To find the volume of the right rectangular prism with dimensions length \( 12 \), width \( 112 \), and height \( 34 \), we can calculate the volume \( V \) using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Substituting in the given dimensions:
\[ V = 12 \times 112 \times 34 \]
Now let's check the provided options for multiplying the edge lengths of the rectangular prism with the original cuboid dimensions you described, particularly focusing on the cube method for packing with unit cubes of edge length \( \frac{1}{4} \).
To find the volume of the prism using the cube method:
-
Confirm the number of cubes: The total number of cubes is given as \( 36 \).
-
Volume of each cube: Each cube has edge length \( \frac{1}{4} \): \[ \text{Volume of each cube} = \left(\frac{1}{4}\right)^3 = \frac{1}{64} \]
-
Total volume from packing with cubes: \[ \text{Volume of the prism} = \text{Number of cubes} \times \text{Volume of each cube} = 36 \times \frac{1}{64} = \frac{36}{64} \] Simplifying this gives: \[ \frac{36}{64} = \frac{9}{16} \]
Now comparing with the given calculation results for the volume:
- The initial calculation involving \( V = 36 \times 164 \) gives \( 3664 \) indeed seems to have been based on \( \text{Volume of cube} = 164 \) which might have a miscalculation based on cube edge length.
Now checking back:
To confirm which multiplication correctly gives \( 916 \) cubic units:
The correct dimensions of your options must be investigated for: \[ 12 \times 1.5 \times \frac{3}{4} \]
Converting fractions:
- Consider that numbers must equal \( (12 \times (1 + \frac{1}{2}) \times \frac{3}{4}) \).
- So \( 12 \cdot \frac{3}{2} \cdot \frac{3}{4} = 12 \times 1.5 \times 0.75 \).
To calculate: \[ 12 \cdot 1.5 = 18 \] \[ 18 \cdot 0.75 = 13.5 \text { units} \]
Thus, no option currently satisfies \( 966 \) directly as before; you are provided choices; hence, strong focus on comparing \( \text{cubic units vs prism dimensions} \).
Final verification with options \( 12 \times 112 \times 34 \) indicates best match from original volume perhaps unsettled based on dimensions investigation; but indeed checking \( v = 36 \times 164 = 916 \), \( 11.5\) cubic should maintain tight fractions.
The answer: Focus on which of the rows consistently delivers the original dimension count upto \( V = 12 \times 11.5 \) should suffice ( \text{Final answer corresponds best with cube unit evaluation.}