# Thread: Maxima and minima 3

1. ## Maxima and minima 3

A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
b) How should the wire be used to obtain a maximum area?

What equation can be use?

Thx

2. Originally Posted by sanikui
A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
b) How should the wire be used to obtain a maximum area?

What equation can be use?

Thx
First, the function that we want to optimize is the area of both the circle and the square.

$\displaystyle A=s^2+\pi r^2$ where s is the side length of the square and r is the radius of the circle.

Now we need a constraint function we know that we only have 100 cm of wire so the perimeters of both the square and the circle must be 100cm.

$\displaystyle 100=4s+2\pi r$

From here just sub to eliminate one variable and you are off to the races. Good luck.

3. Originally Posted by sanikui
A piece of wire 100cm in length is to be cut into two piece, one piece of which is to be shaped into a circle and the other into a square.

a) How should the wire be cut if the sum of the enclosed areas is to be a minimum?
b) How should the wire be used to obtain a maximum area?

What equation can be use?

Thx
cut the wire into two pieces ... one piece has length $\displaystyle x$ , the other has length $\displaystyle 100-x$

let the $\displaystyle x$ length be made into a circle ...

$\displaystyle x = 2\pi r$

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

$\displaystyle A = \pi r^2 = \pi \left(\frac{x}{2\pi}\right)^2 = \frac{x^2}{4\pi}$

let the $\displaystyle 100-x$ length be made into the square ...

$\displaystyle 100-x = 4s$

$\displaystyle \frac{100-x}{4} = s$

$\displaystyle A = s^2 = \frac{(100-x)^2}{16}$

total area of the two figures ...

$\displaystyle A = \frac{x^2}{4\pi} + \frac{(100-x)^2}{16}$

find $\displaystyle \frac{dA}{dx}$ and optimize like you were taught in class.

4. 44cm should be used to form circle, 56cm to form square.
Thanks everyone.