Question

Farmer Ed has 8500 meters of fencing and wants to enclose a rectangular plot that borders on a river. If farmer Ed does not fence the side along the river, what is the largest are that can be enclosed?

Width = 2x
Length = 8,500-2x

Answer 1

Let x and y be width and length of the plot to be fenced. Therefore,

Area (A) = xy m^2

If the side bordering the river is x, then
Circumference (C) = 8500 = x+2y => x = 8500-2y

Substituting for w in the equation for A;
A = (8500-2y)y = 8500y -2y^2

For maximum area, derivative of A with respect to y is 0. That is;
dA/dy = 0 = 8500 -4y => 4y = 8500 => y = 8500/4 = 2125 m
And, x = 8500-2y = 8500 -2*2125 = 4250 m

Therefore,
One side = 4,250 m and second side = 2,125 m.
Largest area, A = 4250*2125 = 9,031,250 m^2