Question

A 2 meters piece of wire is cut into two pieces and one piece is bent into a square and the other is bent into an equilateral triangle. where, if anywhere, should the wire be cut so that the total area enclosed by both is minimum and maximum?

Answer 1

To determine the wire cut that will result in the minimum and maximum total area enclosed by both the square and the equilateral triangle, we need to use the formulas for the perimeters of these shapes. By setting up an equation and solving for the side lengths, we can calculate the areas and determine the optimal wire cut.

Step-by-step explanation:

To find the wire cut that will result in the minimum and maximum total area enclosed by both the square and the equilateral triangle, we need to consider the formulas for the perimeter of a square and an equilateral triangle.

The perimeter of a square is 4 times the length of one side, while the perimeter of an equilateral triangle is 3 times the length of one side. Since the total length of the wire is 2 meters, we can represent the length of one side of the square as x and the length of one side of the equilateral triangle as 2 - x (since the remaining wire will be used for the triangle).

By using the formulas for the perimeters, we can set up the following equation: 4x + 3(2 - x) = 2. Solving this equation will give us the lengths of the sides for the square and the equilateral triangle, and we can then calculate the respective areas and determine where to cut the wire to minimize and maximize the total area enclosed by both shapes.