In this section, we will investigate the question: When two variables are related, how are their rates of change related?

Section 4.5 Related Rates

The Idea

There are two or more variables that all depend on a time variable t that are also related to each other.

For example, when a pebble is dropped into a pond, the radius and area of the circular ripple made both grow as time elapses and are related to each other by the equation .

We want to determine the rate of change in one of these quantities at a specific moment in time when all other values are known.

The Process

This can be achieved by implicitly differentiating the relating equation with respect to t and then algebraically solving for the desired rate of change.

For example, in the example from the previous slide,

Example 1

Air is being pumped into a spherical balloon at a rate of 100cm3/s. How fast is the radius of the balloon changing when its diameter is 50cm?

Example 2

A ladder 10 feet long rests against the side of a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1ft/s, how quickly is the top of the ladder moving when the base of the ladder is 6 feet from the wall?

Example 3

A water tank has the shape of an inverted cone whose height is always twice its radius. The tank is being filled with water at a rate of 2m3/min. Find the rate at which the water level is rising when it is 3m deep.

Example 4

A man is walking along a straight path at a speed of 4ft/s. A searchlight is located on the ground 20 feet from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 feet from the point on the path closest to the light?

Example 5

A man starts walking north at 4ft/s from a point P. Five minutes later, a woman starts walking south at a rate of 5ft/s from a point that is 500 feet due east of P. At what rate is the distance between them increasing 15 minutes after the woman started walking?

Example 6

At noon, ship A is 100km due west of ship B. Ship A is sailing south at 25km/hr and ship B is sailing west at 15km/hr? How fast is the distance between them changing at 3:00PM?

Example 7

A Ferris wheel, 40 feet in diameter, revolves at a rate of 8 radians per minute. How fast is a passenger moving vertically when (s)he is 12 feet higher than the center of the Ferris wheel and rising?

Example 8

A highway patrol plane is flying 1 mile above a straight stretch of road at a speed of 100mph. The pilot sees an oncoming car on the road, and with radar, determines the distance between the plane and the car to be 1.8 miles and decreasing at a rate of 140mph. What is the speed of the car?