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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][tex]$h(12-2h)^2 \, \text{in}^3$[/tex][/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][tex]$h$[/tex][/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][tex]$h$[/tex][/tex] is the side length of the cut-out squares.



Answer :

GinnyAnswer
To determine the volume of an open-faced box made from a 12 in by 12 in sheet of tin, we need to understand how the box is constructed and use the appropriate volume formula.

Step-by-step solution:

1. Understand the problem:
- We start with a 12 in by 12 in sheet of tin.
- Squares of side length [tex]\( h \)[/tex] are cut out from each corner of the square sheet.
- The remaining flaps are folded up to form the sides of the box.

2. Determine the new dimensions:
- After cutting out the squares, the new length and width of the base of the box will both be reduced by twice the side length of the cut-out squares.
- Thus, the length and width of the base will be [tex]\( 12 - 2h \)[/tex].

3. Height of the box:
- The height of the box will be equal to the side length of the cut-out squares, [tex]\( h \)[/tex].

4. Calculate the volume of the box:
- The volume [tex]\( V \)[/tex] of the box is given by the product of the base area and the height.
- The base area is [tex]\( (12 - 2h) \times (12 - 2h) = (12 - 2h)^2 \)[/tex].
- The height of the box is [tex]\( h \)[/tex].

5. Formula for the volume:
- Combine these to find the volume: [tex]\( V = \text{base area} \times \text{height} = (12 - 2h)^2 \times h \)[/tex].

6. Conclusion:
- The volume of the box is given by:
[tex]\[ V = h(12 - 2h)^2 \][/tex]

After evaluating the given options, the correct formula that describes the volume of the open-faced box is:

The volume of the box is given by [tex]\( h(12 - 2h)^2 \)[/tex] in^3, where [tex]\( h \)[/tex] is the side length of the cut out squares. Therefore, the correct answer is:

Formula 1: [tex]\( h(12 - 2h)^2 \)[/tex].

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