Answer :
To find the correct formula for the volume of the cone, let's start by recalling the volume formula for both a cone and a pyramid.
1. Volume of a Cone:
The volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height.
2. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} \][/tex]
Now, let us consider the given information: the volume of the cone is [tex]\(\frac{\pi}{4}\)[/tex] times that of the volume calculation involving the shape it fits inside, which in this case is a pyramid. We need to determine which of the given expressions correctly represents this relationship.
3. Conversion:
The expressions to evaluate this relationship are:
- [tex]\(\frac{\pi}{4}(2 r^2 h)\)[/tex]
- [tex]\(\frac{\pi}{4}(4 r^2 h)\)[/tex]
- [tex]\(\frac{\pi}{4}\left(\frac{r^2 h}{3}\right)\)[/tex]
- [tex]\(\frac{\pi}{4}\left(\frac{4 r^2 h}{3}\right)\)[/tex]
First, calculate the volume of the cone directly using [tex]\(\frac{1}{3} \pi r^2 h\)[/tex]:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
Next, consider each expression provided:
a. For the first expression [tex]\(\frac{\pi}{4}(2 r^2 h)\)[/tex]:
This would simplify to:
[tex]\[ V_1 = \frac{\pi}{4} \times 2 r^2 h = \frac{\pi}{2} r^2 h \][/tex]
This is not consistent with the volume of a cone.
b. For the second expression [tex]\(\frac{\pi}{4}(4 r^2 h)\)[/tex]:
[tex]\[ V_2 = \frac{\pi}{4} \times 4 r^2 h = \pi r^2 h \][/tex]
This is still not the formula for the volume of the cone.
c. For the third expression [tex]\(\frac{\pi}{4} \left(\frac{r^2 h}{3}\right)\)[/tex]:
[tex]\[ V_3 = \frac{\pi}{4} \times \frac{r^2 h}{3} = \frac{\pi r^2 h}{12} \][/tex]
This still doesn't match the cone volume formula.
d. For the fourth expression [tex]\(\frac{\pi}{4} \left(\frac{4 r^2 h}{3}\right)\)[/tex]:
[tex]\[ V_4 = \frac{\pi}{4} \times \frac{4 r^2 h}{3} = \frac{\pi}{3} r^2 h \][/tex]
This matches the cone's volume.
Thus, the correct expression representing the volume of the cone that is [tex]\(\frac{\pi}{4}\)[/tex] times the volume of the pyramid it fits inside is:
[tex]\[ \boxed{\frac{\pi}{4} \left(\frac{4 r^2 h}{3}\right)} \][/tex]
This ensures that the volume relationship aligns with the cone's volume [tex]\(\frac{1}{3} \pi r^2 h\)[/tex].
1. Volume of a Cone:
The volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height.
2. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} \][/tex]
Now, let us consider the given information: the volume of the cone is [tex]\(\frac{\pi}{4}\)[/tex] times that of the volume calculation involving the shape it fits inside, which in this case is a pyramid. We need to determine which of the given expressions correctly represents this relationship.
3. Conversion:
The expressions to evaluate this relationship are:
- [tex]\(\frac{\pi}{4}(2 r^2 h)\)[/tex]
- [tex]\(\frac{\pi}{4}(4 r^2 h)\)[/tex]
- [tex]\(\frac{\pi}{4}\left(\frac{r^2 h}{3}\right)\)[/tex]
- [tex]\(\frac{\pi}{4}\left(\frac{4 r^2 h}{3}\right)\)[/tex]
First, calculate the volume of the cone directly using [tex]\(\frac{1}{3} \pi r^2 h\)[/tex]:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
Next, consider each expression provided:
a. For the first expression [tex]\(\frac{\pi}{4}(2 r^2 h)\)[/tex]:
This would simplify to:
[tex]\[ V_1 = \frac{\pi}{4} \times 2 r^2 h = \frac{\pi}{2} r^2 h \][/tex]
This is not consistent with the volume of a cone.
b. For the second expression [tex]\(\frac{\pi}{4}(4 r^2 h)\)[/tex]:
[tex]\[ V_2 = \frac{\pi}{4} \times 4 r^2 h = \pi r^2 h \][/tex]
This is still not the formula for the volume of the cone.
c. For the third expression [tex]\(\frac{\pi}{4} \left(\frac{r^2 h}{3}\right)\)[/tex]:
[tex]\[ V_3 = \frac{\pi}{4} \times \frac{r^2 h}{3} = \frac{\pi r^2 h}{12} \][/tex]
This still doesn't match the cone volume formula.
d. For the fourth expression [tex]\(\frac{\pi}{4} \left(\frac{4 r^2 h}{3}\right)\)[/tex]:
[tex]\[ V_4 = \frac{\pi}{4} \times \frac{4 r^2 h}{3} = \frac{\pi}{3} r^2 h \][/tex]
This matches the cone's volume.
Thus, the correct expression representing the volume of the cone that is [tex]\(\frac{\pi}{4}\)[/tex] times the volume of the pyramid it fits inside is:
[tex]\[ \boxed{\frac{\pi}{4} \left(\frac{4 r^2 h}{3}\right)} \][/tex]
This ensures that the volume relationship aligns with the cone's volume [tex]\(\frac{1}{3} \pi r^2 h\)[/tex].
