The area of the shaded region is [tex]\rm (150\sqrt{3} \ - 75\pi ) \ feet^2[/tex] option first is correct.
It is given that a circle is inscribed in a regular hexagon with sides of 10 feet.
It is required to find the shaded area (missing data is attached shown in the picture).
What is a circle?
It is defined as the combination of points that and every point has an equal distance from a fixed point ( called the center of a circle).
We have a hexagon with a side length of 10 feet.
We know the area of the hexagon is given by:
[tex]\rm A = \frac{3\sqrt{3} }{2} a^2[/tex] where a is the side length.
[tex]\rm A = \frac{3\sqrt{3} }{2} 10^2[/tex] ⇒ [tex]150\sqrt{3}[/tex] [tex]\rm feet^2[/tex]
We have the shortest length = x feet and from the figure:
2x = 10
x = 5 feet
The radius of the circle r = longer leg
[tex]\rm r = x\sqrt{3} \Rightarrow 5\sqrt{3}[/tex] feet
The area of the circle a = [tex]\pi r^2[/tex] ⇒ [tex]\pi (5\sqrt{3} )^2 \Rightarrow 75\pi \ \rm feet^2[/tex]
The area of the shaded region = A - a
[tex]\rm =(150\sqrt{3} \ - 75\pi ) \ feet^2[/tex]
Thus, the area of the shaded region is [tex]\rm (150\sqrt{3} \ - 75\pi ) \ feet^2[/tex]
Learn more about circle here:
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