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Sagot :
To determine the area of the shaded region, we need to follow a few distinct steps: calculating the area of the regular hexagon, the area of the inscribed circle, and then finding the difference between these two areas.
### 1. Determine the Area of the Regular Hexagon
A regular hexagon with side length [tex]\( s \)[/tex] can be divided into 6 equilateral triangles. The area [tex]\( A \)[/tex] of a regular hexagon with side length [tex]\( s \)[/tex] is given by:
[tex]\[ A = \frac{3\sqrt{3}}{2}s^2 \][/tex]
Plugging in [tex]\( s = 10 \)[/tex] feet:
[tex]\[ A = \frac{3\sqrt{3}}{2} \cdot 10^2 = \frac{3\sqrt{3}}{2} \cdot 100 = 150\sqrt{3} \text{ square feet} \][/tex]
### 2. Determine the Radius of the Inscribed Circle
The radius of the inscribed circle in a regular hexagon is the same as the height of one of the equilateral triangles formed in step 1. In a [tex]\( 30^\circ - 60^\circ - 90^\circ \)[/tex] triangle, the relationship between sides is as follows:
- The shortest leg (half the length of a side of the equilateral triangle) is [tex]\( x \)[/tex].
- The longest leg (the height) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse (side length of the hexagon) is [tex]\( 2x \)[/tex].
Thus, for our triangle, where the side (hypotenuse) is 10 feet,
[tex]\[ 10 = 2x \implies x = 5 \text{ feet (half the side length)} \][/tex]
The height (and hence the radius [tex]\( r \)[/tex] of the inscribed circle) is:
[tex]\[ r = x\sqrt{3} = 5\sqrt{3} \text{ feet} \][/tex]
### 3. Determine the Area of the Inscribed Circle
The area [tex]\( A_c \)[/tex] of a circle is given by:
[tex]\[ A_c = \pi r^2 \][/tex]
With the radius [tex]\( r = 5\sqrt{3} \)[/tex] feet:
[tex]\[ A_c = \pi (5\sqrt{3})^2 = 75\pi \text{ square feet} \][/tex]
### 4. Determine the Area of the Shaded Region
The shaded region is the difference between the area of the hexagon and the area of the inscribed circle:
[tex]\[ \text{Shaded Area} = A_{\text{hexagon}} - A_{c} \][/tex]
[tex]\[ \text{Shaded Area} = 150\sqrt{3} - 75\pi \text{ square feet} \][/tex]
Hence, the area of the shaded region, which matches one of the provided choices, is:
[tex]\[ 150\sqrt{3} - 75\pi \text{ square feet} \][/tex]
### 1. Determine the Area of the Regular Hexagon
A regular hexagon with side length [tex]\( s \)[/tex] can be divided into 6 equilateral triangles. The area [tex]\( A \)[/tex] of a regular hexagon with side length [tex]\( s \)[/tex] is given by:
[tex]\[ A = \frac{3\sqrt{3}}{2}s^2 \][/tex]
Plugging in [tex]\( s = 10 \)[/tex] feet:
[tex]\[ A = \frac{3\sqrt{3}}{2} \cdot 10^2 = \frac{3\sqrt{3}}{2} \cdot 100 = 150\sqrt{3} \text{ square feet} \][/tex]
### 2. Determine the Radius of the Inscribed Circle
The radius of the inscribed circle in a regular hexagon is the same as the height of one of the equilateral triangles formed in step 1. In a [tex]\( 30^\circ - 60^\circ - 90^\circ \)[/tex] triangle, the relationship between sides is as follows:
- The shortest leg (half the length of a side of the equilateral triangle) is [tex]\( x \)[/tex].
- The longest leg (the height) is [tex]\( x\sqrt{3} \)[/tex].
- The hypotenuse (side length of the hexagon) is [tex]\( 2x \)[/tex].
Thus, for our triangle, where the side (hypotenuse) is 10 feet,
[tex]\[ 10 = 2x \implies x = 5 \text{ feet (half the side length)} \][/tex]
The height (and hence the radius [tex]\( r \)[/tex] of the inscribed circle) is:
[tex]\[ r = x\sqrt{3} = 5\sqrt{3} \text{ feet} \][/tex]
### 3. Determine the Area of the Inscribed Circle
The area [tex]\( A_c \)[/tex] of a circle is given by:
[tex]\[ A_c = \pi r^2 \][/tex]
With the radius [tex]\( r = 5\sqrt{3} \)[/tex] feet:
[tex]\[ A_c = \pi (5\sqrt{3})^2 = 75\pi \text{ square feet} \][/tex]
### 4. Determine the Area of the Shaded Region
The shaded region is the difference between the area of the hexagon and the area of the inscribed circle:
[tex]\[ \text{Shaded Area} = A_{\text{hexagon}} - A_{c} \][/tex]
[tex]\[ \text{Shaded Area} = 150\sqrt{3} - 75\pi \text{ square feet} \][/tex]
Hence, the area of the shaded region, which matches one of the provided choices, is:
[tex]\[ 150\sqrt{3} - 75\pi \text{ square feet} \][/tex]
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