The circumference of a circle is related to its diameter. A circle with larger diameter has larger circumference.

Doubling the diameter of a circle doubles the circumference. Tripling the diameter triples the circumference, and so on.

The fact that the graph of the circumference of a circle as a function of its diameter is a straight line through the origin gives the same information.

We say that the circumference of a circle is proportional to its diameter.

We say that the area of a circle is proportional to the square of its radius and the constant of proportionality is .

In general, we say that a quantity is proportional or directly proportional to another if the second quantity is the product of a constant and the first quantity.

The constant is known as the constant of proportionality.

In the previous example, the constant of proportionality is .

The area of a circle is a function of its radius. In fact, we know that Area = .

Let's see what happens to the area of a circle if we double its radius.

Let r be the radius of the first circle. Then the radius of the second circle is double this amount, or 2r.

So, the original area is and the new area is , or .

So the new area is four times the original area.

So far we have only considered proportions that are examples of increasing functions. That is, as the input values increase, the output values increase as well.

Now let's study proportions that are examples of decreasing functions. That is, as the input values increase, the output values decrease.

For example, suppose you want to take a 100 mile trip. If you travel at a rate of 20 miles per hour, it will take you 5 hours to complete the trip. However, if you travel at a rate of 50 miles an hour, you will only spend 2 hours on the road.

If R is the rate at which you travel, then, t, the number of hours you spend on the road, is given by . As the value of R increases, the denominator of the equation becomes larger and therefore t becomes smaller.

We say that t is inversely proportional to R. In this example, the constant of proportionality is 100.

The weight of an object is inversely proportional to the square of the distance between the object and the center of the earth.

How much will she weigh when she is 1000 miles from the surface of the earth? Round your answer to the nearest 0.001. Click "Submit" when finished.

The radius of the earth is about 4000 miles. Suppose an astronaut weighs 125 pounds on Earth.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education