Question

both circles have the same center. what is the area of the shaded region?
23.6 cm
d=79.8 cm
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 diameter of the inner circle \( d = 79.8 \) cm, so the radius \( r_{inner}=\frac{d}{2}=\frac{79.8}{2} = 39.9 \) cm.

Step2: Find radius of outer circle

The radius of the outer circle \( r_{outer}=r_{inner} + 23.6=39.9+23.6 = 63.5 \) cm.

Step3: Calculate area of shaded region (annulus)

The area of an annulus is \( A=\pi r_{outer}^{2}-\pi r_{inner}^{2}=\pi(r_{outer}^{2}-r_{inner}^{2}) \).
Substitute \( r_{outer} = 63.5 \) and \( r_{inner}=39.9 \):
\[

$$\begin{align*} A&=\pi(63.5^{2}-39.9^{2})\\ &=\pi((63.5 + 39.9)(63.5 - 39.9))\\ &=\pi(103.4\times23.6)\\ &=\pi\times2439.24\\ &\approx3.14159\times2439.24\\ &\approx7661.24 \end{align*}$$

\]

Answer:

\( 7661.24 \)