Question:
A lighthouse is located on an island 3 miles from the closest point on a straight shoreline. If the lighthouse light rotates clockwise at a constant rate of 9 revolutions per minute, how fast does the beam of light move towards the point on the shore closest to the island when it is 52 miles from that point
Answer:
The beam of light moves at
miles/min
Explanation:
This question is illustrated with the attached image
Taking the instructions in the question, one at a time.
A revolution of 9 per minute implies that:


Take tan of the angle in the attachment:

Differentiate both sides with respect to time

Rewrite as:

In calculus:
-- Chain rule
So:

In trigonometry:

So:

Recall that:






Recall that:



Multiply both sides by 3



miles/min
Hence:
The beam of light moves at
miles/min