A rancher has 240 feet of fence with which to enclose three sides of a rectangular plot (the fourth side is a river and will not require fencing). Find the dimensions of the plot 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).)

Length = 120 ft

Width = 60 ft

Area = 60 x 120 = 7,200ft2

Step-by-step explanation:

Hi, to answer this question we have to apply the next formulas:

Perimeter = 2 widths + 1 length = 2w+l

Replacing with the value given:

240 = 2w+l

Solving for l (length)

l = 240-2w

Area of a rectangle (A) = width x length

Substituting l=240-2w

A = w (240-2w)

A = 240w - 2w^2

Taking the first derivative of A to find the largest area:

A' = 240-4w

When a = 0, and solving for w

0=240-4w

4w=240

w = 240/4

w = 60 ft

Replacing w in the other dimension:

l = 240-2 (60)

l = 240-120=120 ft

length = 120 ft

Area = 60 x 120 = 7,200 ft2