Two straight roads intersect at right angles in Kingstonville. Car A is on one road moving toward the intersec

Question

Two straight roads intersect at right angles in Kingstonville. Car A is on one road moving toward the intersection at a speed of 50 mph. Car B is on the other road moving away from the intersection at a speed of 30 mph. We wish to know how fast the distance between the cars is changing when car A is 3 miles from the intersection and car B is 4 miles from the intersection.

Write an equation involving the variables whose rates of change either are given or are to be determined. Then, using the chain rule, implicitly differentiate both sides of the equation with respect to time t.

Answer 1

At that moment, of course, the cars are 5 miles apart.

Car A's position can be described thusly: p(a) = 3 - 50t (the car is 3 miles out at time 0 and is getting closer to the intersection at 50mph; t is measured in hours).
Car B's position is, likewise, p(b) = 4 + 30t.

As points, the cars are at (0, 3-50t) for A and (4+30t, 0) for B, respectively. The distance between those two points would be d(t) = sqrt[(4+30t)^2 + (3-50t)^2)]. If you take the derivative of d(t) you will find out how fast the distance is changing at time t.

A little messy, perhaps, but solvable.

We know that in 3/50 of an hour, car A will be at the intersection; car B would be 4+30(3/50) = 5.8 miles away. This would approximate the rate as (5.8-5)/(3/50) = 13.33 mph as the rate of change in distance per unit time when t=3/50.