To solve this problem, let the length of the wire allocated for the square be $x$ meters, and the length of the wire allocated for the equilateral triangle be $5 - x$ meters.

1. Perimeter and side lengths:

• For the square, the perimeter is $x$, so each side of the square is: $\text{Side of the square} = \frac{x}{4}.$

• For the equilateral triangle, the perimeter is $5 - x$, so each side of the triangle is: $\text{Side of the triangle} = \frac{5 - x}{3}.$

2. Areas:

• The area of the square is: $\text{Area of the square} = \left( \frac{x}{4} \right)^2 = \frac{x^2}{16}.$

• The area of the equilateral triangle is given by: $\text{Area of the triangle} = \frac{\sqrt{3}}{4} \left( \frac{5 - x}{3} \right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{(5 - x)^2}{9} = \frac{\sqrt{3} (5 - x)^2}{36}.$

3. Total Area:

The total area covered by the two pieces is: $\text{Total Area} = \frac{x^2}{16} + \frac{\sqrt{3} (5 - x)^2}{36}.$

4. Optimization:

If we want to maximize or minimize the total area:

• Differentiate the total area expression with respect to $x$.
• Solve $\frac{d(\text{Total Area})}{dx} = 0$ to find critical points.
• Evaluate the total area at these critical points and endpoints ($x = 0$ and $x = 5$) to determine the optimum.

Let me know if you'd like to proceed with the optimization step or analyze a specific aspect!

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Related Questions:

1. What is the length of the wire used for each shape if the total area is minimized?
2. How does the total area change if the wire length is increased to 6 meters?
3. What happens to the area of the triangle if the entire wire is used for it?
4. What is the ratio of areas when the wire is equally divided between the square and triangle?
5. Can this problem be extended to other polygons (e.g., pentagons)?

Tip: Always calculate the side lengths carefully when dealing with perimeters and areas of geometric shapes.