Question

If one side of a rectangular field is to have a river as a natural boundary, find the dimensions
of the largest rectangular field that can be enclosed by using 240 m of fence for the other three sides.

Answer 1

The Perimeter $P$ of the field is given as:

$P=2W+L\\ \text{where L and W are length and width}\\ 240=2W+L\\ L=240-2W\\ A=W \times L\\ A=W(240-2W)\\ A=240W-2W^2\\ A'=240-4W\\ \text{set A' to 0}\\ 0=240-4W\\ W=240/4=60\\ A''=-4\\ \text{This implies that W=60 is the maximum value}\\\text{Substitute 60 for W in L=240-2W}\\ L=240-120\\ L=120m$

Hence the dimension of the largest rectangle field will be $60m \times 120m$.