# Calculus problem on optimization.

• April 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.
• April 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 $x$ ... other piece will have length $40-x$

let $x$ = piece forming a circle

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

$40-x$ = piece forming a square

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

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

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