Question
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem
The volume of a cylinder is 36 cm^3 What is the volume of a sphere if its radius is the same as the cylinders and the height of the cylinder is equal to the spheres diameter?
54 cm^3
18 cm^3
12 cm^3
24 cm^3
The volume of a cylinder is 36 cm^3 What is the volume of a sphere if its radius is the same as the cylinders and the height of the cylinder is equal to the spheres diameter?
54 cm^3
18 cm^3
12 cm^3
24 cm^3
Answers
GPT-4o mini
To solve the problem, we'll start with the formulas for the volumes of a cylinder and a sphere.
1. **Volume of a Cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of a Sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
According to the given information:
- The volume of the cylinder is \( 36 \, \text{cm}^3 \).
- The height of the cylinder equals the diameter of the sphere, which can be expressed as \( h = 2r \).
Substituting \( h \) into the cylinder's volume equation:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]
We know that:
\[
2\pi r^3 = 36
\]
Now, we solve for \( r^3 \):
\[
\pi r^3 = \frac{36}{2} = 18
\]
\[
r^3 = \frac{18}{\pi}
\]
Next, we can find the volume of the sphere using the radius \( r \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 \) from our previous calculation:
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right)
\]
Now, simplify this:
\[
V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \, \text{cm}^3
\]
Thus, the volume of the sphere is **24 cm³**.
So, the correct answer is:
\[
\boxed{24 \, \text{cm}^3}
\]
1. **Volume of a Cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of a Sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
According to the given information:
- The volume of the cylinder is \( 36 \, \text{cm}^3 \).
- The height of the cylinder equals the diameter of the sphere, which can be expressed as \( h = 2r \).
Substituting \( h \) into the cylinder's volume equation:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]
We know that:
\[
2\pi r^3 = 36
\]
Now, we solve for \( r^3 \):
\[
\pi r^3 = \frac{36}{2} = 18
\]
\[
r^3 = \frac{18}{\pi}
\]
Next, we can find the volume of the sphere using the radius \( r \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 \) from our previous calculation:
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right)
\]
Now, simplify this:
\[
V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \, \text{cm}^3
\]
Thus, the volume of the sphere is **24 cm³**.
So, the correct answer is:
\[
\boxed{24 \, \text{cm}^3}
\]
just wanna make sure are you correct-?
GPT-4o mini
Let's go through the calculations step-by-step to ensure everything is correct.
1. **Volume of Cylinder**:
- The volume of the cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
- We know \( V_{\text{cylinder}} = 36 \, \text{cm}^3 \).
2. **Height of the Cylinder**:
- The height \( h \) of the cylinder is equal to the diameter of the sphere, which is \( h = 2r \).
3. **Substituting Height into the Cylinder Volume Formula**:
- Substitute \( h \) in the volume equation:
\[
36 = \pi r^2 (2r)
\]
\[
36 = 2\pi r^3
\]
4. **Solving for \( r^3 \)**:
- Rearranging gives:
\[
\pi r^3 = \frac{36}{2} = 18
\]
\[
r^3 = \frac{18}{\pi}
\]
5. **Volume of the Sphere**:
- Now, use the radius \( r \) to find the volume of the sphere:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
- Substitute \( r^3 \) into the sphere's volume:
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right)
\]
- The \( \pi \) cancels out:
\[
V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \, \text{cm}^3
\]
Therefore, upon reviewing, every step has been confirmed to be correct, and the final volume of the sphere is indeed \( 24 \, \text{cm}^3 \).
The answer is:
\[
\boxed{24 \, \text{cm}^3}
\]
1. **Volume of Cylinder**:
- The volume of the cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
- We know \( V_{\text{cylinder}} = 36 \, \text{cm}^3 \).
2. **Height of the Cylinder**:
- The height \( h \) of the cylinder is equal to the diameter of the sphere, which is \( h = 2r \).
3. **Substituting Height into the Cylinder Volume Formula**:
- Substitute \( h \) in the volume equation:
\[
36 = \pi r^2 (2r)
\]
\[
36 = 2\pi r^3
\]
4. **Solving for \( r^3 \)**:
- Rearranging gives:
\[
\pi r^3 = \frac{36}{2} = 18
\]
\[
r^3 = \frac{18}{\pi}
\]
5. **Volume of the Sphere**:
- Now, use the radius \( r \) to find the volume of the sphere:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
- Substitute \( r^3 \) into the sphere's volume:
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right)
\]
- The \( \pi \) cancels out:
\[
V_{\text{sphere}} = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \, \text{cm}^3
\]
Therefore, upon reviewing, every step has been confirmed to be correct, and the final volume of the sphere is indeed \( 24 \, \text{cm}^3 \).
The answer is:
\[
\boxed{24 \, \text{cm}^3}
\]