Question

A farmer wants to enclose a rectangular field along a river on three sides. If 4,800 feet of fencing is to be used, what dimensions will maximize the enclosed area

Mathematics

Answer 1

so hmmm, notice the picture below

the perimeter is just w + w + l or 2w+l

thus
`\bf \textit{area of a rectangle}\\\\ A=lw\qquad 4800-2w=l\implies A(w)=(4800-2w)w \\\\\\ A(w)=4800w-2w^2`

take the derivative of A(w), then zero it out to get the critical points, and do a first-derivative test on the critical points for any maxima between 0 and 4800