calculus question

Question

Car A is driving south away from an intersection. Car B is approaching the same

intersection and is moving west.

At what rate is the distance between the cars changing at the instant when car A is 40

miles from the intersection while traveling at 55 mph and car B is 30 miles from the

intersection while traveling at 45 mph?

(HINT: since car B is traveling west towards the intersection, the distance from the car

to the intersection is decreasing, meaning its derivative is negative)

Answer 1

The distance, z, between the two cars is the length of the hypotenuse of the right triangle with vertices at the 2 cars and the intersection, with the right angle at the intersection.

Let x be car B's distance from the intersection, y be car A's distance from the intersection. We have x = 30 and dx/dt = - 45. Also, y = 40 and dy/dt = 55. Lastly, z = 50 (by pythag thm) and x² + y² = z².

Differentiating with respect to t, gives 2x·dx/dt + 2y·dy/dt = 2z·dz/dt.

-2,700 + 4,400 = 100·dz/dt

dz/dt = 17 mph