Expert: Ahmed Salami - 11/4/2009

Question
A 20 foot ladder leans against a building. The top of the ladder starts sliding down the side at .5 ft/sec. How fast is the foot of the ladder moving away from the building when the foot of the ladder is 12 ft from the building?

Answer
Hi Kevin,
Let the height of the ladder on the wall at any time be y, and the distance to the foot of the ladder be x.
We know that
x² + y² = 20²
       = 400
Differentiating with respect to x,
2x + 2y(dy/dx) = 0
2y(dy/dx) = -2x
dy/dx = -x/y
Considering rates of change with respect to time t,
dy/dt = dy/dx . dx/dt
As the ladder slides down, y decreases and so dy/dt = -0.5 ft/s
When x = 12
12² + y² = 400
y² = 400 - 144
  = 256
y = 16
So,
-0.5 = (-12/16).(dx/dt)
dx/dt = -0.5/-0.75
     = 2/3
     = 0.67 ft/s

Regards