Question

A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes three revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)

Answer 1

as usual, draw a diagram and you will see that if θ=0 when x=0,

tanθ = x/4
so,
sec^2θ dθ/dt = 1/4 dx/dt
Now, you have θ and dθ/dt, so just solve for dx/dt

Answer 2

At x=1 km

dθ/dt=3 rev/min=6π rad/min

dx/dt=(1^2+4^2)/4 * 6π=80.1 km/min=4806.6 kph