The volumes of a sphere and a right circular cylinder having the same radius are equal. The ratio of the diameter of the sphere to the height of the cylinder is

Options

A. `1:2`
B. `2:1`
C. `2:3`
D. `3:2 `

Answer

To solve the problem, we need to find the ratio of the diameter of a sphere to the height of a right circular cylinder, given that both have the same radius and their volumes are equal.

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- The volume \( V \) of a right circular cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Set the Volumes Equal**:
Since the volumes of the sphere and cylinder are equal, we can set the two volume equations equal to each other:
\[ \frac{4}{3} \pi r^3 = \pi r^2 h \]

3. **Cancel Common Terms**:
We can cancel \( \pi \) from both sides of the equation:
\[ \frac{4}{3} r^3 = r^2 h \]

4. **Isolate \( h \)**:
To isolate \( h \), divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4}{3} r \]

5. **Find the Diameter of the Sphere**:
The diameter \( d \) of the sphere is twice the radius:
\[ d = 2r \]

6. **Set Up the Ratio**:
Now, we need to find the ratio of the diameter of the sphere to the height of the cylinder:
\[ \text{Ratio} = \frac{d}{h} = \frac{2r}{\frac{4}{3} r} \]

7. **Simplify the Ratio**:
The \( r \) terms cancel out:
\[ \text{Ratio} = \frac{2}{\frac{4}{3}} = 2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \]

### Final Answer:
The ratio of the diameter of the sphere to the height of the cylinder is:
\[ \frac{3}{2} \]