Question

You have 2m of wire which you can bend into a square or a circle. You also have the option of cutting the wire and making one piece into a square and the other into a circle. How would you maximize the total area enclosed by the wire?

Answer 1

To maximize the total area enclosed by the wire, you should cut the wire and make one piece into a square and the other into a circle. This will result in a total enclosed area of approximately 0.1415m^2.

Step-by-step explanation:

To maximize the total area enclosed by the wire, we need to find the optimal way to use the 2m of wire. Let's consider two scenarios: bending the wire into a square and cutting the wire to make a square and a circle.

If we bend the wire into a square, each side of the square will be 2m/4 = 0.5m. The area of the square is then (0.5m)^2 = 0.25m^2.

If we cut the wire into two equal pieces, each piece will be 1m long. We can use one piece to form a square with side length 1m/4 = 0.25m, resulting in an area of (0.25m)^2 = 0.0625m^2. We can use the other piece to form a circle with circumference 1m, which gives us a radius of 1m/2π = 0.159m. The area of the circle is then π(0.159m)^2 ≈ 0.079m^2.

Comparing the two scenarios, we can see that cutting the wire and forming a square and a circle will maximize the total enclosed area, which is approximately 0.1415m^2.