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.
