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

(a) Find a function that models the amount F of fencing required, in terms of the width w of the pen.

F (w) =

(b) Find the pen dimensions that require the minimum amount of fencing.

width m

length m

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.