We have a string of length . It does not stretch. But we want to form a rectangle using this string as its boundary. Moreover we want this rectangle’s area is maximized. What to do? (Spoiler alert: solution below!)

Well, if we form a rectangle, then it has sides and satisfying (the perimiter is equal to the length of string).

The area would be

(1)

This seems simple enough. We impose our constraint to write

(2)

Thus our area function becomes

(3).

How do we find the optimal area?

Take its derivative! We find

(4).

Setting this to zero gives us

(4′).

Its solution is

(5).

Plugging this back into (2) to find the value of produces

(6).

Thus the sides to the rectangle *are equal!*

So the optimal rectangle (which maximizes its area with its perimiter fixed) is a **square!**

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