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