To find the area of the hexagonal window, we need to calculate the area of a regular hexagon given the side length, which is 14 inches.
The formula for the area of a regular hexagon with side length [tex]\( a \)[/tex] is:
[tex]\[ \text{Area} = \frac{3 \sqrt{3}}{2} a^2 \][/tex]
Let's go through the steps to calculate it:
1. Identify the side length: [tex]\( a = 14 \)[/tex] inches.
2. Substitute the side length into the area formula:
[tex]\[
\text{Area} = \frac{3 \sqrt{3}}{2} \times (14)^2
\][/tex]
3. Calculate [tex]\( (14)^2 \)[/tex]:
[tex]\[
(14)^2 = 196
\][/tex]
4. Multiply by [tex]\(\frac{3 \sqrt{3}}{2}\)[/tex]:
[tex]\[
\text{Area} = \frac{3 \sqrt{3}}{2} \times 196
\][/tex]
5. Compute the numerical result:
[tex]\[
\text{Area} \approx 509.2 \text{ in}^2
\][/tex]
Thus, the best estimate for the area of the window, considering the given options, is:
[tex]\[ \boxed{509.2 \text{ in}^2} \][/tex]
