calculus
posted by fati on .
The beacon on a lighthouse makes one revolution every 20 seconds. The beacon is 300 feet from
the nearest point, P, on a straight shoreline. Find the rate at which the ray of light moves along
the shore at a point 200 feet from P.

Make a sketch, let the distance between the end of the ray of light and point P be x ft
let the angle formed at that moment be Ø
then,
tanØ = x/300
x = 300tanØ
dx/dt =300sec^2 Ø dØ/dt
let the hypotenuse be h
when x=200
h^2 = 200^2+300^2
h = 100√13
secØ = 100√13/300 = √13/3
sec^2 Ø = 13/9
and we are told dØ/dt = 2π/20 rad/sec
= π/10 rad/sec
dx/dt = 300(13/9)(π/10) = 130π/3 ft/sec