AP Calculus Question. Work please.
Two cars start from the same point. One travels south at 60mi/hr and the other travels west at 25mi/hr. At what rate is the distance between them increasing two hours later?
Let x be the distance from the starting point to the car going west, and y be the distance to the car going south. After 2 hours, x=50 and y=120.
We also are given the constant rate of change of the vehicles (the speed) so `(dx)/(dt)=25,(dy)/(dt)=60` .
The distance between the cars can be found using the distance formula (basically the Pythagorean theorem); so
Then the rate of change of the distance between the cars is:
`(dd)/(dt)=1/2(x^2+y^2)^(-1/2)(2x(dx)/(dt)+2y(dy)/(dt))` using the chain rule.
Substituting the given values we get:
Thus the distance between the cars is increasing at a rate of 65mph at the 2 hour mark.