help using related rates

Question

A boat is pulled in to a dock by a rope with one end attached to the front of the boat, and the other end passing through a pulley attached to the dock at a point 6 feet higher than the front of the boat. The boat is being pulled through the pulley at the rate of 0.7 ft/s. How fast is the boat approaching the dock when 12 feet of rope are out?

Answer 1

construct a right triangle, with altitude = 6 feet

hypotenuse = 12 feet

the base = square root of (12^2 - 6^2) = sqr(144-36) = sqr108= 6sqr3

h^2 = a^2 + b^2

take the derivative with respect to time

2hh' = 2aa' + 2bb'

divide by 2

hh' = aa' + bb' a'= 0

hh' = bb' h' =.7

12(.7) = (6sqr3)(b')

b' = 8.4/6sqr3 = 1.4/sqr3 = 1.4sqr3/3 =about .81

b' = about 0.81 feet per second = speed at which the boat is approaching the dock