Two buses are driving along parallel freeways that are 12 mi apart, one heading east and the other heading west. assuming that each bus drives a constant 50 mph, find the rate (in mph) at which the distance between the buses is changing when they are 37 mi apart, heading toward each other.

The component of speed in the direction of the other bus is the speed of the bus multiplied by the cosine of the angle between its travel path and the direction of the other bus. The sine of the angle is the ratio of the opposite side (12 mi) to the hypotenuse (37 mi). Thus the cosine is

cos (α) = √ (1 - (12/37) ²) = 35/37.

Since the geometry is symmetrical, we can use 50+50 = 100 mph as the closing speed if the buses were traveling directly toward each other. The distance between the buses is decreasing at

(100 mph) · (35/37) ≈ 94.6 mph