Questions: A piece of wire of length 62 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square? (a) To minimize the combined area, the wire should be cut so that 27.274 are used for the circle and 34.726 are used for the square. (Type integers or decimals rounded to the nearest thousandth as needed.) (b) To maximize the combined area, the wire should be cut so that are used for the circle and are used for the square. (Type integers or decimals rounded to the nearest thousandth as needed.)

Solution Steps

The combined area, \(A_{combined}\), is given by \(A_{combined} = \frac{x^2}{4\pi} + \left(\frac{L - x}{4}\right)^2\).

To find the minimum and maximum combined areas, we take the derivative of \(A_{combined}\) with respect to \(x\), set it to zero, and solve for \(x\). $A'_{combined} = \frac{x}{2\pi}+(\frac{x}{8} - \frac{31}{4})$

The length of wire to cut to minimize the combined area is approximately \(x = 27.274\) units, resulting in a minimum combined area of approximately $A_{min}=\frac{27.274^2}{4 \pi} + \frac{(62 - 27.274)^2}{16}=$ 134.564 square units. The length of wire to cut to maximize the combined area is approximately \(x = 34.726\) units, resulting in a maximum combined area of approximately $A_{max}=\frac{34.726^2}{4 \pi} + \frac{(62 - 34.726)^2}{16}=$ 142.454 square units.

Final Answer: