Math help please

Question

1. You have a wire gate that can bent into any shape, to make a play area for you new puppy. When extended, the gate is 5.2m long. What is the largest rectangular area that you can create for your puppy
2. Consider the scenario in question 1. What if you shaped the gate into a circular area? Would you have more or less area? How much more or less?
3. Consider the scenario in question 1. What if you created a rectangular area, where you utilized the fence as your 4th side? What is the maximum area you could create? Show a simple sketch of your shape, with dimensions labeled on the sides.

Answer 1

5.2 m= the perimeter = 5.2. each side of the square = 5.2/4 = 1.3 m. the maximum area=1.3^2=1.69 m^2

the rectangle with maximum area is a square

if you used the fence for a circle, the area would be larger. Circumference= 2pi(r) =5.2, radius =5.2/2pi = 2.6/pi = about 2.6/3.14. the area = pi(r)^2 = pi(2.6/pi)^2= (2.6)^2/pi =1.69(4/pi) = about 1.69(4/3.14) =about 2.15 m^2

area of the circle = about 2.15. area of the square = 1.69

the circle's area is about 0.46 m^2 larger

if 5.2 m gate is used for 3 sides, then 1/2 or 2.6 will be for the side parallel to the fence and 1.3 for each of the other 2 sides, creating an area = 2.6x1.3 = 1.69(2) = 3.38 m^2

LW = A

L=5.2-2W

A = (5.2-2W)W = 5.2W-2W^2

A' = 5.2- 4W = 0

4W = 5.2

W =5.2/4 = 1.3 meters

L =5.2-2(1/3) =5.2-2.6= 2.6 meters

max Area = 1.3x2.6 =3.38 square meters