The Solution:
The correct answer is 206.9 square centimeters.
Let the two concentric circles be represented as below:
We are required to find the area of the ring between the two concentric circles.
Step 1:
We shall find the radii of both concentric circles by using the formula below:
[tex]\text{ Circumference=Perimeter=2}\pi r[/tex]In this case,
For the smaller circle: radius (r) and circumference = 55cm.
[tex]55=2\pi r[/tex]Dividing both sides by 2 pi, we have
[tex]\begin{gathered} \frac{55}{2\pi}=\frac{2\pi r}{2\pi} \\ \\ r=\frac{55}{2\pi}=8.7535\text{ cm} \end{gathered}[/tex]For the bigger circle: radius (R) and circumference = 75cm.
[tex]75=2\pi R[/tex]Solving for R, we get
[tex]\begin{gathered} \frac{2\pi\text{R}}{2\pi}=\frac{75}{2\pi} \\ \\ R=\frac{75}{2\pi}=11.9366\text{ cm} \end{gathered}[/tex]Step 2:
We shall the required area by using the formula below:
[tex]\text{Area}=\pi R^2-\pi r^2=\pi(R^2-r^2)[/tex]In this case,
[tex]\begin{gathered} r=8.7535\operatorname{cm} \\ R=11.9366\operatorname{cm} \end{gathered}[/tex]Substituting these values in the formula above, we get
[tex]\text{Area}=\pi(11.9366^2-8.7535^2)=65.8587\pi=206.901\approx206.9cm^2[/tex]Therefore, the correct answer is 206.9 square centimeters.
