Calculus rate problem

Question

Hello, Ive been struggling with this question.

A kite 100 ft above the ground moves horizontally at a speed of 7 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out?

Answer 1

Draw a figure!

Let L = length of kite string and s = distance from the tether to a point directly under the kite.

L² = 10⁴ + s²

L(dL/dt) = s(ds/dt) => dL/dt = (s/L)(ds/dt)

θ=arc sin(100/L)

dθ/dt = (-100/sqrt(L²-10⁴)(dL/dt)

Now you have a relationship between the rate of change of θ and the rate of change of s which is what you need to solve the problem. Good luck!