Certainly! Let's work through the problem step by step.
1. Define the Problem:
- We're starting with a 12-inch by 12-inch square tin sheet.
- We need to create an open-faced box by cutting squares from each corner and folding up the sides.
- We need to express the volume of this box in terms of the side length of the cut-out squares, [tex]\( h \)[/tex].
2. Visualize the Situation:
- If [tex]\( h \)[/tex] is the side length of the cut-out square, then each corner of the sheet will have a square of side length [tex]\( h \)[/tex] removed.
- After cutting out these squares, the sheet will be folded up along the cuts to form the sides of the box.
3. Determine the New Dimensions of the Base:
- Original side length of the sheet is 12 inches.
- After removing [tex]\( h \)[/tex] from each side, the length and width of the base of the box will each be reduced by [tex]\( 2h \)[/tex]. Therefore, the new dimensions of the base are [tex]\( (12 - 2h) \)[/tex] by [tex]\( (12 - 2h) \)[/tex].
4. Height of the Box:
- The height of the box will actually be the side length of the cut-out square, [tex]\( h \)[/tex], since the height is formed by folding up the cut squares.
5. Determine the Volume of the Box:
- The volume [tex]\( V \)[/tex] of a box is given by the product of its base area and its height: [tex]\( V = \text{Base Area} \times \text{Height} \)[/tex].
- The base area is the area of a square with side length [tex]\( (12 - 2h) \)[/tex], which is [tex]\( (12 - 2h)^2 \)[/tex].
- The height of the box is [tex]\( h \)[/tex].
Putting it all together, we can write the volume [tex]\( V \)[/tex] of the box as:
[tex]\[ V = h \times (12 - 2h)^2 \][/tex]
Thus, the volume of the open-faced box, expressed in terms of the side length of the cut-out squares [tex]\( h \)[/tex], is:
[tex]\[ V = h(12 - 2h)^2 \][/tex]
This formula shows how the volume of the box will change as the side length [tex]\( h \)[/tex] of the cut-out squares changes. The units of the volume are cubic inches ([tex]\( \text{in}^3 \)[/tex]).
