A 13 foot ladder leaning against a wall makes an angle of degree radians with the ground. The base of the ladder is pulled away from the wall at a rate of 2ft/sec. How fast is degree changing when the base of the ladder is 5 feet way.

Answer 1

if the base is x from the wall,

cosθ = x/13
-sinθ dθ/dt = 1/13 dx/dt
when x=5, sinθ = 12/13

-12/13 dθ/dt = 1/13 (2)
dθ/dt = -1/6

note the "-" sign: the angle decreases as the ladder slips down.

Answer 2

5, 12, 13 right triangle

call angle ladder to ground T
cos T = x/13 where x is base of ladder to base of wall

-sin T dT/dt = (1/13) dx/dt

at x = 5, t = 0, dx/dt = 5
cos T = 5/13 so T = 67.4 degrees or 1.18 radians

sin T = sin 67.4 = 12/13 = .923

-.923 dT/dt = (1/13)(2)
so
dT/dt = -.167 radians/second
times 180/pi = -9.55 degrees/second