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...

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?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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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

`d=sqrt(x^2+y^2)=(x^2+y^2)^(1/2)`

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:

`(dd)/(dt)=1/2(50^2+120^2)^(-1/2)(2(50)(25)+2(120)(60))`

`=16900/(2sqrt(16900))`

`=16900/260`

`=65`

Thus the distance between the cars is increasing at a rate of 65mph at the 2 hour mark.

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