Question

a. if a circle has a circumference of 12π cm, what is its area?
b. if a circle of radius r and a square with a side of length b have equal areas, express r in terms of b.
a. the area of the circle is 36π cm².
(simplify your answer, including any radicals. type an exact answer, using π as needed.)
b. the radius of the circle is given by r = □
(type an expression using b as a variable. simplify your answer, including any radicals. type an exact answer, using π as needed)

Explanation:

Step1: Find the radius of the circle

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(C=12\pi\), we set up the equation \(12\pi=2\pi r\). Dividing both sides by \(2\pi\), we get \(r = 6\) cm.

Step2: Calculate the area of the circle

The formula for the area of a circle is \(A=\pi r^{2}\). Substituting \(r = 6\) into the formula, we have \(A=\pi\times6^{2}=36\pi\) \(cm^{2}\).

Step3: Relate the area of the circle and the square

The area of the circle is \(A_{circle}=\pi r^{2}\), and the area of the square is \(A_{square}=b^{2}\). Since \(A_{circle}=A_{square}\), we have \(\pi r^{2}=b^{2}\).

Step4: Solve for \(r\)

First, divide both sides of the equation \(\pi r^{2}=b^{2}\) by \(\pi\): \(r^{2}=\frac{b^{2}}{\pi}\). Then take the square root of both sides. Since \(r>0\) (radius is a non - negative quantity), \(r=\frac{b}{\sqrt{\pi}}=\frac{b\sqrt{\pi}}{\pi}\).

Answer:

a. \(36\pi\) \(cm^{2}\)
b. \(r = \frac{b\sqrt{\pi}}{\pi}\)