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: Find radius of inner circle

The diameter of the inner circle is 24 in, so the radius \( r_{inner} = \frac{24}{2} = 12 \) in.

Step2: Find radius of outer circle

The radius of the outer circle is the radius of the inner circle plus 10 in, so \( r_{outer} = 12 + 10 = 22 \) in.

Step3: Calculate area of outer circle

The formula for the area of a circle is \( A = \pi r^2 \), so the area of the outer circle is \( A_{outer} = \pi (22)^2 = 484\pi \) square inches.

Step4: Calculate area of inner circle

The area of the inner circle is \( A_{inner} = \pi (12)^2 = 144\pi \) square inches.

Step5: Calculate area of shaded region

The area of the shaded region (annulus) is the area of the outer circle minus the area of the inner circle: \( A_{shaded} = A_{outer} - A_{inner} = 484\pi - 144\pi = 340\pi \).
Using \( \pi \approx 3.14159 \), we get \( A_{shaded} \approx 340 \times 3.14159 \approx 1068.14 \) square inches.

Answer:

1068.14