A 36-in. piece of string is cut into two pieces. One piece is used to form a circle while the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum? Help!

Thank you!

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- Dec 7th 2008, 05:33 PMnathan02079Minimizing Area
A 36-in. piece of string is cut into two pieces. One piece is used to form a circle while the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum? Help!

Thank you! - Dec 7th 2008, 05:44 PMskeeter
one piece has length x ... make that length into a circle

other piece has length (36-x) ... make that length into a square

circumference, $\displaystyle C = x = 2\pi r$

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

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

side of the square, $\displaystyle s = \frac{36-x}{4}$

square area, $\displaystyle s^2 = \frac{(36-x)^2}{16}$

total area ...

$\displaystyle A = \frac{x^2}{4\pi} + \frac{(36-x)^2}{16}$