To find the area of a circle, we need to use the formula for the circumference to first determine the radius. The formulas related to the circle we will use are:
-
Circumference (C) of a circle: \[ C = 2\pi r \] where \( r \) is the radius.
-
Area (A) of a circle: \[ A = \pi r^2 \]
Given that the circumference \( C = 75.4 \) inches, we can rearrange the circumference formula to find the radius:
\[ r = \frac{C}{2\pi} = \frac{75.4}{2\pi} \]
Now, calculating \( r \):
\[ r = \frac{75.4}{2 \times 3.14159} \approx \frac{75.4}{6.28318} \approx 12.0 \text{ inches} \]
Next, we can now find the area using the radius:
\[ A = \pi r^2 = \pi (12.0)^2 = \pi (144) \approx 3.14159 \times 144 \approx 452.4 \text{ square inches} \]
Now rounding to the nearest tenth, the area of the circle is:
\[ \boxed{452.4} \text{ sq in} \]
Thus, the correct answer from the options provided is 452.4 sq in.