Find a function that models the area A of a circle in terms of its circumference C

To model the area A of a circle in terms of its circumference C, we start from the basic formulas for the area and circumference of a circle:

• The area A of a circle is given by the formula: A = πr², where r is the radius.
• The circumference C of a circle is given by the formula: C = 2πr.

To express the area in terms of the circumference, we need to solve the circumference formula for r:

1. From the circumference formula, we can isolate r: r = C / (2π).
2. Now, we substitute this expression for r into the area formula:

A = π(C / (2π))²

Next, we simplify this:

1. A = π * (C² / (4π²))
2. A = (C² / (4π))

Thus, the function that models the area A of a circle in terms of its circumference C is:

A = (C² / (4π))

This equation shows that the area of the circle increases with the square of the circumference, scaled by the factor of 4π.