To determine what the factors of the expression \(6(x+2)(x)\) represent in the context of the box dimensions, let's analyze the box properties given and the mathematical expression:
1. Volume of the Box:
The volume \( V \) of a rectangular box is given by multiplying its length \( L \), width \( W \), and height \( H \):
[tex]\[
V = L \times W \times H
\][/tex]
2. Given Expression for Volume:
The given expression for the volume of the box is:
[tex]\[
6(x+2)(x)
\][/tex]
3. Height of the Box:
We know from the problem statement that the height of the box is always 6 inches. Hence, one of the factors in the expression, specifically the 6, represents the height of the box.
4. Length and Width of the Box:
Let \( x \) be the width of the box in inches.
- According to the problem, the length of the box is always 2 inches more than the width. Hence, the length \( L \) can be expressed as \( x + 2 \).
Given:
- Width, \( W = x \)
- Length, \( L = x + 2 \)
5. Representing Factors:
In the expression \( 6(x+2)(x) \):
- The factor \( 6 \) represents the height of the box.
- The factor \( (x+2) \) represents the length of the box.
- The factor \( x \) represents the width of the box.
Therefore, the factors \( 6 \), \( (x+2) \), and \( x \) collectively represent the height, length, and width of the box respectively.
Conclusion:
The correct interpretation of what the factors represent in the given context is:
(b) the length, width, and height of the box.
