To solve this problem, we minimize the total area of an equilateral triangle and a circle formed by cutting a 6-meter wire into two pieces. Let's restate the solution process:

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1. Define variables:

• Let $x$ be the length of the wire used to form the equilateral triangle.
• The remaining length $6 - x$ is used to form the circle.

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2. Formulas for areas:

Equilateral Triangle:

• Side length of the triangle: $s = \frac{x}{3}$
• Area of the triangle: $A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \left( \frac{x}{3} \right)^2 = \frac{\sqrt{3}}{36} x^2$

Circle:

• Circumference of the circle: $6 - x$
• Radius of the circle: $r = \frac{6 - x}{2\pi}$
• Area of the circle: $A_{\text{circle}} = \pi r^2 = \pi \left( \frac{6 - x}{2\pi} \right)^2 = \frac{(6 - x)^2}{4\pi}$

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3. Total area:

The total area of the two shapes is: $A_{\text{total}} = A_{\text{triangle}} + A_{\text{circle}}$ Substitute: $A_{\text{total}} = \frac{\sqrt{3}}{36} x^2 + \frac{(6 - x)^2}{4\pi}$

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4. Minimize $A_{\text{total}}$:

To minimize the total area, differentiate $A_{\text{total}}$ with respect to $x$, set the derivative to zero, and solve for $x$: $\frac{dA_{\text{total}}}{dx} = \frac{\sqrt{3}}{18} x - \frac{2(6 - x)}{4\pi}$

Solve this equation for $x$.

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Let’s compute the numerical solution for the critical point and verify whether it minimizes the total area. Verify and numerically solve for x dA_dx = diff(A_total, x) critical_x = solve(Eq(dA_dx, 0), x) critical_x_numeric = [point.evalf() for point in critical_x]

Check the critical points

critical_x_numericThe critical point is approximately $x \approx 3.74$ meters. This means:

• Use 3.74 meters of the wire for the equilateral triangle.
• Use the remaining 6 - 3.74 = 2.26 meters for the circle.

This allocation minimizes the total area of both figures.

Verification:

1. The critical point satisfies the minimization conditions.
2. The solution ensures optimal distribution of the wire to reduce total storage space.

Would you like further insights or a detailed explanation of the steps? Here are some follow-up questions to explore:

1. How is the area influenced if more wire is allocated to the circle than the triangle?
2. Can this approach be applied to different total wire lengths?
3. How does the result change if the circle and triangle have additional constraints?
4. What is the maximum area configuration, and how is it derived?
5. How does the geometry of each shape affect their relative storage efficiency?

Tip: Always double-check the derivative and its second-order behavior to confirm the minimum or maximum nature of critical points.