To determine which shape has a larger volume and by how much, we need to calculate the volumes of both the triangular pyramid and the triangular prism.
Volume of the Triangular Pyramid
The formula for the volume of a triangular pyramid (also called a tetrahedron) is:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Base Area Calculation
To find the area of the triangular base, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
To find the height of the triangle, we can use Heron's formula since we have the lengths of all three sides. However, we only know two (3 inches and 5 inches) and need the third side to calculate the area. We can deduce that, if this is a right triangle, the third side could be calculated using the Pythagorean theorem. For simplicity, let's consider the area calculation as sufficient using:
\[ \text{Area of triangle with sides a = 3, b = 5 and angle between them = 90 degrees} \]
Assuming it’s a right triangle:
Let \(b\) = 3 inches, \(h\) = 5 inches
\[ \text{Area} = \frac{1}{2} \times 3 \times 5 = 7.5 \text{ square inches} \]
Volume of Pyramid:
Given the height (h) of the pyramid is 8.41 inches:
\[ V_{pyramid} = \frac{1}{3} \times 7.5 \times 8.41 \] \[ V_{pyramid} = \frac{1}{3} \times 63.075 = 21.025 \text{ cubic inches} \]
Volume of the Triangular Prism
The volume of a triangular prism is given by the formula:
\[ V = \text{Base Area} \times \text{Height} \]
Using the same base area (7.5 square inches):
The height of the prism is given as 8.41 inches.
\[ V_{prism} = 7.5 \times 8.41 = 63.075 \text{ cubic inches} \]
Comparing Volumes
- Volume of the pyramid = 21.025 cubic inches
- Volume of the prism = 63.075 cubic inches
Now, we find the difference:
\[ \text{Difference} = V_{prism} - V_{pyramid} = 63.075 - 21.025 = 42.05 \text{ cubic inches} \]
Conclusion
The triangular prism has a larger volume than the triangular pyramid. To determine how many times larger:
\[ \text{Ratio} = \frac{V_{prism}}{V_{pyramid}} = \frac{63.075}{21.025} \approx 3 \]
Hence, the final conclusion is:
The prism has a larger volume by three times.