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