**Example 1:** Air is being pumped into a spherical balloon such that its radius increases at a rate of .75 in/min. Find the rate of change of its volume when the radius is 5 inches.

The volume ( *V*) of a sphere with radius *r* is

Differentiating with respect to *t*, you find that

The rate of change of the radius *dr/dt* = .75 in/min because the radius is increasing with respect to time.

At *r* = 5 inches, you find that

hence, the volume is increasing at a rate of 75π cu in/min when the radius has a length of 5 inches.

**Example 2:** A car is traveling north toward an intersection at a rate of 60 mph while a truck is traveling east away from the intersection at a rate of 50 mph. Find the rate of change of the distance between the car and truck when the car is 3 miles south of the intersection and the truck is 4 miles east of the intersection.

- Let
*x*= distance traveled by the truck

*y*= distance traveled by the car

*z*= distance between the car and truck

The distances are related by the Pythagorean Theorem: *x* ^{2} + *y* ^{2} = *z* ^{2} (Figure 1

Figure 1A diagram of the situation for Example 2.

The rate of change of the truck is *dx/dt* = 50 mph because it is traveling away from the intersection, while the rate of change of the car is *dy/dt* = −60 mph because it is traveling toward the intersection. Differentiating with respect to time, you find that

hence, the distance between the car and the truck is increasing at a rate of 4 mph at the time in question.