Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 64 meters cubed.
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
StartFraction 64 Over 3 EndFraction meters cubed
32 meters cubed
StartFraction 128 Over 3 EndFraction meters cubed
64 meters cubed
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
StartFraction 64 Over 3 EndFraction meters cubed
32 meters cubed
StartFraction 128 Over 3 EndFraction meters cubed
64 meters cubed
Answers
GPT-4o mini
To find the volume of the sphere given that both the sphere and cylinder have the same radius \( r \) and height \( h \), and that the volume of the cylinder is 64 cubic meters, we can use the formulas for the volumes of the two shapes.
1. **Volume of a Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Given that \( V_{\text{cylinder}} = 64 \) m³, we have:
\[
\pi r^2 h = 64
\]
2. **Volume of a Sphere**:
The volume \( V \) of a sphere is given by:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Next, we need to express either \( h \) or \( r \) from the cylinder's volume formula, to find the sphere's volume.
3. **Expressing \( h \) in terms of \( r \)**:
From the cylinder's volume:
\[
h = \frac{64}{\pi r^2}
\]
4. **Using the radius \( r \)**:
With the sphere's volume formula \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \), we can substitute \( h \) if needed, but since we want the volume in terms of just \( r \), we can proceed to compare it with the cylinder's volume.
Since the sphere and the cylinder share the same \( r \) and \( h \), we can use \( h \) from the cylinder in the sphere's volume.
However, it turns out that this problem doesn't require \( h \). The critical part is using the relationship between the two volumes.
**Given the direct choices for the volume of the sphere**, let's calculate the sphere's volume based on the cylinder's volume through the ratio of volumes:
The volume of a sphere inscribed in a cylinder (having the same radius and height) can be expressed by knowing that the ratio of their volumes is:
\[
V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}}
\]
and since \( V_{\text{cylinder}} = 64 \text{ m}^3 \):
\[
V_{\text{sphere}} = \frac{2}{3} \times 64 = \frac{128}{3} \text{ m}^3
\]
Thus, the volume of the sphere is:
\[
\frac{128}{3} \text{ meters cubed}
\]
So, the correct answer is:
\(\frac{128}{3} \text{ meters cubed}\)
1. **Volume of a Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Given that \( V_{\text{cylinder}} = 64 \) m³, we have:
\[
\pi r^2 h = 64
\]
2. **Volume of a Sphere**:
The volume \( V \) of a sphere is given by:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Next, we need to express either \( h \) or \( r \) from the cylinder's volume formula, to find the sphere's volume.
3. **Expressing \( h \) in terms of \( r \)**:
From the cylinder's volume:
\[
h = \frac{64}{\pi r^2}
\]
4. **Using the radius \( r \)**:
With the sphere's volume formula \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \), we can substitute \( h \) if needed, but since we want the volume in terms of just \( r \), we can proceed to compare it with the cylinder's volume.
Since the sphere and the cylinder share the same \( r \) and \( h \), we can use \( h \) from the cylinder in the sphere's volume.
However, it turns out that this problem doesn't require \( h \). The critical part is using the relationship between the two volumes.
**Given the direct choices for the volume of the sphere**, let's calculate the sphere's volume based on the cylinder's volume through the ratio of volumes:
The volume of a sphere inscribed in a cylinder (having the same radius and height) can be expressed by knowing that the ratio of their volumes is:
\[
V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}}
\]
and since \( V_{\text{cylinder}} = 64 \text{ m}^3 \):
\[
V_{\text{sphere}} = \frac{2}{3} \times 64 = \frac{128}{3} \text{ m}^3
\]
Thus, the volume of the sphere is:
\[
\frac{128}{3} \text{ meters cubed}
\]
So, the correct answer is:
\(\frac{128}{3} \text{ meters cubed}\)