Question

You have 100 yards of fencing available to create an enclosure for farm animals. As the

builder, you can choose one of the design options from below.
4 points
Option 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 area
of each enclosure to the nearest tenth of a square yard.

Answer 1

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

2. area of a hexagon when 1 side is length x is
`(3x^2√(3))/(2)`
so if permiter is 100 then length of one side is 100/6 or about 16.666666666666
the area will be
`(3(16.6666666)^2√(3))/(2)`≈721.7 square yards

3. area of a regular octagon with side lenghts x is
`2(1+√(2))x^2`
so if 8 sides then side length of each is 100/8 or 12.5
area will be
`2(1+√(2))12.5^2`≈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