To compare the volumes of the triangular prism and the triangular pyramid, we can use the formulas for the volumes of each shape.
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Volume of the triangular prism (V_prism):
- The formula for the volume of a prism is given by: \[ V_{\text{prism}} = B \cdot h \]
- Where \(B\) is the area of the base and \(h\) is the height (or length of the prism).
- From the illustration, we have:
- Area of the triangular base \(B = 10 , \text{inches}^2\)
- Length \(h = 7 , \text{inches}\)
- Thus, \[ V_{\text{prism}} = 10 \cdot 7 = 70 , \text{inches}^3 \]
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Volume of the triangular pyramid (V_pyramid):
- The formula for the volume of a pyramid is given by: \[ V_{\text{pyramid}} = \frac{1}{3} B \cdot h \]
- Again, using the area of the base and the height of the pyramid:
- Area of the triangular base \(B = 10 , \text{inches}^2\)
- Perpendicular height \(h = 7 , \text{inches}\)
- Thus, \[ V_{\text{pyramid}} = \frac{1}{3} \cdot 10 \cdot 7 = \frac{70}{3} , \text{inches}^3 \approx 23.33 , \text{inches}^3 \]
Now, we can compare the two volumes:
- Volume of the prism: \(70 , \text{inches}^3\)
- Volume of the pyramid: \(\frac{70}{3} , \text{inches}^3\)
To find how the volumes relate in size: \[ \frac{V_{\text{prism}}}{V_{\text{pyramid}}} = \frac{70}{\frac{70}{3}} = 3 \] This means the volume of the prism is 3 times the volume of the pyramid.
Conclusion: The correct response is: The volume of the prism is three times as large as the volume of the pyramid.