Question

A piece of wire 4 m long is cut into two pieces. One piece is bent into the shape of a circle of radius r and the other is bent into a square of side s. How should the wire be cut so that the total area enclosed is:

a) a maximum? r= and s= .

b) a minimum? r= and s=

Answer 1

you have 2 equations to play with:

\(2 \pi r + 4s = 4\)
\(A = \pi r^2 + s^2\)

and you want to max/min A

you can use lagrange multipliers or use the first equation to make this a single variable calculus problem for A which you solve setting A' = 0.

determining the nature of the stationary point might be a bit trickier using the neater lgrannge but the you might also know that the circle gives you the largest area for a given perimeter/ circumference, which is why, for example, polar bears are the shape they are.