A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 2400 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

x = 600 m

y = 1200 m

Amax = 720000 m²

Step-by-step explanation:

Let call x the smaller side of the rectangular plot and y the largest (we assume we have one y side bounded by a river: Then

A (p) Area of the plot x*y

A (p) = x*y

And perimeter of the plot (to be fenced) is:

P (p) = 2*x + y = 2400 ⇒ y = 2400 - 2*x

Area of rectangular plot as function of x:

A (x) = x * (2400 - 2x)

Taking derivatives on both sides of the equation

A' (x) = (2400 - 2x) + (-2) * x ⇒ A' (x) = (2400 - 2x) - 2x

A' (x) = 0 ⇒ 2400 - 4x = 0 ⇒ 4x = 2400

x = 600 m

And y = 2400 - 2*x

y = 2400 - 1200

y = 1200 m

And the largest enclosed area is Amax = 1200*600

Amax = 720000 m²