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: The formula for the volume \( V \) of a pyramid is given by:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we need to calculate the base area of the triangular base. Given the sides of the triangular base as 3 inches and 5 inches, we can find the area using Heron's formula.
-
Calculate the semi-perimeter \( s \):
- Since we don't have the third side, let's denote it as \( c \). We'll use this later.
-
To calculate the area of the triangle, we need to know the third side or use a right triangle assumption. If side \( c \) is known, then:
\[ s = \frac{3 + 5 + c}{2} \]
For simplicity, if we assume it’s a right triangle (with sides 3 and 5), the area is calculated as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6 \text{ square inches} \quad \text{(Where c=4 is the height)} \]
Now, the volume of the pyramid can be calculated:
\[ V_{\text{pyramid}} = \frac{1}{3} \times 6 \times 8.41 = \frac{50.46}{3} \approx 16.82 \text{ cubic inches} \]
Volume of the triangular prism: The formula for the volume \( V \) of a prism is given by:
\[ V = \text{Base Area} \times \text{Height} \]
Using the base area from above:
\[ V_{\text{prism}} = 6 \times 8.41 = 50.46 \text{ cubic inches} \]
Comparing the volumes:
- Volume of the pyramid: \( \approx 16.82 \) cubic inches
- Volume of the prism: \( 50.46 \) cubic inches
To find how much larger the prism's volume is compared to the pyramid's:
\[ 50.46 - 16.82 = 33.64 \text{ cubic inches} \]
Since \( 50.46 \) is about three times \( 16.82 \), therefore the correct choice is:
The prism has a larger volume by three times.