The purpose of this set of supplemental notes is to provide you with some more worked examples of related rates of change problems. There are certain steps one should take in solving these types of problems. Here are the steps.

1.

Draw a picture of the situation.

2.

Determine what information is given.

3.

Determine what information needs to be found.

4.

Determine the equation.

5.

Take the derivative with respect to t of the equation.

6.

Plug in the values and solve for the unknown.

EXAMPLE 1:

Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates:

Find the rates at which the box's (a) volume, (b) surface area, and (c) diagonal length s are changing at the instant when x = 4, y = 3, and z = 2.

 SOLUTION:

Here are the formulas for the volume, surface area, and diagonal length s.

We want to find dV/dt, dA /dt, and ds/dt when x = 4, y = 3, and z = 2.

PART A

EXAMPLE 2:

A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft away from her?

SOLUTION:

EXAMPLE 3:

A spherical balloon is inflated with helium at a rate of 100p ft ³/min. How fast is the balloon's radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing?

SOLUTION:

EXAMPLE 4:

Suppose that an ostrich 5-ft tall is walking at a speed of 4 ft/sec directly toward a street light 10 ft high. How fast is the tip of the ostrich's shadow moving along the ground? At what rate is the ostrich's shadow decreasing in length?

SOLUTION:

EXAMPLE 5:

A 10-ft ladder is leaning against a wall. The bottom of the ladder begins to slide away from the wall at a speed of 1 mi/hr. How fast is the top of the ladder moving when it is 4 ft above the ground?

SOLUTION:

Since the ladder is in feet, and the rate the ladder is sliding down would make more sense in ft/ sec we must convert 1 mi /hr to ft /sec.

EXAMPLE 6:

At what rate is the area of an equilateral triangle increasing if its base is 10 cm and is increasing at 0.5 cm /sec?

First of all, an equilateral triangle is a triangle in which its three sides are equal. So if the base is x, then so are the other two sides.

The area of a triangle is A = 0.5(base)(height).

We are given dx/dt = 0.5 cm /sec, and we need to find dA/dt.

Before we can start, we need to determine the height of this triangle in terms of its base. I will use the Pythagorean theorem to do this.

EXAMPLE 7:

The base of a rectangle is increasing at 4 cm /sec, while its height is decreasing at 3 cm /sec. At what rate is its area changing when its base is 20 cm and its height is 12 cm?

The area of the rectangle is decreasing at a rate of 12 cm ²/sec.

EXAMPLE 8:

Two ships are sailing toward the same small island. One ship, the Pinta, is east of the island and is sailing due west at 15 mph. The other ship, the Nina, is north of the island and is sailing due south at 20 mph. At a certain time the Pinta is 30 miles from the island and the Nina is 40 miles from it. Are the two ships drawing closer together or farther apart at that time? At what rate?

The two boats are getting closer together.