Let be the sides of the square ends and be the length of the package.

Total area of the solid .

Find in terms of .

Volume of the two hemispheres = Volume of one sphere.

Formula for the volume of the sphere .

Formula for the volume of the cylinder .

Total volume of the solid .

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Solve for .

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Substitute in .

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Differentiate on each side with respect to .

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Find the critical numbers by equating derivative to zero.

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is minimum when .

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

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