Calculus 1 problem

Question

A revolving searchlight on an island 8 miles from shore turns at the rate of 5 revolutions per minute in the clockwise direction. At what speed is the light beam travelling along the straight shoreline the instant it makes an angle of 45◦ with the shoreline? (Recall: one revolution = 2π radians.)

Answer 1

First we find the angular speed of the light house.

If one revolution is 2π then 5 revolutions is 10π.

The angular speed, w is 10π radians per minute(60 seconds).

w = π/6 rad/sec

The tangential velocity, v_T of the lighthouse is 8mi * π/6 rad/sec

v_T = 4π/3 mi/sec

The speed along the shoreline is v = v_Tcos(45º)

speed along the shoreline, v = 2.96 mi/sec