Question

Of all the rectangles having a given perimeter, which rectangle has the largest area.

Answer 1

which rectangles?!?!

Answer 2

you need pictures of the rectangle...my guess is the rectangle that is the biggest has the largest area

Answer 3

Suppose you have a rectangle with length \(x\) and width \(y\). (For the rectangle to exist, you must have both \(0<x<\infty\) and \(0<y<\infty\).)

The perimeter of any such rectangle, \(P\), is
\[P=2x+2y\]
and the area, \(A\), is
\[A=xy\]
Given that \(P\) is fixed to some value ("... given perimeter"), you can attempt to find the optimal dimensions of the rectangle such that \(A\) is maximized. There are a few ways to do this, but for simplicity I'll assume you're working at a Calc 1 level.

First, write \(A\) in terms of only one variable. You do this by solving for either \(x\) or \(y\) in the perimeter equation. Which one you choose doesn't matter, you'll get the same result either way.

Next, substitute this expression into the area equation, and take the derivative with respect to the variable you picked. Find the critical points and extrema - in this case, you should only have one extreme value, and it should be a maximum that indicates the largest area that can be obtained.