Question

both circles have the same center. the circumference of the inner circle is 56.52 inches. what is the area of the shaded region? write your answer as a whole number or a decimal rounded to the nearest hundredth. square inches

Explanation:

Step1: Find the radius of the inner circle

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(C = 56.52\) inches and \(\pi\approx3.14\), we solve for \(r\):
\(r=\frac{C}{2\pi}=\frac{56.52}{2\times3.14}=\frac{56.52}{6.28} = 9\) inches.

Step2: Find the area of the outer circle

The formula for the area of a circle is \(A=\pi R^{2}\). The radius of the outer circle \(R = 15\) inches. So \(A_{outer}=\pi\times15^{2}=3.14\times225 = 706.5\) square inches.

Step3: Find the area of the inner circle

Using the radius of the inner circle \(r = 9\) inches in the area formula \(A=\pi r^{2}\), we get \(A_{inner}=\pi\times9^{2}=3.14\times81=254.34\) square inches.

Step4: Find the area of the shaded region

The area of the shaded region \(A = A_{outer}-A_{inner}\). So \(A=706.5 - 254.34=452.16\) square inches.

Answer:

\(452.16\)