Question

armer Ed has 2500 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 area that can be​ enclosed?

Answer 1

781250 Square Meters

Explanation:

Let the dimensions of the rectangular plot be x and y

Farmer Ed wants to enclose three sides of a rectangular plot with a fencing of 2500 meters.

Therefore: Perimeter, P=x+2y=2500

We want to find the largest area that can be enclosed.

Area of the plot, A(x,y)=xy

Substitute x=2500-2y

A(y)=(2500-2y)y

`A(y)=2500y-2y^2`

To maximize A, we first find its derivative

`A'(y)=2500-4y\\$Setting A'=0\\2500-4y=0\\2500=4y\\y=625 meters\\Recall: x=2500-2y\\x=2500-2(625)=1250meters`

The largest area that can be enclosed(at x=1250m,y=625m) is:

1250 X 625

=781250 Square Meters