Someone had helped me yesterday, but I need it broken down more because I honestly don't understand any of this?

Question

A plane flying horizontally at an altitude of 2 mi and a speed of 420 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station.

Answer 1

Using the Pythagorean Theorem from your Algebra I days, the distance to the plane, z, after the plane has gone x miles since being directly overhead, is

z^2 = 2^2 + x^2

we want the rate at which the distance is changing, dz/dt. So, taking derivatives, and remembering the chain rule,

2z dz/dt = 0 + 2x dx/dt

We know that when x=5,

z^2 = 2^2+5^2 = 29, so
z = √29

So, when x=5, we have, using dx/dt=420 mi/hr

2√29 dz/dt = 2*5*420
dz/dt = 4200/(2√29) = 389.96 mi/hr

Answer 2

Thank you very much! I was thinking this was a ratio problem for some reason.