A ladder 10 ft leans against a vertical wall. The upper end slips down the wall at 5 ft/sec. How fast is the ladder turning when it takes an angle of 20 degrees with the ground.

Answer 1

let the top of the ladder by y ft up on the wall
let the angle made at the foot be Ø

sinØ = y/10
10 sinØ = y
10 cosØ dØ/dt = dy/dt

given: when Ø = 20° , dy/dt = -5 ( y is decreasing)

10 cos20° dØ/dt = -5
dØ/dt = -1/(2cos20°) = -.532 radians/sec

or angle is decreasing at .532 radians / sec
which is appr. 30.5° / sec

(notice the angle is changing very rapidly at that moment, since the ladder has almost fallen)