To solve this problem, we need to determine the function that represents the volume of the larger popcorn box when the length and width are increased by [tex]\( x \)[/tex] inches, while the height remains the same.
We start with the original dimensions of the box:
- Base length = 6 inches
- Base width = 4 inches
- Height = 8 inches
When the length and width each increase by [tex]\( x \)[/tex] inches, the new dimensions will be:
- New base length = [tex]\( 6 + x \)[/tex] inches
- New base width = [tex]\( 4 + x \)[/tex] inches
- Height remains = 8 inches
The volume [tex]\( V \)[/tex] of a rectangular prism (or box) is given by the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Substituting the new dimensions into the volume formula, we get:
[tex]\[ V(x) = (6 + x) \times (4 + x) \times 8 \][/tex]
Now, we simplify to recognize it as one of the given options. The function representing the volume is:
[tex]\[ V(x) = 8 \times (6 + x) \times (4 + x) \][/tex]
This simplified form matches the option (2):
[tex]\[ V(x) = (6 + x)(4 + x)(8) \][/tex]
Therefore, the correct function that represents the volume [tex]\( V(x) \)[/tex] of the larger box is:
[tex]\[ V(x) = (6 + x)(4 + x)(8) \][/tex]
So, the correct answer is:
(2) [tex]\( V(x) = (6 + x)(4 + x)(8) \)[/tex]
