Question

both circles have the same center. 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: Calculate the area of the larger circle

The formula for the area of a circle is \(A = \pi r^{2}\). For the larger circle, \(r = 19.9\) inches. So, \(A_{1}=\pi\times(19.9)^{2}=\pi\times396.01\approx3.14\times396.01 = 1243.4714\) square inches.

Step2: Calculate the area of the smaller circle

For the smaller circle, the radius \(r = 19.9 - 7.5=12.4\) inches. Using the area formula \(A=\pi r^{2}\), we have \(A_{2}=\pi\times(12.4)^{2}=\pi\times153.76\approx3.14\times153.76 = 483.8064\) square inches.

Step3: Calculate the area of the shaded region

The area of the shaded region (annulus) is \(A = A_{1}-A_{2}\). So, \(A=1243.4714 - 483.8064=759.665\approx759.67\) square inches.

Answer:

\(759.67\) square inches