$\"\"$

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Since the cross section is circular, perimeter of cylindrical package is $\"\"$ is equals to $\"\"$ .

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$\"\"$

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$\"\"$ .

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Volume of cylindrical package is $\"\"$ .

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

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$\"\"$

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For maximum volume, $\"\"$ .

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$\"\"$

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$\"\"$ and $\"\"$

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$\"\"$ and $\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$ .

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If $\"\"$ , then $\"\"$ .

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By second derivative test, the volume is minimum at $\"\"$ .

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If $\"\"$ , then $\"\"$ .

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By second derivative test, the volume is maximum at $\"\"$ .

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$\"\"$

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If $\"\"$ , then $\"\"$ .

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The volume is $\"\"$ .

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Maximum volume : $\"\"$ .

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Thus, the maximum volume of cylindrical package is $\"\"$ .

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$\"\"$

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The maximum volume of cylindrical package is $\"\"$ .