derivates and related rates?

Question

Cars A and B leave a town at the same time. Car A heads due south at a rate of 80km/hr and car B heads due west at a rate of 60 km/hr. how fast is the distance between the cars increasing after 3 hours

I have completed the problem, the answer I get is 84.515
but I don't know if it is right....
my question is this, when do you substitue given numbers in this case, before or after you take the first derivative?

Answer 1

You substitute after you take the derivative, otherwise your differentiation won't capture the dependence of all of the variables on time.

In this case, you want dD/dt, where

D = sqrt(y^2+x^2)

where y is the distance car A has traveled, and x is the distance car B has traveled.

Differentiate with respect to t:

dD/dt = (1/2)(1/sqrt(y^2+x^2))(2y dy/dt + 2x dx/dt).

Now plug in dy/dt = 80 and dx/dt = 60. Since t=3, car A will have traveled y = 3*80 = 240 km, and car B will have traveled x = 3*60 = 180 km, so plug in those values too.

You get

dD/dt = (1/2)(1/sqrt(240^2 + 180^2))(480*80 + 360*60) = 100 km/hr.