shape should be and its area and dimensions. Justify your answer using mathematical reasoning.

Question

Tex has a home on the range (where the deer and antelope play). He wants to build a
rectangular corral for his horses. He only has 170 feet of fencing. What size of corral
should be built to make sure the horses have the most room? List the dimensions and
area. Justify how you know the corral is as large as possible. Explain how you figured it
out.
Tex gets a better idea. He has a barn on his ranch that is 80 feet by 120 feet. He decides
to build the corral using one side of the barn and the 170 feet of fencing. What should be
the dimensions of the corral now? What is the area? Explain your mathematical
reasoning.
A year later, Tex needs a second corral. This time he has 240 feet of fencing. He doesn’t
want to use the other side of the barn, because it is near a small pond. He picks out a new
location and realizes that he does not need to make the corral a rectangle. He designs a
corral in the shape of a hexagon. What are the lengths of the sides and what is the area of
the corral? Explain how you found your answer.
Tex thinks that maybe another shape would make an even larger area for his corral.
Determine what the shape should be and its area and dimensions. Justify your answer
using mathematical reasoning.

Answer 1

largest area is with a square, each side 170/4 = 42 1/2 feet long. Area = 42.5^2 = 1806.25 square feet

If he uses the barn for one side, then one side = 105, the other sides 65 feet.

Area = A = s(170-s)/2 = (1/2)(170s-s^2)

A' = 65 - s = 0

s=65, 170-s = 105

A=65(105) = 6925 square feet for maximum area using one side of the barn

hexagon has 6 sides. 240/6= 40 feet for each side.

Area = 1600(1+sqr2) = 38624 square feet

or more generally, if F = total fence length then the area, A = (F/6)^2(1+sqr2)

a circle makes the largest area for a given fence length. Area = (pi)r^2

circumference = 240 = 2pir, with r = 120/pi, so Area = pi(120/pi)^2 = 14400/pi square feet