A piece of wire 10 feet long is cut into two pieces. One piece is bent into the shape of a circle and the other into the shape of the square. How should the wire be cut so that the combined area of the two figures is as small as possible?

i'm sorry, this is about optimization problems.

x and (10 - x)

x is 2 pi r
r = x/(2 pi)
Ac = pi r^2 = pi x^2/(4pi^2) = x^2/(4pi)

(10 -x) = 4*side = 4 s
so
s = (10-x)/4
As = s^2 = (100-20 x + x^2)/16

A= Ac+As = x^2/4pi + (1/16)(x^2-20x+100)

dA/dx = 0 for max or min
0= x/2pi +1/16 (2x-20)
0= .16 x + .125 x - 1.25
0= .285 x - 1.25
x = 4.4

5inches

To find the optimal way to cut the wire, we need to consider the formulas for the circumference of a circle and the perimeter of a square.

Let's assume that x represents the length of the wire used for the circle. Hence, the length of the wire used for the square would be (10 - x).

The circumference of a circle is given by the formula: C = 2πr, where r is the radius. Since the entire wire is used for the circle, the circumference must be equal to the total length. Therefore, 2πr = x.

The perimeter of a square is given by the formula: P = 4s, where s is the length of a side. Again, as the remaining wire is used for the square, the perimeter is equal to the remaining length. Therefore, 4s = (10 - x).

To find the optimal solution, we need to minimize the total area, which is the sum of the circle's area and the square's area. The area of a circle is given by the formula: A = πr^2, and the area of a square is given by the formula: A = s^2.

We can substitute the values for r and s using the formulas we derived earlier:

For the circle:
C = 2πr
2πr = x
r = x / (2π)

For the square:
P = 4s
4s = (10 - x)
s = (10 - x) / 4

Now, we can calculate the area of each figure:

Circle's area:
A_circle = πr^2 = π(x/(2π))^2 = x^2 / (4π)

Square's area:
A_square = s^2 = ((10 - x)/4)^2 = (10 - x)^2 / 16

The combined area is the sum of the circle's area and the square's area:

A_combined = A_circle + A_square = x^2 / (4π) + (10 - x)^2 / 16

To minimize the combined area, we can take the derivative of A_combined with respect to x and set it equal to zero:

dA_combined/dx = 0

By solving this equation, we can find the optimal value of x that minimizes the combined area.