$\"\"$

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(a)

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Observe the figure :

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Consider the first rectangular solid.

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Dimensions of the first rectangular solid are $\"\"$ , $\"\"$ and $\"\"$ inches.

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Surface area of the rectangular solid is $\"\"$ .

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Substitute corresponding values in the above formula.

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$\"\"$

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Therefore, surface area of the solid is $\"\"$ sq.inches.

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Consider the second regular cube.

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Side of the cube is $\"\"$ inches.

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Surface area of the cube is $\"\"$ .

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Substitute $\"\"$ in the above formula.

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$\"\"$ .

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Therefore, surface area of the cube is $\"\"$ sq.inches.

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Consider the third rectangular solid.

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Dimensions of the first rectangular solid are $\"\"$ , $\"\"$ and $\"\"$ inches.

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Surface area of the rectangular solid is $\"\"$ .

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Substitute corresponding values in the above formula.

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$\"\"$

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Therefore, surface area of the solid is $\"\"$ sq.inches.

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Hence it is verified that the each of the solid has a surface area of $\"\"$ sq.inches.

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$\"\"$

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(b)

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Volume of the first rectangular solid is $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in above formula.

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$\"\"$

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Volume of the first rectangular solid is $\"\"$ cubic inches.

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Volume of the second regular cube is $\"\"$ .

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Substitute $\"\"$ in the above formula.

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$\"\"$

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Volume of the cube is $\"\"$ cubic inches.

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Volume of the third rectangular solid is $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in above formula.

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$\"\"$

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Volume of the third rectangular solid is $\"\"$ cubic inches.

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$\"\"$

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(c)

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Surface area of the rectangular solid is $\"\"$ .

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If the base is square then $\"\"$ .

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Surface area of the rectangular solid with square base is

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$\"\"$

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Surface area of the rectangular solids with square base is $\"\"$ sq.inches.

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

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$\"\"$

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Volume of the rectangular cube with square base is $\"\"$ .

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

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$\"\"$

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Apply derivative on each side with respective to $\"\"$ .

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$\"\"$

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For maximum volume equate $\"\"$ to zero . \ \

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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Therefore, dimensions of the rectangular solids with square base are $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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(a) Verified that the each of the solid has a surface area of $\"\"$ sq.inches.

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(b) Volume of the first rectangular solid is $\"\"$ cubic inches.

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Volume of the cube is $\"\"$ cubic inches.

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Volume of the third rectangular solid is $\"\"$ cubic inches.

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(c) Dimensions of the rectangular solids with square base are $\"\"$ , $\"\"$ and $\"\"$ .