Question

car is approaching right angle intersection from the north at 70 mph and a truck is traveling to the east at 60 mph. when the car is 1.5 miles north of the intersection and the truck is 2 miles to the east, at what rate, in miles per hour, is the distance between the car and truck is changing?

Answer 1

The rate at which the distance between the car and truck is changing is approximately 92.20 mph.

Step-by-step explanation:

To find the rate at which the distance between the car and truck is changing, we can use the concept of relative velocity. Let's consider a right triangle formed by the car, the truck, and the distance between them. The distance between the car and truck can be found using the Pythagorean theorem: d = sqrt((1.5)^2 + (2)^2) = 2.5 miles.

Now, we can differentiate this distance with respect to time to find the rate of change.

Since the car and truck are moving in different directions, their velocities can be represented as vectors. The car's velocity vector is pointing north with a magnitude of 70 mph, and the truck's velocity vector is pointing east with a magnitude of 60 mph. To calculate the rate at which the distance between them is changing, we can use the formula: rate = sqrt((velocity of car)^2 + (velocity of truck)^2).

Substituting the values into the formula, we get: rate = sqrt((70 mph)^2 + (60 mph)^2)

= sqrt(4900 + 3600)

= sqrt(8500)

≈ 92.20 mph.