You have a wire that is 71 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 when A is a minimum is 31.21 cm.

Step-by-step explanation:

If we construct a square of side "a" and a circle of diameter "D", we divide the length of the wire in the sum of both perimeters.

The perimeter of the square is

The perimeter of the circle is

The length of the wire is

Then we can express a in function of D:

The area A is the sum of the area of the circle and the square. We can then replace a by D as the independent variable:

To optimize the area we derive and equal to 0:

The area is mimimum when the D=9.94.

Then, the circunference for this diameter is:

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step-by-step explanation:

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