I don't know how to do related rates with the correct "derivative with respect to time" when the variables are not constant.
A girl flies a kite at a height of 300 feet, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 feet from her?
Best Answer
let $x$ be the horizontal component and $y$ the vertical component of the kite of length $k$, then from Pythagoras' $$k^2=x^2+y^2$$ Apply implicit differentiation with respect to time and you get $$2k\cdot\cfrac{dk}{dt}= 2x\cdot\cfrac{dx}{dt}+ 2y\cdot\cfrac{dy}{dt} $$ The kite flies only horizontally, thus there is no variation of $y$ with respect to time and $\cfrac{dy}{dt}=0$.
Find $x$ using Pythagras', the goal is to look for $\cfrac{dk}{dt}$, so with the values you were given $$\cfrac{dx}{dt}=25ft\cdot s^{-1},\ \ k=500ft$$
You should be able to complete it.