$\"\"$

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The sum of perimeters of an equilateral traingle and a square is $\"\"$ .

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Area of the square is $\"\"$

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Area of the equilateral traingle is $\"\"$

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Let us consider side of the equilateral triangle that produces minimum area is $\"\"$ .

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Let us consider side of the square that produces minimum area is $\"\"$

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Total area : $\"\"$

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Sum of perimeters of an equilateral traingle and a square is $\"\"$ .

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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Find the minimum area.

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$\"\"$

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Evaluate $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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The dimensions of the triangle minimum area is $\"\"$

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The dimensions of the square minimum area is $\"\"$ .

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$\"\"$

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The dimensions of the triangle for minimum area is $\"\"$ .

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The dimensions of the square for minimum area is $\"\"$ .