Answer :
To determine whether a prism has two times the volume of a pyramid with the same base and altitude, we need to analyze the volumes of both the prism and the pyramid.
1. Volume of a Prism: The formula for the volume of a prism is given by:
[tex]\[ V_{\text{prism}} = B \times h \][/tex]
where [tex]\( B \)[/tex] is the area of the base and [tex]\( h \)[/tex] is the height (altitude).
2. Volume of a Pyramid: The formula for the volume of a pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h \][/tex]
where [tex]\( B \)[/tex] is the area of the base and [tex]\( h \)[/tex] is the height (altitude).
3. Comparison: According to the question, the volume of the prism is compared to two times the volume of a pyramid with the same base and height.
Let's calculate a specific example to better understand this:
- Base area, [tex]\( B = 10 \)[/tex] square units.
- Height, [tex]\( h = 15 \)[/tex] units.
Using these values:
- Volume of the prism:
[tex]\[ V_{\text{prism}} = B \times h = 10 \times 15 = 150 \text{ cubic units} \][/tex]
- Volume of the pyramid:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h = \frac{1}{3} \times 10 \times 15 = 50 \text{ cubic units} \][/tex]
Now, let's see if the volume of the prism is twice the volume of the pyramid:
[tex]\[ 2 \times V_{\text{pyramid}} = 2 \times 50 = 100 \text{ cubic units} \][/tex]
4. Conclusion:
- The volume of the prism is [tex]\( 150 \)[/tex] cubic units.
- Twice the volume of the pyramid is [tex]\( 100 \)[/tex] cubic units.
Since [tex]\( 150 \neq 100 \)[/tex], the statement that "a prism has two times the volume of a pyramid with the same base and altitude" is False.
1. Volume of a Prism: The formula for the volume of a prism is given by:
[tex]\[ V_{\text{prism}} = B \times h \][/tex]
where [tex]\( B \)[/tex] is the area of the base and [tex]\( h \)[/tex] is the height (altitude).
2. Volume of a Pyramid: The formula for the volume of a pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h \][/tex]
where [tex]\( B \)[/tex] is the area of the base and [tex]\( h \)[/tex] is the height (altitude).
3. Comparison: According to the question, the volume of the prism is compared to two times the volume of a pyramid with the same base and height.
Let's calculate a specific example to better understand this:
- Base area, [tex]\( B = 10 \)[/tex] square units.
- Height, [tex]\( h = 15 \)[/tex] units.
Using these values:
- Volume of the prism:
[tex]\[ V_{\text{prism}} = B \times h = 10 \times 15 = 150 \text{ cubic units} \][/tex]
- Volume of the pyramid:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h = \frac{1}{3} \times 10 \times 15 = 50 \text{ cubic units} \][/tex]
Now, let's see if the volume of the prism is twice the volume of the pyramid:
[tex]\[ 2 \times V_{\text{pyramid}} = 2 \times 50 = 100 \text{ cubic units} \][/tex]
4. Conclusion:
- The volume of the prism is [tex]\( 150 \)[/tex] cubic units.
- Twice the volume of the pyramid is [tex]\( 100 \)[/tex] cubic units.
Since [tex]\( 150 \neq 100 \)[/tex], the statement that "a prism has two times the volume of a pyramid with the same base and altitude" is False.
