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
This seems simple enough. We impose our constraint to write
Thus our area function becomes
How do we find the optimal area?
Take its derivative! We find
Setting this to zero gives us
Its solution is
Plugging this back into (2) to find the value of produces
Thus the sides to the rectangle are equal!
So the optimal rectangle (which maximizes its area with its perimiter fixed) is a square!