A 12' ladder is leaning against a wall making an angle theta with the ground. The ladder begins to slide down the wall. what is the rate of change of the height of the ladder, h, with respect to the change in the angle of the ladder with the floor when the ladder reaches a height of 9 feet?

Answer 1

if the distance from wall is x,
x^2 + h^2 = 144
x = √(144-h^2)
when h=9, x = √(144-81) = √63
sinθ = h/12

the wording of the question is odd. Usually they ask for the rate of change of x when h=9. In that case,

2x dx/dt + 2h dh/dt = 0
but we don't have dh/dt.

So, it appears they want dh/dθ

h/12 = sinθ
h = 12sinθ
dh/dθ = 12cosθ = 12(x/12) = x = √63

however, rather than asking for "the rate of change of h with respect to θ" you say they want "rate of change of h with respect to the change of θ". Don't know how to interpret that