Question

(a) Write the area A of a circle as a function of the circumference C. (b) Find the (instantaneous) rate of change of the area A with respect to the circumference C. (c) Evaluate the rate of change of \(A\) at \(C=\pi\) and \(C=6 \pi\) (d) If \(C\) is measured in inches and \(A\) is measured in square inches, what units would be appropriate for \(d A / d C ?\)

Short answer

The area as a function of the circumference is \(A(C) = \frac{C^2}{4\pi}\). The rate of change of the area with respect to the circumference is \(dA/dC = \frac{C}{2\pi}\). The rate of change at \(C = \pi\) is \(1/2\) and at \(C = 6\pi\) is \(3\). The units for \(dA/dC\) are inches.

Step-by-step solution

Expressing area as a function of circumference

From the formula for circumference \(C = 2\pi r\), we can solve for \(r\), giving us \(r = \frac{C}{2\pi}\). Substituting \(r\) into the formula for the area \(A = \pi r^2\), we get \(A(C) = \pi * \left(\frac{C}{2\pi}\right)^2 = \frac{C^2}{4\pi}\).

Computing the rate of change of area with respect to circumference

To find the rate of change of \(A\) with respect to \(C\), we differentiate \(A(C)\) with respect to \(C\). The derivative of \(\frac{C^2}{4\pi}\) with respect to \(C\) is \(dA/dC = \frac{C}{2\pi}\).

Evaluating the rate of change at specific circumferences

Substitute \(C = \pi\) and \(C = 6\pi\) into the derivative to find the rate of change at these specific values. For \(C = \pi\), we have \(dA/dC = \frac{\pi}{2\pi} = \frac{1}{2}\). For \(C = 6\pi\), we have \(dA/dC = \frac{6\pi}{2\pi} = 3\).

Determining the appropriate units

The units of the rate of change \(dA/dC\) are given by the units of \(A\) divided by the units of \(C\), which in this case, are square inches per inch, or simply, inches.