Question

ou have a wire that is 32 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?

The circumference of the circle is cm.

Give your answer to two decimal places

Answer 1

I'm not too sure but first divide by two then take away 2 - 4 centimeters from the circle to the square.

Answer 2

C = Circumference of circle
P = Perimeter of Square
C + S = 32\[A = P^2/16 + C^2/4\pi\]\[A = (32 - C)^2/16 + C^2/4\pi\]\[A = 32^2/16 - 64C/16 + C^2/16 + C^2/4\pi\]\[A = 64 - 4C + C^2 * (4+\pi)/16\]To solve for a minimum, take a derivative and find when it is 0. Therefore: \[0 = -4 + C*(4+\pi)/8\pi\]\[C = 32\pi/(4+\pi) = 14.07\]

Answer 3

its a minimization problem. you want to create some equations that represent whats going on, take derivatives, and plug answers back in.

Let x be the length of wire for the square, and y the length of wire for the circle.

we know that x + y = 32. Thats one equation.

The area of the square will be:\[\left(\frac{x}{4}\right)^2\]The area of the circle will be:\[\pi\left(\frac{y}{2\pi}\right)^2\]. Add those two areas together, substitute y for x or x for y from the first equation, and take the first derivative.