To determine the area of the largest circular mirror Faith can hang, we first need to find the radius of the mirror given the circumference of the frame.
The formula for the circumference \(C\) of a circle is given by:
\[ C = 2\pi r \]
where \(r\) is the radius of the circle.
We know the circumference of the frame is 40 inches, so we can set up the equation:
\[ 40 = 2\pi r \]
To solve for \(r\), we divide both sides by \(2\pi\):
\[ r = \frac{40}{2\pi} = \frac{20}{\pi} \]
Next, we want to find the area \(A\) of the circle, which is given by the formula:
\[ A = \pi r^2 \]
Substituting \(r\) into the area formula:
\[ A = \pi \left(\frac{20}{\pi}\right)^2 \]
Calculating \(\left(\frac{20}{\pi}\right)^2\):
\[ \left(\frac{20}{\pi}\right)^2 = \frac{400}{\pi^2} \]
Now substituting back into the area formula:
\[ A = \pi \cdot \frac{400}{\pi^2} = \frac{400\pi}{\pi^2} = \frac{400}{\pi} \]
The final area of the largest mirror Faith can hang is:
\[ \frac{400}{\pi} \text{ square inches} \]
This is approximately:
\[ \frac{400}{3.14159} \approx 127.32 \text{ square inches} \]
Thus, the area of the largest mirror Faith can hang is:
\[ \frac{400}{\pi} \text{ square inches} \approx 127.32 \text{ square inches} \]