Certainly! Let's go through the problem step by step.
1. Understand the Setup:
We have a piece of construction paper with a length of 15 inches and a width of 10 inches. We need to create an open-top box by cutting out squares from each corner of the paper and then folding up the sides.
2. Define Variables:
- Let [tex]\( x \)[/tex] be the side length (in inches) of the squares cut out from each of the four corners.
3. Determine the Dimensions After Cutting:
- After cutting out the squares, the new length of the box will be the original length minus twice the cut side length (since squares are cut from both ends): [tex]\( 15 - 2x \)[/tex].
- Similarly, the new width of the box will be the original width minus twice the cut side length: [tex]\( 10 - 2x \)[/tex].
4. Understand the Volume of the Box:
- When we fold up the sides after cutting, the height of the box will be equal to the side length of the squares we cut out: [tex]\( x \)[/tex].
5. Formulate the Volume Expression:
- The volume [tex]\( V(x) \)[/tex] of the box can be calculated using the formula for the volume of a rectangular prism: [tex]\( \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \)[/tex].
- Substituting in our dimensions, we get:
[tex]\[
V(x) = \text{(new length)} \times \text{(new width)} \times \text{(height)}
\][/tex]
[tex]\[
V(x) = (15 - 2x) \times (10 - 2x) \times x
\][/tex]
6. Final Expression for Volume:
- Therefore, the expression for the volume of the open-top box as a function of the side length [tex]\( x \)[/tex] of the square cutouts is:
[tex]\[
V(x) = x \times (10 - 2x) \times (15 - 2x)
\][/tex]
Thus, the expression for the volume [tex]\( V(x) \)[/tex] in terms of the side length [tex]\( x \)[/tex] of the square cutouts is:
[tex]\[
V(x) = x (10 - 2x) (15 - 2x)
\][/tex]
