A lighthouse is located on a small island 3 km away from the

A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Calculus 1 Question)

Solution

Denote x: The distance along the shoreline.
Now,

x/3 = tan()

Thus,
dx/d = 3 sec²()
= 3(10/3)² .................................... when x=1
= 10/3

By the chain rule:
dx/dt = d/dt dx/d
= 8 radians/min (10/3) km ............ 4 rpm = 8 radians/min
= 80/3 km/min

The required answer: 80/3 km/min