Homework Help: calculus

Posted by fati on Saturday, June 16, 2012 at 2:09am.

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.

calculus - Reiny, Saturday, June 16, 2012 at 9:29am
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 Ų
then,
tanŲ = x/300
x = 300tanŲ
dx/dt =300sec^2 Ų dŲ/dt

let the hypotenuse be h
when x=200
h^2 = 200^2+300^2
h = 100√13
secŲ = 100√13/300 = √13/3
sec^2 Ų = 13/9
and we are told dŲ/dt = 2π/20 rad/sec
= π/10 rad/sec

dx/dt = 300(13/9)(π/10) = 130π/3 ft/sec

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