Question

both circles have the same center. the circumference of the inner circle is 71.592 millimeters. what is the area of the shaded region? 10 mm c=71.592 mm write your answer as a whole number or a decimal rounded to the nearest hundredth. square millimeters

Explanation:

Step1: Find radius of inner circle

The formula for the circumference of a circle is \( C = 2\pi r \). We know \( C = 71.592 \) mm and \( \pi \approx 3.14 \). So, \( r=\frac{C}{2\pi}=\frac{71.592}{2\times3.14} \)
\( \frac{71.592}{6.28}=11.4 \) mm.

Step2: Find radius of outer circle

From the diagram, the width of the shaded region (ring) is 10 mm. So outer radius \( R = 11.4 + 10 = 21.4 \) mm.

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 \( R = 21.4 \), \( r = 11.4 \)
\( R^{2}-r^{2}=(21.4)^{2}-(11.4)^{2}=(21.4 - 11.4)(21.4 + 11.4)=10\times32.8 = 328 \)
Then \( A = 3.14\times328 = 1029.92 \) square millimeters.

Answer:

1029.92