To determine if the right rectangular prism has the same volume as the prism packed with 24 unit cubes, we need to calculate the volume of both shapes.
Volume of the Right Rectangular Prism: The formula for the volume \( V \) of a right rectangular prism is: \[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Length \( = 1 , \text{m} \)
- Width \( = \frac{2}{3} , \text{m} \)
- Height \( = 1 \frac{1}{3} , \text{m} = \frac{4}{3} , \text{m} \)
Now, calculate the volume: \[ V = 1 \times \frac{2}{3} \times \frac{4}{3} \] Calculating the product step-by-step: \[ V = 1 \times \frac{2}{3} = \frac{2}{3} \] Then multiply by the height: \[ V = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} , \text{m}^3 \]
Volume of the Prism Packed with Unit Cubes: The volume of each unit cube is given by the formula: \[ \text{Volume} ; \text{of} ; \text{one} ; \text{cube} = \text{edge length}^3 \] Given the edge length of each cube is \( \frac{1}{3} , \text{m} \): \[ \text{Volume of one cube} = \left(\frac{1}{3}\right)^3 = \frac{1}{27} , \text{m}^3 \] Since there are 24 cubes, the total volume of the prism is: \[ \text{Total Volume} = 24 \times \frac{1}{27} = \frac{24}{27} = \frac{8}{9} , \text{m}^3 \]
Comparison: The volume of the right rectangular prism is \( \frac{8}{9} , \text{m}^3 \) and the total volume for the prism packed with 24 unit cubes is also \( \frac{8}{9} , \text{m}^3 \).
Thus, the two volumes are equal.
Conclusion: Yes, a right rectangular prism with length 1m, width \( \frac{2}{3} \)m, and height \( 1 \frac{1}{3} \)m has the same volume as a prism packed with 24 unit cubes with an edge length of \( \frac{1}{3} \)m.