[Related Rates] How fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

Question

A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

Answer 1

Let H be the distance from the floor to the top of the ladder and L = distance from the wall to the bottom of the ladder.

Let H²+L²=100, then H=√100-25=5√3. Take the derivative to find 2HdH/dt+ 2LdLdt=0. Thus dL/dt=-H/LdH/dt. Note dL/dt is the speed of the bottom and dH/dt is the speed of the top. So, dL/dt=-5√3/5*2=2√3 ft/sec