To compare the volumes of the triangular prism and the triangular pyramid, we can use the formulas for the volumes of both shapes.
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Volume of the Triangular Prism: \[ V_{\text{prism}} = B \cdot h \] where \( B \) is the area of the triangular base and \( h \) is the length (or height) of the prism. Given \( B = 10 , \text{in}^2 \) and \( h = 7 , \text{in} \): \[ V_{\text{prism}} = 10 \cdot 7 = 70 , \text{in}^3 \]
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Volume of the Triangular Pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} B \cdot h \] where \( B \) is the area of the triangular base and \( h \) is the height of the pyramid. Given \( B = 10 , \text{in}^2 \) and \( h = 7 , \text{in} \): \[ V_{\text{pyramid}} = \frac{1}{3} \cdot 10 \cdot 7 = \frac{70}{3} , \text{in}^3 \approx 23.33 , \text{in}^3 \]
Now, we can compare the two volumes:
- Volume of the prism: \( 70 , \text{in}^3 \)
- Volume of the pyramid: \( \frac{70}{3} , \text{in}^3 \)
To find out how they compare: \[ \frac{V_{\text{prism}}}{V_{\text{pyramid}}} = \frac{70}{\frac{70}{3}} = 70 \cdot \frac{3}{70} = 3 \]
Thus, the volume of the prism is 3 times larger than the volume of the pyramid.
The correct response is: The volume of the prism is three times as large as the volume of the pyramid.