Question

The light beacon of a lighthouse is situated a distance 2.00 km from a long, straight shoreline (shortest distance to the shoreline). The light rotates at a rate of 3.00 rev/minute. At a certain time, the illuminated spot on the shoreline is 4.00 km from the lighthouse. How fast is the illuminated spot moving at that point?

Answer 1

In triangle ABC

tanθ = BC/AB

tanθ = x/2

taking derivative both side relative to "t"

sec²θ (dθ/dt) = (0.5) (dx/dt)

sec²θ w = (0.5) v                                eq-1

when x = 4 ,

tanθ = 4/2

θ = tan⁻¹ (2) = 63.4 deg

using eq-1

sec²θ w = (0.5) v  

sec²63.4 (3) (2pi/60) = (0.5) v  

v = 3.13 m/s