Question

Jun has 120 meters of fencing to make a rectangular enclosure. She also wants to use some fencing to split the enclosure into two parts with a fence parallel to two of the sides. What dimensions should the enclosure have to have the maximum possible area?

Answer 1

Lenght = 20m

Width = 30m

Explanation:

Perimeter of a rectangular enclosure = 2(L+W)

Adding a length of the fence parallel to one side of the sudden to split the closure, the total is 2(L+W) + L = 120

Area of a rectangular closure = L*W

To find the dimension that would maximize the area solve for L and W.

2(L+W) + L = 120

2L + 2W + L = 120

3L + 2W = 120

2W = 120 - 3L

W = (120 - 3L)/2

Put the value of W into Area formula

A = L*W

A = L *(120 - 3L) /2

= L(60 - 1.5L)

= 60L - 1.5L^2

= -1.5L^2 + 60L

This is a quadratic equation. Compare to ax^2 + bx + c

L = -b/2a

a = -1.5, b= 60, c= 0

L = -60/2(-1.5)

= -60/-3

= 20m

Put L = 20 into the value of W

W = (120 - 3L) /2

W= (120 - 3*20) / 2

= (120 - 60)/2

= 60 / 2

= 30m

The dimensions are 20m by 30m