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Formula for the volume of the rectangular prism is $\"\"$ ,

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where $\"\"$ is length, $\"\"$ is width and $\"\"$ is height of the prism.

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Volume of the design $\"\"$ cubic inches.

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$\"\"$

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The base of the design is square. So $\"\"$ .

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

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

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$\"\"$

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$\"\"$

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$\"\"$ .

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Surface area of the prism $\"\"$ .

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

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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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Find the absolute minimum point by using graph.

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Graph the equation $\"\"$ and locate the absolute minimum point.

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$\"\"$

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Observe the graph:

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The absolute minimum point is at $\"\"$ .

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Length of the design box is $\"\"$ ft.

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Width of the design box is $\"\"$ ft.

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

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$\"\"$

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$\"\"$ ft

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Height of the design box is $\"\"$ ft.

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Therefore, the three dimensions are $\"\"$ inches and the design box is a cube.

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The function for the surface area of the design in terms of the length of one side of the base is $\"\"$ .

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Graph of the function:

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$\"\"$

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\ \ The absolute minimum occured when $\"\"$ in.

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The three dimensions are $\"\"$ inches and the design box is a cube.