At a given moment, a plane passes directly above a radar station at an altitude of 8 km going 600 km/h. How fast is the distance between the plane and the radar station changing 36 seconds later?

Question

I have all the grunt work of this problem done, I found the derivative, I just don't understand how to solve for x. I know it has something to do with a proportion to an hour, but I can't figure it out in order to plug it into my function.

Answer 1

Hi, if x is how far the plane moves, x is v_pt and dx/dt is v_p (Note, you have to convert km/hr to km/s by dividing by 3600)

The distance away from the radar is s = sqrt((8km)²+ x²) (just the hypotenuse of right triangle formed by altitude and x)

It's actually easier to implicitly differentiate s² = 64 + x²

2s ds/dt = 2x dx/dt

ds/dt = (x/s) dx/dt (ds/dt is what the question is asking)

At t = 36 s, x = v_pt, s = sqrt(64 + x²), and dx/dt = v_p

I hope that helped!