A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in by 10 in. By cutting out square corners with side length x and folding up the pieces.

Write an equation for the volume (V) of the box in
terms of x.

You want an equation for the volume (V) in terms of x, the side length of the square corner removed from an 8" by 10" piece of cardboard. After the square is removed, the cardboard is folded to make an open-top box.

Depth

The depth of the box is the x dimension of the flap left after the corner is removed.

Bottom dimensions

The side flap in the 8" direction will have a remaining dimension of (8 -2x) inches.

The side flap in the 10" direction will have a remaining dimension of (10-2x) inches.

Then the bottom area is (8 -2x)(10 -2x).

Volume

The volume is the product of the depth and the area of the bottom of the box:

V = x(8 -2x)(10 -2x)

This can be expanded to the cubic ...

V = 4x(4 -x)(5 -x) = 4x(x² -9x +20)

V = 4x³ -36x² +80x

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Additional comment

The volume is maximized when x = 3-(√21)/3 ≈ 1.472 inches.

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A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in by 10 in. By cutting out square corners with side length x and folding up the pieces. Write an equation for the volume (V) of the box in terms of x.

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Depth

Bottom dimensions

Volume

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A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in by 10 in. By cutting out square corners with side length x and folding up the pieces.

Write an equation for the volume (V) of the box in
terms of x.