Question
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A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3
. What is the volume of the whole object?(1 point)
Responses
18 m3
18 m cubed
36 m3
36 m cubed
There is not enough information to say.
There is not enough information to say.
24 m3
24 m cubed
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A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3
. What is the volume of the whole object?(1 point)
Responses
18 m3
18 m cubed
36 m3
36 m cubed
There is not enough information to say.
There is not enough information to say.
24 m3
24 m cubed
Skip to navigation
Answers
GPT-4o mini
To find the volume of the whole object, which consists of a hemisphere placed on top of an inverted cone, we need to calculate the volume of the hemisphere and add it to the volume of the cone.
1. **Volume of the cone**: It's given that the volume of the cone is \( 12 \, \text{m}^3 \).
2. **Volume of the hemisphere**: The formula for the volume of a hemisphere is given by:
\[
V = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere.
3. **Relation between cone and hemisphere**: We know that the height of the cone equals the diameter of the hemisphere. Therefore, if the radius of the hemisphere is \( r \), then:
- The diameter of the hemisphere = \( 2r \) (which is equal to the height of the cone).
4. **Volume of the cone formula**: The volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( h \) is the height of the cone. Here \( h = 2r \).
5. Plugging in \( h = 2r \) into the cone volume formula gives:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
6. We have established that \( V_{\text{cone}} = 12 \, \text{m}^3 \), which implies:
\[
\frac{2}{3} \pi r^3 = 12
\]
7. To find \( r^3 \), rearranging gives:
\[
r^3 = \frac{12 \cdot 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi}
\]
8. Now, substituting \( r^3 \) back into the hemisphere volume formula gives:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 \, \text{m}^3
\]
9. **Total volume of the object**:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 \, \text{m}^3 + 12 \, \text{m}^3 = 24 \, \text{m}^3
\]
So, the volume of the whole object is \( \text{24 m}^3 \).
**Answer: 24 m cubed**
1. **Volume of the cone**: It's given that the volume of the cone is \( 12 \, \text{m}^3 \).
2. **Volume of the hemisphere**: The formula for the volume of a hemisphere is given by:
\[
V = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere.
3. **Relation between cone and hemisphere**: We know that the height of the cone equals the diameter of the hemisphere. Therefore, if the radius of the hemisphere is \( r \), then:
- The diameter of the hemisphere = \( 2r \) (which is equal to the height of the cone).
4. **Volume of the cone formula**: The volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( h \) is the height of the cone. Here \( h = 2r \).
5. Plugging in \( h = 2r \) into the cone volume formula gives:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
6. We have established that \( V_{\text{cone}} = 12 \, \text{m}^3 \), which implies:
\[
\frac{2}{3} \pi r^3 = 12
\]
7. To find \( r^3 \), rearranging gives:
\[
r^3 = \frac{12 \cdot 3}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi}
\]
8. Now, substituting \( r^3 \) back into the hemisphere volume formula gives:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi \left(\frac{18}{\pi}\right) = \frac{2 \cdot 18}{3} = 12 \, \text{m}^3
\]
9. **Total volume of the object**:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 12 \, \text{m}^3 + 12 \, \text{m}^3 = 24 \, \text{m}^3
\]
So, the volume of the whole object is \( \text{24 m}^3 \).
**Answer: 24 m cubed**