Question

both circles have the same center. the circumference of the inner circle is 77.872 centimeters. what is the area of the shaded region? write your answer as a whole number or a decimal rounded to the nearest hundredth. square centimeters

Explanation:

Step1: Find radius of inner circle

The formula for the circumference of a circle is \( C = 2\pi r \). Given \( C = 77.872 \) cm, we solve for \( r \):
\( r=\frac{C}{2\pi}=\frac{77.872}{2\times3.14}=\frac{77.872}{6.28}=12.4 \) cm.

Step2: Find radius of outer circle

The width of the ring (shaded region) is 8.3 cm, so outer radius \( R = r + 8.3 = 12.4 + 8.3 = 20.7 \) cm.

Step3: Calculate area of shaded region (annulus)

The area of an annulus is \( A=\pi R^{2}-\pi r^{2}=\pi(R^{2}-r^{2}) \).
Substitute \( R = 20.7 \) and \( r = 12.4 \):
\( R^{2}-r^{2}=(20.7)^{2}-(12.4)^{2}=428.49 - 153.76 = 274.73 \).
Then \( A = 3.14\times274.73 = 862.6522 \approx 862.65 \) (or as a whole number, 863 if rounded, but let's check precision). Wait, let's recalculate \( R^{2}-r^{2} \) more accurately:
\( 20.7^2 = 20.7\times20.7 = 428.49 \), \( 12.4^2 = 12.4\times12.4 = 153.76 \), difference is \( 428.49 - 153.76 = 274.73 \). Multiply by \( \pi \) (3.14): \( 274.73\times3.14 = 862.6522 \), which rounds to 862.65 or 863. But let's verify the steps again.

Wait, maybe using \( \pi = 3.14 \) is approximate. Let's use the exact formula. Alternatively, maybe I made a mistake in radius. Wait, \( C = 77.872 \), \( 77.872\div3.14 = 24.8 \), then divided by 2 is 12.4. That's correct. Then outer radius is 12.4 + 8.3 = 20.7. Then area of outer circle: \( \pi R^2 = 3.14\times20.7^2 = 3.14\times428.49 = 1345.4586 \). Area of inner circle: \( \pi r^2 = 3.14\times12.4^2 = 3.14\times153.76 = 482.8064 \). Subtract: \( 1345.4586 - 482.8064 = 862.6522 \approx 862.65 \) square centimeters.

Answer:

The area of the shaded region is approximately \(\boldsymbol{862.65}\) square centimeters (or 863 if rounded to the nearest whole number).