Let the dimension for the space be by , where denotes length and denotes width.
From given we have , also the area can be expressed as . Our goal is to maximize the area .
Using the given perimeter constraint, we can write . Then let's rewrite the area as:
Now the area is expressed as a function of and it is a quadratic function whose graph is a parabola open down. Hence it will have a maximum. More specific, we have
which implies that when , we obtain maximum area . Thus the dimension should be .