You have 100 yards of fencing available to create an enclosure for farm animals. As thebuilder, you can choose one of the design options from below.4 pointsOption 1: Create an enclosure shaped like a regular pentagon.Option 2: Create an enclosure shaped like a regular hexagon.Option 3: Create an enclosure shaped like a regular octagon.Decide which option will give the animals the most space. Show your work and give the areaof each enclosure to the nearest tenth of a square yard.

1. area of a pentagon when 1 side is x is [tex]\frac{x^2}{4}\sqrt{5(5+2\sqrt{5})}[/tex]
so if perimiter is 100 then one side is 100/5 or 20
area will be [tex]\frac{20^2}{4}\sqrt{5(5+2\sqrt{5})}[/tex]≈688.2 square yards

2. area of a hexagon when 1 side is length x is [tex]\frac{3x^2\sqrt{3}}{2}[/tex]
so if permiter is 100 then length of one side is 100/6 or about 16.666666666666
the area will be [tex]\frac{3(16.6666666)^2\sqrt{3}}{2}[/tex]≈721.7 square yards

3. area of a regular octagon with side lenghts x is [tex]2(1+\sqrt{2})x^2[/tex]
so if 8 sides then side length of each is 100/8 or 12.5
area will be [tex]2(1+\sqrt{2})12.5^2[/tex]≈754.4 square yards


octagon wil have most space
you can tell because the more sides it has, the more it gets to a circle and the more area it encloses with a given perimiter