John has 100 feet of fencing and would like to make a vegetable garden along the already existing backyard fence with a total area of 1200ft2. What is the largest width of the vegetable garden that can result in the desired area of 1200ft2.

The largest width of the vegetable garden that can result in the desired area of 1200 ft² is 60 ft.

Step-by-step explanation:

The given parameters are;

The available fencing John has = 100 feet

The area available in the backyard fence = 1,200 ft²

The number of sides of the vegetable garden = 3 sides

Therefore, given that the area = Length × Width

Whereby one side of the garden is the existing backyard fence, we have;

The length of the remaining 3 sides = 100 ft

From which we have;

L + W + L = 100

W = 100 - 2·L

The area = L × W = L ×(100 - 2·L) = 100·L - 2·L²

The desired area = 1200 ft²

∴ 100·L - 2·L² = 1200

0  = 2·L² -100·L+ 1200

2·L² -100·L+ 1200 = 0

L² - 50·L+ 600 = 0

(L - 20)(L - 30) = 0

L = 20 or 30

When L = 20, W = 100 - 2×20 = 60

When L = 30, W = 100 - 2×30 = 40

We have;

L = 20 ft, W = 60 ft

L = 30 ft, W = 40 ft

Therefore, the largest width, W, that will result in an area of 1200 ft² is W = 60 ft

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