1. Maximizing and Minimizing Area

A wire 4 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
For the square:
For the circle:

Where should the wire be cut to maximize the total area?
For the square:
For the circle:

I'm completely stumped. Can someone help me?

2. Originally Posted by iheartthemusic29
A wire 4 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
For the square:
For the circle:

Where should the wire be cut to maximize the total area?
For the square:
For the circle:

I'm completely stumped. Can someone help me?
let $\displaystyle x$ = length of wire for the circle
$\displaystyle 4-x$ = length of wire for the square

$\displaystyle 2\pi r = x$

$\displaystyle r = \frac{x}{2\pi}$

circle area, $\displaystyle A_c = \frac{x^2}{4\pi}$

side length of square is $\displaystyle \frac{4-x}{4}$

square area, $\displaystyle A_s = \frac{(4-x)^2}{16}$

total area, $\displaystyle A = A_c + A_s$ ... find $\displaystyle \frac{dA}{dx}$ and minimize