To solve the problem, let's break it down into several key steps:
1. Determine the radius of the inscribed circle:
The radius of the inscribed circle of a regular hexagon can be found by considering a 30-60-90 triangle that is formed by splitting one of the equilateral triangles within the hexagon.
In a 30-60-90 triangle, we know:
- The shortest leg (half the side length of the equilateral triangle) is [tex]\( x = \frac{10}{2} = 5 \)[/tex] feet.
- The longer leg (which is the height of the equilateral triangle and also the radius [tex]\( r \)[/tex]) is [tex]\( x \sqrt{3} = 5 \sqrt{3} \)[/tex] feet.
Therefore, the radius [tex]\( r \)[/tex] of the circle is:
[tex]\[
r = 5 \sqrt{3} \text{ feet}
\][/tex]
2. Calculate the area of the regular hexagon:
The area [tex]\( A_h \)[/tex] of a regular hexagon with side length [tex]\( s \)[/tex] can be calculated using the formula:
[tex]\[
A_h = \frac{3 \sqrt{3}}{2} s^2
\][/tex]
Given [tex]\( s = 10 \)[/tex] feet, we get:
[tex]\[
A_h = \frac{3 \sqrt{3}}{2} \times 10^2 = \frac{3 \sqrt{3}}{2} \times 100 = 150 \sqrt{3} \text{ square feet}
\][/tex]
3. Calculate the area of the inscribed circle:
The area [tex]\( A_c \)[/tex] of a circle with radius [tex]\( r \)[/tex] is given by:
[tex]\[
A_c = \pi r^2
\][/tex]
Since [tex]\( r = 5 \sqrt{3} \)[/tex] feet, we get:
[tex]\[
A_c = \pi (5 \sqrt{3})^2 = \pi \times 25 \times 3 = 75 \pi \text{ square feet}
\][/tex]
4. Find the area of the shaded region:
The area of the shaded region [tex]\( A_s \)[/tex] is the area of the hexagon minus the area of the circle:
[tex]\[
A_s = A_h - A_c = 150 \sqrt{3} - 75 \pi \text{ square feet}
\][/tex]
Given the problem and the answer choices, the answer corresponding to:
[tex]\[
(150 \sqrt{3} - 75 \pi) \pi^2
\][/tex]
Therefore, the correct answer is:
[tex]\[
150 \sqrt{3} - 75 \pi
\][/tex]
