How fast must you release the string of your kite if the kite that you are flying is 40 meters high, 50 meters away from you and moving horizontally away at rate of 30 meters per minute?

Answer 1

" 50 m away from you"
-measured along the string, or measured along the ground?

There is not enough information , there are 3 rates involved, but you only give one.
Are we ignoring the fact that the string cannot form a straight line ?
Is the kite always 40 m high?

Answer 2

well, since the kite is flying horizontally, I think we can assume that it stays 40m high.

And we usually go with a straight string in elementary problems like this.

So, if the kite is x feet away horizontally, the straight-line distance to the kite is

d^2 = x^2 + y^2 = x^2 + 1600
Conveniently, at the moment in question, we have a nice tidy 3-4-5 triangle, so x=30.

Now we just look for rates of change:

2d dd/dt = 2x dx/dt
100 dd/dt = 60 * 30
dd/dt = 18 m/s