.73205*(600-h)=h

3.73205*600-3.73205h=h

2239.23-3.73205h=h

2239.23=h+3.73205h

2239.23=4.73205h

h=2239.23/4.73205

h=473.205059 m

the height of the cloud is 473.205059 m