Calculus Related Rates

Question

A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 10 feet below the level of the pulley. If the rope is pulled through the pulley at a rate of 16 ft/min, at what rate will the boat be approaching the dock when 110 ft of rope is out?

Answer 1

c² = a²+b²

c²-a² = b²

(c²-a²)^1/2 = b

Plugin a=10:

(c²-10²)^1/2 = b

(c²-100)^1/2 = b

derive both sides of the equation with respect to time (t):

(1/2)(c²-100)^-1/2(2c•dc/dt) = db/dt

c(c²-100)^-1/2 (dc/dt) = db/dt

plugin the value of c=100 and dc/dt=16:

110(110²-100)^-1/2 (16) = db/dt

db/dt ≈16.07 ft./min.