$\"\"$

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Let $\"\"$ be the sides of the square ends and $\"\"$ be the length of the package.

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Total area of the solid $\"\"$ .

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

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Volume of the two hemispheres = Volume of one sphere.

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Formula for the volume of the sphere $\"\"$ .

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Formula for the volume of the cylinder $\"\"$ .

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Total volume of the solid $\"\"$ .

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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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$\"\"$ .

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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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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is minimum when $\"\"$ .

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The radius of the cylinder that produces the minimum surface area is $\"\"$ .

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$\"\"$

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$\"\"$ .