Question

Two cars are heading toward the same intersection. Car A travels due east at 20 miles per hour, and Car B travels due north at 50 miles per hour. At what rate, in miles per hour, is the distance between the two cars decreasing when Car A is 0.375 miles from the intersection and Car B is 0.5 miles from the intersection?

Answer 1

the distance between them is decreasing at a rate of 2 mph.

Explanation:

List all given rates and the unknown rate.

Write the formula that relates the variables in the problem: x, y, and s.

There’s a right triangle in the diagram, so you use the Pythagorean Theorem:

For this problem, x and y are the legs of the right triangle, and s is the hypotenuse, so

Differentiate with respect to t.

Use the Pythagorean Theorem again to solve for s.

x = 0.4

y = 0.3

You can reject the negative answer because s obviously has a positive length. So s = 0.5.

Now you’ve got everything you need to substitute into the differentiation result and solve for ds/dt.

This negative answer means that the distance, s, is decreasing. Thus, when car A is 3 blocks north of the intersection and car B is 4 blocks east of the intersection, the distance between them is decreasing at a rate of 2 mph.