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 1.1 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? (That is, find the angle's rate of change when the bottom of the ladder is 8 ft from the wall.)

let:

X = the distance of the bottom of the ladder from the wall at any time

dX/dt = rate of travel of the bottom of the ladder = 1.1 ft/sec

A = the angle of the ladder with the ground at anytime

dA/dt = rate of change of the angle in radians per second

X = 10 cos A

dX/dt = - 10 sin A dA/dt = 1.1

dA/dt = - 1.1 / (10 sinA)

When X = 6; cosA = 6/10; sinA = 8/10

Therefore:

dA/dt = - 1.1 / (10 x 0.8) = - 0.1375 radiant per second.