Please help me with my homework ASAP! (help with part c)

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.

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