Calculus - related rates

Question

20 foot ladder that is resting on a wall. The ladder is resting at 5 ft away from the wall. The bottom is pulling away at 24 ft/sec. Find dy/dt and dthetha/dt.

Answer 1

In this situation, x and y are related via the Pythagorean theorem:

x² + y² = (20ft)²

Solve for y:

y = (400 - x²)^1/2

Now lets take the derivative (start with the chain rule):

dy = (-2x)(400 - x²)^-1/2dx/2 = -x(400 - x²)^-1/2dx

To get velocities, divide both sides by dt:

dy/dt = -x(400 - x²)^-1/2dx/dt

So, when x = 5 and dx/dt = 24,

dy/dt = -5(400 - 25)^-1/2(24) = -6.2

If theta is the angle of the ladder from the ground, we can find dtheta/dt with the similar procedure but using the relationship

x = (20ft)cos(theta)

which I leave as an exercise for the student.