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

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 their volumes are equal and they have the same radius.

### Step-by-step Solution:

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

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

3. **Cancel Common Terms**:
We can simplify the equation by canceling \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = r^2 h \]
Next, we can divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ \frac{4}{3} r = h \]

4. **Express Height in Terms of Radius**:
From the equation \( \frac{4}{3} r = h \), we can express the height \( h \) in terms of the radius \( r \):
\[ h = \frac{4}{3} r \]

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

6. **Calculate the Ratio of Diameter to Height**:
Now we need to find the ratio of the diameter of the sphere to the height of the cylinder:
\[ \text{Ratio} = \frac{d}{h} = \frac{2r}{\frac{4}{3} r} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{2r \cdot 3}{4r} = \frac{6}{4} = \frac{3}{2} \]

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