A large solid metallic cylinder whose radius and height are equal to each other is to be melted and 48 identical solid balls are to be recast from the liquid metal so formed. What is the ratio of the radius of a ball to the radius of the cylinder?

Options

A. `1:16`
B. `1:12`
C. `1:8`
D. `1:4 `

Answer

To solve the problem of finding the ratio of the radius of a ball to the radius of the cylinder, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Dimensions of the Cylinder**:
- Let the radius and height of the cylinder be \( r \). Thus, both the radius and height are equal.

2. **Calculate the Volume of the Cylinder**:
- The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
- Since the height \( h \) is equal to the radius \( r \), we can substitute \( h \) with \( r \):
\[ V = \pi r^2 \cdot r = \pi r^3 \]

3. **Calculate the Volume of One Ball**:
- The volume \( V \) of a sphere (ball) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- Here, \( R \) is the radius of one ball.

4. **Total Volume of 48 Balls**:
- Since there are 48 identical balls, the total volume of the balls is:
\[ V_{total} = 48 \times \frac{4}{3} \pi R^3 = 64 \pi R^3 \]

5. **Equate the Volumes**:
- Since the volume of the cylinder is equal to the total volume of the balls, we can set them equal to each other:
\[ \pi r^3 = 64 \pi R^3 \]

6. **Cancel Out \( \pi \)**:
- Dividing both sides by \( \pi \):
\[ r^3 = 64 R^3 \]

7. **Solve for the Ratio of Radii**:
- To find the ratio \( \frac{R}{r} \), we can rearrange the equation:
\[ \frac{R^3}{r^3} = \frac{1}{64} \]
- Taking the cube root of both sides gives:
\[ \frac{R}{r} = \frac{1}{4} \]

8. **Final Ratio**:
- Therefore, the ratio of the radius of a ball to the radius of the cylinder is:
\[ \frac{R}{r} = \frac{1}{4} \]

### Conclusion:
The ratio of the radius of a ball to the radius of the cylinder is \( \frac{1}{4} \).