Since the total length of the walls must be 800 linear feet, then the perimeter of the rectangle should be equal to 800 ft. Then:
[tex]2x+2y=800[/tex]
Isolate y from the equation:
[tex]\begin{gathered} 2x+2y=800 \\ \Rightarrow x+y=400 \\ \Rightarrow y=400-x \end{gathered}[/tex]
Notice that the maximum length of a single side can be 400 ft. In that case, the side parallel to the first one will have a measure of 400ft and the height will be 0. In that case, the area is equal to 0.
If we start making the first side of 400 ft slightly lower, then the other one will start to grow and then the area will also start to grow, until both sides have the same length. From that point, if we keep making one side larger, it will be a similar case than when the other side was larger and the previous one a bit smaller. In other words: It doesn't matter if the larger side is the width or the height, the area will grow only when those two lengths get closer.
Therefore, the maximum area of a fixed perimeter rectangle is reached when the sides are equal. If the four sides of the rectangle have the same length, then each one should have a measure of 800ft/4 = 200 ft. The figure will be a square and the area will be:
[tex]200\cdot200=40,000[/tex]
Therefore, the maximum square footage of the store will be 40,000.
