Question

A rectangular field is to be bounded by a fence on 3 sides and a straight stream on the 4th side. Find the dimensions of the field with maximum area that can be enclosed using 1000 ft of rope. This is in the applied maximum and minimum chapter btw.

Answer 1

Check the picture below.

$\bf \stackrel{area}{A}=length\cdot width\implies A(w)=(1000-2w)w \\\\\\ \boxed{A(w)=1000w-2w^2} \\\\\\ \cfrac{dA}{dw}=1000-4w\implies 0=1000-4w\implies 4w=1000 \\\\\\ w=\cfrac{1000}{4}\implies \boxed{w=\stackrel{width}{250}} \\\\\\\ [1000-2(250)]\implies 1000-500\implies \boxed{l=\stackrel{length}{500}}$

and if you run a first-derivative test on say x = 249 and x = 251, you'll notice the slope is positive and negative respectively, meaning, is a maximum.