Answer:
There is a general rule that if you want the most possible area relative to the border length of a rectangle, then that rectangle should be a square.
Roughly speaking, because the numbers must add up to a fixed amount, the more one increases, the more the other decreases. You can see that by simply testing a few values:
25 * 25 = 625
24 * 26 = 624
23 * 27 = 621
etc. The further you move from having equal sides, the less area those sides encompass.
If you want to see this proven, we just need a bit of introductory calculus. We'll start with the basic equation:
a = w * h
and the information we're given:
w + h = 50
Let's rearrange that:
h = 50 - w
Now we can substitute that into the area equation:
a = w * (50 - w)
a = 50w - w²
This equation shows the relationship between the area of the rectangle and its width. You'll notice that if you were to graph it, it would form an upside-down parabola. We need to find the place where that parabola peaks.
We can do that by taking the derivative of that equation (i.e., a version of that equation that tells us what its rate of change is):
da/dw = 50 - 2w
And solve that for zero, as the peak has a slope of zero:
0 = 50 - 2w
2w = 50
w = 25
Proving that the width of 25, and thus the height of 25 is indeed the perfect distance.
