Calculus problem

Question

A man on a dock is pulling in a rope that is fastened to the bow of a boat. If the mans hands are 12 feet higher than the point where the rope is attached to the boat, and if he is retrieving the rope at a rate of 3 feet per second, how fast is the boat approaching the dock when 20 feet of rope are still out?

Answer 1

Let r be the length of the rope between the boat and the man. Let x be the horizontal distance from the boat

to the dock. This forms a right triangle with r as the hypotenuse.

r²=x²+12²

2r dr/dt = 2x dx/dt

dx/dt = r/x dr/dt

When r = 20, x = √(20²-144) = 16

dx/dt = (20/16)3 = 60/16 = 15/4 feet/sec

Answer 2