Question

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?

Answer 1

x = 600 m

y = 1200 m

Amax = 720000 m²

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