Question

A lighthouse is located on a small island 2.5 km from the nearest point on a straight shoreline. Its light makes 3 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 kilometers from the point directly opposite the lighthouse

Answer 1

draw a picture

let Θ be the angle between the line to the nearest point and the light beam
... s is the distance from the nearest point to the light beam

tan(Θ) = s / 2.5 km

dΘ/dt = 6 π / min

differentiating ... d[tan(Θ)]/dt = .4 ds/dt ... sec^2(Θ) dΘ/dt = .4 ds/dt

ds/dt = 2.5 sec^2(Θ) dΘ/dt = 2.5 km * [(2.5^2 + 1^2) / 2.5^2] * 6 π / min