To determine which shape has the larger volume, we can calculate the volumes of both the triangular pyramid and the triangular prism.
Volume of the Triangular Pyramid (V_pyramid):
The formula for the volume of a triangular pyramid is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
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Calculate the base area of the triangular base:
For a triangle, the area can be calculated using Heron's formula, which requires the lengths of all three sides. However, we will use the simpler formula for the area of a triangle with base and height.We can find the height of the triangle with sides of lengths 3 inches and 5 inches by using the fact that the third side (the side opposite to the height) can be calculated using the Pythagorean theorem, or we can assume this triangle is a right triangle for the sake of volume. But since we don't have the third side, we will proceed to calculate the area assuming we calculate the height from the 3-inch side as the base:
However, without loss of generality we can find an approximate area using a general triangle area formula (as an assumption of a right triangle can mislead): Let's say we assume a height proportion effectively gives: The height corresponding to base 3 can be estimated using proportional segments but we will revert to: Direct area approximation:
Using \( A = \frac{base \times height}{2} \) where the height derived via \( \sqrt{\text{(max)\text{/2 height triangle)}} \) could set: \[ \text{Base Area} \approx \frac{3 \times 5}{2} = 7.5 \text{ sq inches (assumed)} \]
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Volume of the pyramid using height of 8.41 inches: \[ V_{pyramid} = \frac{1}{3} \times 7.5 \times 8.41 \approx 19.83 \text{ cubic inches} \]
Volume of the Triangular Prism (V_prism):
The formula for the volume of a triangular prism is:
\[
V = \text{Base Area} \times \text{Length}
\]
Using the same triangle base area of 7.5 sq inches from above, and the length being 8.41 inches: \[ V_{prism} = 7.5 \times 8.41 \approx 63.07 \text{ cubic inches} \]
Comparison:
Now we compare the two volumes:
\[
V_{prism} \approx 63.07 \text{ cubic inches}, \quad V_{pyramid} \approx 19.83 \text{ cubic inches}
\]
Thus, the prism has a larger volume. To find out how much larger it is: \[ 63.07 - 19.83 = 43.24 \]
In terms of ratios: \[ \frac{ \text{V}{prism}}{ \text{V}{pyramid}} = \frac{63.07}{19.83} \approx 3.17 \text{ times larger} \]
Conclusion: The prism has a larger volume by more than three times, specifically in cubic inch approximation of differences given large scales.
Therefore, the correct response is: The prism has a larger volume by approximately three times.