How fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall?

Question

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.6 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall? (That is, find the angle's rate of change when the bottom of the ladder is 6 ft from the wall.)

Answer 1

cosθ = x/10; then by differentiation we have: - sinθdθ/dt = (1/10)dx/dt; dθ/dt = -(1/10)dx/dt/sinθ;

sinθ = y/10 = √(100 - 36)/10 = 0.8; dθ/dt = -(1/10)·0.6ft/s/0.8 = - 0.075 rad/s