To determine the area of a regular octagon given the measurements of its apothem and its perimeter, we can use the formula for the area of a regular polygon. The formula is:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{apothem} \times \text{perimeter} \][/tex]
Here are the given values:
- Apothem ([tex]\( a \)[/tex]) = 10 inches
- Perimeter ([tex]\( P \)[/tex]) = 66.3 inches
Now, plug these values into the formula to find the area:
[tex]\[ \text{Area} = \frac{1}{2} \times 10 \times 66.3 \][/tex]
First, calculate the product of the apothem and the perimeter:
[tex]\[ 10 \times 66.3 = 663 \][/tex]
Then, multiply this product by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 663 = 331.5 \text{ square inches} \][/tex]
Finally, round the area to the nearest square inch:
[tex]\[ \text{Area} \approx 332 \text{ square inches} \][/tex]
Therefore, the area of the octagon, rounded to the nearest square inch, is 332 square inches.
The correct answer is:
[tex]\[ \boxed{332} \][/tex]
