To compare the volumes of the right triangular prism and the right cylinder, let's start with the formulas for their volumes.
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The volume \( V \) of a right triangular prism is given by: \[ V_{\text{prism}} = \text{Area of base} \times \text{Height} \] Let \( A \) be the area of the triangular base. Thus, \[ V_{\text{prism}} = A \times 2 \]
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The volume \( V \) of a right cylinder is given by: \[ V_{\text{cylinder}} = \text{Area of base} \times \text{Height} \] The base area is also \( A \) (since the cross-sectional areas are congruent), and the height is 6 units. Therefore, \[ V_{\text{cylinder}} = A \times 6 \]
Now, we can express the volumes in terms of \( A \):
- Volume of the triangular prism: \[ V_{\text{prism}} = 2A \]
- Volume of the cylinder: \[ V_{\text{cylinder}} = 6A \]
To compare the volumes, we can set up the relationship: \[ V_{\text{prism}} = 2A \] \[ V_{\text{cylinder}} = 6A \]
Now, we compare \( V_{\text{prism}} \) and \( V_{\text{cylinder}} \): \[ V_{\text{cylinder}} = 3 \times V_{\text{prism}} \] This means that the volume of the cylinder is three times the volume of the prism.
This leads us to the conclusion:
The volume of the triangular prism is not equal to the volume of the cylinder.