Lighthouse is located on an island 4 km away from P on a straight shoreline and its light makes 7 revolutions per min. How fast is the beam of light moving along the shoreline when it is 1 km from P?

Answer 1

I assume that the island is directly across from P and we are talking about the light speed along the coast one mile to the right of P, away from P as the light rotates left to right on the coast. Let's draw a right triangle from the light house to P (4 km), right along the shore from P to the light is hitting A, (1km) and then close the triangle with a line from A to the light house, L, which is the hypotenuse. Let's call the angle at the light house θ. When θ = 0, the light is at P. The process is symmetric in magnitude - I've picked the movement away in order that everything is postive.

We know that the light house rotates at 7 rev/ min or 14 π rad/min

We also know the position of the light for any θ: x = (4 km )tan(θ)

v is dx/dt, so differentiate the above expression for x: v = dx/dt = 4sec²(θ) dθ/dt from chain rule.

Plug in for dθ/dt and solve for θ for sides 1 and 4, specifically.

Hope that helped.