A rancher has 240 meters of fence with which to enclose three sides of a rectangular garden (the fourth side is a cliff wall and will not require fencing). Find the dimensions of the garden with the largest possible area. (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side).)

The dimensions are length 120 and width 60.

Step-by-step explanation:

2W + L = 240 where L = length and W = width.

The area A = LW.

From the first equation L = 240 - 2W, so substituting this in A = LW gives:

A = W (240 - 2W)

A = 240W - 2W^2

Finding the derivative

A' = 240 - 4W

This = 0 for local maxima / minima:

240 - 4W = 0

W = 240/4

W = 60.

Note the second derivative = - 4 so this is a maxima.

Length L = 240 - 2 (60) = 120 m.