Question

Two straight roads meet at right angles at an intersection. A car is traveling due

south towards the intersection at a constant speed of 60 miles per hour. A second
car is traveling due west towards the same intersection at a constant speed of 70
miles per hour. How fast is the distance between the cars changing at the instant
when the southbound car is 30 miles from the intersection and the westbound car
is 40 miles from the intersection?

Answer 1

The distance between the cars is changing at a rate of -100 miles per hour.

Step-by-step explanation:

To find how fast the distance between the cars is changing, we can use the concept of relative motion. Since the cars are on perpendicular roads, we can consider their velocities as the legs of a right triangle. We can use the Pythagorean theorem to find the distance between the cars and then differentiate it with respect to time to find the rate of change.

Let's call the distance between the cars D. We have a right triangle with one leg as 30 miles and the other leg as 40 miles. Using the Pythagorean theorem, we can find that D is √(30^2 + 40^2) = 50 miles.

Now, let's differentiate D with respect to time:
d(D)/dt = (d/ dt) (√(30^2 + 40^2)) = (1/2) (1/√(30^2 + 40^2)) (d(30^2 + 40^2)/dt) = (1/2) (1/50) (2(30)(d(30)/dt) + 2(40)(d(40)/dt))

Now, we need to find the rate of change of each car's distance from the intersection. The southbound car is moving only south, so d(30)/dt = -60 mph. The westbound car is moving only west, so d(40)/dt = -70 mph. Substituting these values, we get:
d(D)/dt = (1/2) (1/50) (2(30)(-60) + 2(40)(-70)) = -100 mph.

Therefore, the distance between the cars is changing at a rate of -100 miles per hour.