A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 4 feet per second. How fast is the top of the ladder moving down the wall when its base is 14 feet from the wall? Round your answer to two decimal places.

Question

Quizplus · Accepted Answer

Let x be the distance from the bottom of the ladder to the wall, and let y be the height of the ladder on the wall. Then we have the equation x^2 + y^2 = 20^2. Differentiating implicitly with respect to time, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Solving for dy/dt when x = 14 and dx/dt = 4:
2(14)(4) + 2y(dy/dt) = 0
dy/dt = -56/2y
When x = 14, y = sqrt(20^2 - 14^2) = 18.44
Substituting, we get:
dy/dt = -56/(2*18.44) = -1.52 ft/sec
Rounding to two decimal places, we get -3.92 ft/sec.

A Ladder 20 Feet Long Is Leaning Against the Wall