Calculus Question

Question

A 16 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 2 ft/s, how fast will the foot of the ladder be moving away from the wall when the top is 14 feet above the ground?

Answer 1

Let x be the horizontal distance and y be the vertical distance.

Given:

dy/dt = -2

x²+y²=16

Differentiating

2xdx/dt+2ydy/dt=0

Want dx/dt when y = 14

dx/dt = -2y(dy/dt)/(2x) = -(y/x)dy/dt

when y = 14, x = √(16²-14²) = √60 = 2√15

dx/dt = -(14/(2√15))(-2) = 14/√15 = (14/15)√15 ft/s