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}}](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7Barea%7D%7BA%7D%3Dlength%5Ccdot%20width%5Cimplies%20A%28w%29%3D%281000-2w%29w%0A%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7BA%28w%29%3D1000w-2w%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7BdA%7D%7Bdw%7D%3D1000-4w%5Cimplies%200%3D1000-4w%5Cimplies%204w%3D1000%0A%5C%5C%5C%5C%5C%5C%0Aw%3D%5Ccfrac%7B1000%7D%7B4%7D%5Cimplies%20%5Cboxed%7Bw%3D%5Cstackrel%7Bwidth%7D%7B250%7D%7D%0A%5C%5C%5C%5C%5C%5C%5C%0A%5B1000-2%28250%29%5D%5Cimplies%201000-500%5Cimplies%20%5Cboxed%7Bl%3D%5Cstackrel%7Blength%7D%7B500%7D%7D)
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.
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.