To find the dimensions of the larger rectangular prism that can be filled with 288 smaller cubes measuring \(\frac{1}{2} , \text{cm}\) on each side, we first need to calculate the volume of the larger prism in cubic centimeters.
The volume of one small cube is: \[ \text{Volume of one cube} = \left(\frac{1}{2} , \text{cm}\right)^3 = \frac{1}{8} , \text{cm}^3. \]
Thus, the total volume of the 288 cubes is: \[ \text{Total volume} = 288 \times \frac{1}{8} = 36 , \text{cm}^3. \]
Now we need to find the dimensions of a rectangular prism (length \( l \), width \( w \), height \( h \)) such that: \[ l \times w \times h = 36 , \text{cm}^3. \]
The possible dimensions must also satisfy the volumes’ requirement of using whole numbers and pairs from the given options.
The options provided are:
- \(1/8\)
- \(36\)
- \((2, 3, 6)\)
- \((2, 6, 12)\)
From this list, we can combine dimensions that yield a product of \(36\):
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For the dimensions \((2, 3, 6)\): \[ 2 \times 3 \times 6 = 36. \]
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For the dimensions \((2, 6, 12)\): \[ 2 \times 6 \times 12 = 144 \quad \text{(not valid)}. \]
Thus, the only valid set of dimensions that corresponds to the volume of \(36 , \text{cm}^3\) is \((2, 3, 6)\).
To recap, a possible set of dimensions of the larger rectangular prism to fit all 288 smaller cubes is: \[ 2 , \text{cm}, , 3 , \text{cm}, , 6 , \text{cm}. \]