Question

Steve likes to entertain friends at parties with "wire tricks." Suppose he takes a piece of wire 56 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. What should the lengths of the wires be so that the total area of the circle and square combined is as small as possible

Answer 1

To find the wire lengths that would minimize the combined area of a circle and square, we express the circumference of the circle and perimeter of the square in terms of the wire length and calculate the areas. The goal is to find an allocation of the wire to the circle and square such that their total area is minimized.

Step-by-step explanation:

To minimize the combined area of a circle and square made from a piece of wire 56 inches long, we first need to determine the relationship between the circle's circumference and the square's perimeter to their respective areas. Let x inches be the length of the wire used for the circle, and 56 - x inches be the length for the square. The circumference of the circle is equal to x, which means the radius r of the circle can be expressed as r = x / (2π). The area of the circle Ac is then πr2 or π(x2) / (4π2). Similarly, the perimeter of the square is 56 - x, making each side of the square (56 - x) / 4 inches, and the area of the square As is [(56 - x) / 4]2.

Now we want to minimize the sum of the areas At = Ac + As. By differentiating At with respect to x and setting the derivative equal to zero, we can find the value of x that minimizes the total area. However, the exact calculation for the lengths would require the use of calculus, which is beyond the scope of a simple answer. The main concept is that by properly allocating the length of the wire to the circle and square, we can achieve the smallest possible combined area.