In the range of $ (0,10]$, or $[80,90)$ feet.
Work Step by Step
Step 1. The perimeter of a rectangle is given by $2(l+w)=180$; we have $l+w=90$ Step 2. Let $x=w$; we have $l=90−w=90−x$, with $0\lt x\lt90$ Step 3. The area is given by $A=lw=x(90−x)=−x^2+90x$ Step 4. Let $A\leq800$; we have $−x^2+90x\leq800$, or $x^2−90x+800\geq0$ Step 5. Factor the above inequality; we have $(x−10)(x−80)\geq0$ and the boundary points are $x=10,80$. Step 6. Using the test points to examine signs of the left side across the boundary points, we have $(0)...(+)...(10)...(−)...(80)...(+)...(90)$ Thus the solutions are $ (0,10]$, or $[80,90)$ feet Step 7. We conclude that the length of a side should be in the range of $ (0,10]$ or $[80,90)$ feet.