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An open-faced box is to be constructed by cutting squares out of the corners of a 12 in by 12 in sheet of tin and folding up the sides. Which of the following describes the volume of such a box?

A. The volume of the box is given by [tex]h(12-2h)^2 \, \text{in}^3[/tex], where [tex]h[/tex] is the side length of the cut-out squares.
B. The volume of the box is given by [tex]h^2(12-2h) \, \text{in}^3[/tex], where [tex]h[/tex] is the side length of the cut-out squares.
C. The volume of the box is given by [tex](12-2h)^3 \, \text{in}^3[/tex], where [tex]h[/tex] is the side length of the cut-out squares.
D. The volume of the box is given by [tex]h^3 \, \text{in}^3[/tex], where [tex]h[/tex] is the side length of the cut-out squares.

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To determine which of the given formulas correctly describes the volume of an open-faced box that is constructed by cutting squares out of the corners of a 12-inch by 12-inch sheet of tin and folding up the sides, we will analyze each option provided.

First, let us summarize the process involved in creating the box:

1. Cut squares with a side length [tex]\( h \)[/tex] from each corner of the 12-inch by 12-inch tin sheet.
2. Fold up the sides to form an open-faced box.

The dimensions of the resulting box will be:
- Length and width: [tex]\( 12 - 2h \)[/tex] (because cutting [tex]\( h \)[/tex] from each side removes [tex]\( 2h \)[/tex] from the total length and width)
- Height: [tex]\( h \)[/tex] (as the height is equal to the side length of the cut-out squares)

Given these dimensions, the volume [tex]\( V \)[/tex] of the box can be calculated as:
[tex]\[ V = \text{Length} \times \text{Width} \times \text{Height} \][/tex]
[tex]\[ V = (12 - 2h) \times (12 - 2h) \times h \][/tex]
[tex]\[ V = h (12 - 2h)^2 \][/tex]

Now, let’s evaluate each of the provided formulas against this correct formula:
1. The first formula: [tex]\( h(12 - 2h)^2 \)[/tex]
- This aligns exactly with our derived formula, so it is correct.

2. The second formula: [tex]\( h^2(12 - 2h) \)[/tex]
- While this formula does involve [tex]\( h \)[/tex] and [tex]\( 12 - 2h \)[/tex], the arrangement is different. When we test this with [tex]\( h = 2 \)[/tex]:
- Calculated volume: [tex]\( 2^2(12 - 2 \times 2) = 4 \times 8 = 32 \)[/tex]

3. The third formula: [tex]\( (12 - 2h)^3 \)[/tex]
- This formula does not correctly incorporate [tex]\( h \)[/tex], the height of the box, as it only modifies the base dimensions.
- Calculated volume: [tex]\( (12 - 2 \times 2)^3 = 8^3 = 512 \)[/tex]

4. The fourth formula: [tex]\( h^3 \)[/tex]
- This formula assumes the volume is solely a function of the height cubed, which does not take into account the reduction in length and width.
- Calculated volume: [tex]\( 2^3 = 8 \)[/tex]

Given our calculations and assessments, the formula that correctly describes the volume of the box is:
[tex]\[ \boxed{h(12 - 2h)^2} \][/tex]
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