Question

A rancher wants to build a rectangular pen with an area of 36 m2.

Answer 1

a).  If the width is 'w' and the area is 36, then the length is  36/w.
The amount of fencing required is 2 lengths + 2 widths (the perimeter).
That's  2w + 72/w  or  (2/w)(w²+36)  or  2(w + 36/w) .

b).  The shape that requires the minimum amount of fencing is a circle
with area = 36 m² .  The radius of the circle is about 3.385 meters, and
the fence around it is about 21.269 meters.

If the pen must be a rectangle, then the rectangle with the smallest perimeter
that encloses a given area is a square.  For 36 m² of area, the sides of the
square are each 6 meters, and the perimeter needs 24 meters of fence to
enclose it.

I don't know how to prove either of these factoids without using calculus.