To determine if the right rectangular prism has the same volume as the prism packed with 10 unit cubes, we will calculate the volume of each.
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Volume of the right rectangular prism: The volume \( V \) of a rectangular prism is given by the formula: \[ V = \text{length} \times \text{width} \times \text{height} \] For the rectangular prism with length \( \frac{1}{2} , \text{m} \), width \( \frac{1}{4} , \text{m} \), and height \( \frac{5}{4} , \text{m} \), we get: \[ V = \frac{1}{2} \times \frac{1}{4} \times \frac{5}{4} \] Calculating this: \[ V = \frac{1 \cdot 1 \cdot 5}{2 \cdot 4 \cdot 4} = \frac{5}{32} , \text{m}^3 \]
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Volume of the 10 unit cubes: Each unit cube has an edge length of \( \frac{1}{4} , \text{m} \). The volume of one unit cube is: \[ V_{\text{cube}} = \text{edge length}^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{64} , \text{m}^3 \] Therefore, the total volume for 10 unit cubes is: \[ V_{\text{total cubes}} = 10 \times \frac{1}{64} = \frac{10}{64} = \frac{5}{32} , \text{m}^3 \]
Now, comparing the two volumes:
- Volume of the rectangular prism: \( \frac{5}{32} , \text{m}^3 \)
- Volume of the prism packed with 10 unit cubes: \( \frac{5}{32} , \text{m}^3 \)
Since both volumes are equal, the answer is:
Yes, they both have a volume of \( \frac{5}{32} \) cubic meters.