# Calculus problem on optimization.

• Apr 12th 2012, 05:01 PM
temporary
Calculus problem on optimization.
Hi I was wondering if anyone can help with this it just doesn't click with me...

A 40 cm string is cut in 2. one piece forms a square and the other a circle. find what the lengths of the pieces has to be so that the total of the areas is minimal.
• Apr 12th 2012, 05:47 PM
skeeter
Re: Calculus problem on optimization.
Quote:

Originally Posted by temporary
Hi I was wondering if anyone can help with this it just doesn't click with me...

A 40 cm string is cut in 2. one piece forms a square and the other a circle. find what the lengths of the pieces has to be so that the total of the areas is minimal.

let one piece of string have length $\displaystyle x$ ... other piece will have length $\displaystyle 40-x$

let $\displaystyle x$ = piece forming a circle

$\displaystyle x = 2\pi r$ , derive an expression for the circle area in terms of $\displaystyle x$

$\displaystyle 40-x$ = piece forming a square

$\displaystyle 40-x$ = Perimeter of the square , derive an expression for the square area in terms of $\displaystyle x$

total area, $\displaystyle A$ = circle area as a function of $\displaystyle x$ + square area as a function of $\displaystyle x$

find $\displaystyle \frac{dA}{dx}$ and determine the value of $\displaystyle x$ that minimizes the total area