Question
Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere?(1 point)
The volume of a cone is 27 cm3
. What is the volume of a cylinder that shares the same radius and height as the cone?(1 point)
The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
Use the image to answer the question.
A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3
, what is the volume of the whole object?
The volume of a cone is 27 cm3
. What is the volume of a cylinder that shares the same radius and height as the cone?(1 point)
The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
Use the image to answer the question.
A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3
, what is the volume of the whole object?
Answers
Answered by
GPT-4o mini
Let's break down each question one by one.
### Question 1
**Ratio of volume of cone to volume of sphere:**
1. The volume of a cone \( V_c \) is given by the formula:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. The volume of a sphere \( V_s \) is given by the formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
3. Given that the height of the cone \( h = 2r \), substituting this into the volume formula for the cone gives us:
\[
V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
4. Now we can write the ratio of the volume of the cone to the volume of the sphere:
\[
\text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} = \frac{2}{4} = \frac{1}{2}
\]
### Question 2
**Volume of the cylinder with the same radius and height as the cone:**
1. Given the volume of the cone is \( 27 \, \text{cm}^3 \), and using the formula for the volume of a cone:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
we can let \( h = 2r \), so:
\[
27 = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Solving for \( r^3 \):
\[
27 = \frac{2}{3} \pi r^3 \implies r^3 = \frac{27 \cdot 3}{2\pi} = \frac{81}{2\pi}
\]
The height \( h \) would then be \( 2r \), and using the radius:
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]
Since we have \( r^3 = \frac{81}{2\pi} \):
\[
V_{\text{cylinder}} = 2\pi \cdot \frac{81}{2\pi} = 81 \, \text{cm}^3
\]
### Question 3
**Volume of the cone when the height is equal to the sphere's diameter:**
1. The volume of a sphere is given as \( 72 \, m^3 \).
2. Using the volume formula for a sphere:
\[
V_s = \frac{4}{3} \pi r^3 \implies 72 = \frac{4}{3} \pi r^3 \implies r^3 = \frac{72 \cdot 3}{4\pi} = \frac{54}{\pi}
\]
3. The diameter of the sphere is \( 2r \), thus the height of the cone will be \( h = 2r \).
4. Now substituting \( h \) in the volume formula for a cone (\( V_c \)):
\[
V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Substituting \( r^3 = \frac{54}{\pi} \):
\[
V_c = \frac{2}{3} \pi \cdot \left(\frac{54}{\pi}\right) = \frac{2 \cdot 54}{3} = 36 \, m^3
\]
### Question 4
**Volume of the whole object (cone + cylinder):**
1. Given the volume of the cone is \( 50 \, cm^3 \).
2. The cylinder also shares the same radius and height as the cone, which we can find using:
\[
V_{\text{cylinder}} = 2\pi r^3
\]
where \( V_c = \frac{1}{3} \pi r^2 h \) and \( h \) is equal to the cone's height.
3. Since we only know the cone's volume at this point, to find the cylinder's volume, we’d calculate based on the same proportions as previously or from initial values.
4. The total volume is:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 + V_{\text{cylinder}}
\]
(*Note: To calculate the volume of the cylinder, we need the height which is \( h = 2r \); we'll need radius from cone info above \( r^3 \) relationship, inferred during earlier calculation.*)
To summarize the final results:
1. Ratio of cone to sphere = \( \frac{1}{2} \)
2. Volume of cylinder = \( 81 \, cm^3 \)
3. Volume of the cone = \( 36 \, m^3 \)
4. Total Volume = \( 50 + V_{\text{cylinder}} \)
(need cylinder volume estimation based on data to finalize).
### Question 1
**Ratio of volume of cone to volume of sphere:**
1. The volume of a cone \( V_c \) is given by the formula:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. The volume of a sphere \( V_s \) is given by the formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
3. Given that the height of the cone \( h = 2r \), substituting this into the volume formula for the cone gives us:
\[
V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
4. Now we can write the ratio of the volume of the cone to the volume of the sphere:
\[
\text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} = \frac{2}{4} = \frac{1}{2}
\]
### Question 2
**Volume of the cylinder with the same radius and height as the cone:**
1. Given the volume of the cone is \( 27 \, \text{cm}^3 \), and using the formula for the volume of a cone:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
we can let \( h = 2r \), so:
\[
27 = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Solving for \( r^3 \):
\[
27 = \frac{2}{3} \pi r^3 \implies r^3 = \frac{27 \cdot 3}{2\pi} = \frac{81}{2\pi}
\]
The height \( h \) would then be \( 2r \), and using the radius:
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]
Since we have \( r^3 = \frac{81}{2\pi} \):
\[
V_{\text{cylinder}} = 2\pi \cdot \frac{81}{2\pi} = 81 \, \text{cm}^3
\]
### Question 3
**Volume of the cone when the height is equal to the sphere's diameter:**
1. The volume of a sphere is given as \( 72 \, m^3 \).
2. Using the volume formula for a sphere:
\[
V_s = \frac{4}{3} \pi r^3 \implies 72 = \frac{4}{3} \pi r^3 \implies r^3 = \frac{72 \cdot 3}{4\pi} = \frac{54}{\pi}
\]
3. The diameter of the sphere is \( 2r \), thus the height of the cone will be \( h = 2r \).
4. Now substituting \( h \) in the volume formula for a cone (\( V_c \)):
\[
V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Substituting \( r^3 = \frac{54}{\pi} \):
\[
V_c = \frac{2}{3} \pi \cdot \left(\frac{54}{\pi}\right) = \frac{2 \cdot 54}{3} = 36 \, m^3
\]
### Question 4
**Volume of the whole object (cone + cylinder):**
1. Given the volume of the cone is \( 50 \, cm^3 \).
2. The cylinder also shares the same radius and height as the cone, which we can find using:
\[
V_{\text{cylinder}} = 2\pi r^3
\]
where \( V_c = \frac{1}{3} \pi r^2 h \) and \( h \) is equal to the cone's height.
3. Since we only know the cone's volume at this point, to find the cylinder's volume, we’d calculate based on the same proportions as previously or from initial values.
4. The total volume is:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 + V_{\text{cylinder}}
\]
(*Note: To calculate the volume of the cylinder, we need the height which is \( h = 2r \); we'll need radius from cone info above \( r^3 \) relationship, inferred during earlier calculation.*)
To summarize the final results:
1. Ratio of cone to sphere = \( \frac{1}{2} \)
2. Volume of cylinder = \( 81 \, cm^3 \)
3. Volume of the cone = \( 36 \, m^3 \)
4. Total Volume = \( 50 + V_{\text{cylinder}} \)
(need cylinder volume estimation based on data to finalize).