The value of a certain recangular solid is 12x. If the dimensions of the solid are the intigers x,y,andz, what is the greatest possible value of z?

Options

A. `24`
B. `12`
C. `6`
D. `4`

Answer

To find the greatest possible value of \( z \) in the given rectangular solid with volume \( 12x \) and dimensions \( x, y, z \), we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a rectangular solid is given by the formula:
\[ V = x \times y \times z \]
According to the problem, the volume is also given as \( 12x \). Therefore, we can set up the equation:
\[ x \times y \times z = 12x \]

### Step 2: Simplify the Equation
Since \( x \) is a common factor on both sides of the equation, we can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ y \times z = 12 \]

### Step 3: Identify Integer Values
We need to find integer values for \( y \) and \( z \) such that their product equals 12. The pairs of integers that multiply to 12 are:
- \( (1, 12) \)
- \( (2, 6) \)
- \( (3, 4) \)
- \( (4, 3) \)
- \( (6, 2) \)
- \( (12, 1) \)

### Step 4: Maximize \( z \)
To find the greatest possible value of \( z \), we will look at the pairs we identified:
- If \( y = 1 \), then \( z = 12 \)
- If \( y = 2 \), then \( z = 6 \)
- If \( y = 3 \), then \( z = 4 \)
- If \( y = 4 \), then \( z = 3 \)
- If \( y = 6 \), then \( z = 2 \)
- If \( y = 12 \), then \( z = 1 \)

Among these pairs, the maximum value of \( z \) occurs when \( y = 1 \) and \( z = 12 \).

### Conclusion
Thus, the greatest possible value of \( z \) is:
\[ \boxed{12} \]