To find the volumes of the prism and the pyramid, we first need to calculate the area of the triangular base. The sides of the triangle are given as 16 meters, 14 meters, and 8 meters, and we have the height from the side measuring 16 meters to the opposite vertex, which is 7 meters.
Step 1: Calculate the area of the triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can use the side measuring 16 meters as the base and the height of 7 meters:
\[ A = \frac{1}{2} \times 16 , \text{m} \times 7 , \text{m} = \frac{1}{2} \times 112 , \text{m}^2 = 56 , \text{m}^2 \]
Step 2: Calculate the volume of the prism
The volume \( V \) of a prism is given by the formula:
\[ V = \text{Base Area} \times \text{Height} \]
Given the area of the base is \( 56 , \text{m}^2 \) and the height of the prism is 15 meters:
\[ V_{\text{prism}} = 56 , \text{m}^2 \times 15 , \text{m} = 840 , \text{m}^3 \]
Step 3: Calculate the volume of the pyramid
The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Using the same base area of \( 56 , \text{m}^2 \) and the height of the pyramid as 15 meters:
\[ V_{\text{pyramid}} = \frac{1}{3} \times 56 , \text{m}^2 \times 15 , \text{m} = \frac{1}{3} \times 840 , \text{m}^3 = 280 , \text{m}^3 \]
Final Answers
- The volume of the prism is 840 m³.
- The volume of the pyramid is 280 m³.