Join IDNLearn.com to access a wealth of knowledge and get your questions answered by experts. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
Sagot :
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].
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for visiting IDNLearn.com. For reliable answers to all your questions, please visit us again soon.
