How fast must she let out the string when the kite is 500ft away from her?

Question

A girl flies a kite at a height of 300 ft, the wind is carrying the kite horizontally away from her at a rate of 25ft/sec.

Answer 1

Let's start with the kite directly overhead at 300 ft above us. It moves sideways at the rate dx/dt = 25 ft/s where x is how far it has moved sideways in time t. The length of the line will be the diagonal length:

l = ((300)² + x²)^1/2

We'll take the derivative in order to obtain dl/dt:

di/dt = 1/2 (90000 + x²)^-1/2 (2x dx/dt) using chain rule

There is a slight ambiguity to the problem, but I think that the 500 ft away is along the diagonal rather than in x (We can take advantage of the 3-4-5 right triangle: 300 up, 500 along diagonal, and 400 = x)

dl/dt = 1/2(90000ft² + 160000ft²)^-1/2 (2(400ft)(25ft/s)) = 8 ft/s

Hope that helps.