To find out which shape has the larger volume, we need to calculate the volumes of both the triangular pyramid and the triangular prism.
Volume of the Triangular Pyramid:
The volume \( V \) of a pyramid is given by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we need to calculate the area of the triangular base. The triangular base has edges measuring 3 inches and 5 inches. To find the area of the triangle, we can use Heron's formula:
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Calculate the semi-perimeter \( s \): Since we don't have the length of the third side, we can't directly use Heron's formula. However, we can find an approximate area using the base and height.
Since we don't have the complete triangle defined, we can find the area as: If we assume that the triangle that forms the base is isosceles, the height can be derived from basic trigonometry, but let's assume that the height corresponding to the side of 5 inches is 3 inches (as it can give us an approximate area).
\( A \) = \(\frac{1}{2} \times \text{Base} \times \text{Height}\) \( A \) = \(\frac{1}{2} \times 5 \times 3\) \( A = \frac{15}{2} = 7.5 \text{ square inches}\)
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Now we can calculate the volume of the pyramid: \[ V = \frac{1}{3} \times 7.5 \times 8.41 \] \[ V = \frac{1}{3} \times 63.075 = 21.025 \text{ cubic inches} \]
Volume of the Triangular Prism:
The volume \( V \) of a prism is given by the formula: \[ V = \text{Base Area} \times \text{Height} \]
Using the same area of the triangular base \( (7.5 \text{ square inches}) \): \[ V = 7.5 \times 8.41 = 63.075 \text{ cubic inches} \]
Comparing Volumes:
- Volume of the pyramid: \( 21.025 \text{ cubic inches} \)
- Volume of the prism: \( 63.075 \text{ cubic inches} \)
Now, to find out how much larger the volume of the prism is compared to the volume of the pyramid: \[ 63.075 - 21.025 = 42.05 \text{ cubic inches} \]
To find out how much larger the volume of the prism is compared to the volume of the pyramid, we can divide the volume of the prism by the volume of the pyramid: \[ \frac{63.075}{21.025} \approx 3 \]
Overall, the triangular prism has a larger volume than the triangular pyramid by approximately three times.
Thus, the correct response is: The prism has a larger volume by three times.