A rocket fired straight in the air is being tracked by a radar station 3 miles from the launching pad. If the rocket is travelling at 2 miles per second, how fast is the distance between the rocket and the tracking station changing at the moment when the rocket is 4 miles in the air?

Question

D^2=9+y^2

Answer 1

I assume you meant fired straight up in the air. In that case, at time t seconds, the rocket has gone 2t miles.

So, the distance D is found, as you showed, by

D^2 = 9+y^2 = 9+4t^2

when the rocket is 4 miles up, t=2, and we have a 3-4-5 triangle, so D=5.

2D dD/dt = 8t
when t=2, we have

2(5) dD/dt = 8(2)
dD/dt = 8/5