$\"\"$

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Let $\"\"$ be the length of each side of the base, $\"\"$ be the height , and $\"\"$ be the volume.

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Since the rectangular solid has a square base, volume of the solid is $\"\"$ .

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Find the function $\"\"$ in terms of $\"\"$ .

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Surface area $\"\"$ .

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$\"\"$ .

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$\"\"$

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Solve for $\"\"$ .

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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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$\"\"$ .

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$\"\"$

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$\"\"$

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

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$\"\"$ .

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Find the critical numbers by equating derivative to zero.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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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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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ is maximum when $\"\"$ and $\"\"$ .

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$\"\"$

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The dimensions are $\"\"$ .