Question

two spotlights, one blue and the other white, are placed 6.0 m apart on a track on the ceiling of a ballroom. A stationary observer standing on the ballroom floor notices that the angle of elevation is 45degrees to the blue spotlight and 70 degrees to the white one. How high, is the ceiling of the ballroom?

Answer 1

Let h be the ceiling height. Horizontal distances from the observer to the blue and white spotlights are d_b and d_w. From the angles,
tan45 = h/d_b = 1 ⇒ d_b = h,
tan70 = h/d_w ⇒ d_w = h / tan70.

The lights are 6.0 m apart along the track, and the white one (70°) is closer, so
d_b − d_w = 6.0 ⇒ h − h/tan70 = 6.0 ⇒ h(1 − 1/tan70) = 6.0.

With tan70° ≈ 2.7475:
1 − 1/tan70 ≈ 0.63603 ⇒ h ≈ 6.0 / 0.63603 ≈ 9.43 m.

So the ceiling is about 9.4 m high.