To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. An observer D = 510 m away measures the angle of elevation to the spot of light to be 45°. Find the height h of the cloud cover. (Round your answer to the nearest meter.)

The height of the cloud = 394.55 m

Explanation:

The observer is 500m away from the spotlight.

Let x be the distance from the observer to the interception of the segment of the height, h with the floor. The equations are thus:

Tan 45° = h/x ... eq1

Tan 75° = h/(500- x ) ... eq2

From eq 1, Tan 45° = 1, therefore eq1 becomes:

h = x ... eq3

Put eq3 into eq2

Tan 75° = h/(500- h)

h = ( 500 - h ) Tan 75°

h = 500Tan 75° - hTan75°

h + h Tan 75° = 500 Tan 75°

h ( 1 + Tan 75° ) = 500 Tan75°

h = 500Tan75°/ (1 + Tan 75°)

h= 1866.02 / 4.73

h = 394.55m