A solid sphere with radius 6 cm was totally submerged in water kept in a cylindrical vessel. The water level in the vessel rose by 4.5 cm. Find the radius of the cylindrical vessel.

Answer

To find the radius of the cylindrical vessel when a solid sphere of radius 6 cm is submerged in it, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a solid sphere submerged in a cylindrical vessel, and we know the radius of the sphere (6 cm) and the height the water level rises (4.5 cm).

2. **Volume of the Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, \( r = 6 \) cm.

3. **Calculate the Volume of the Sphere**:
Substitute the radius into the volume formula:
\[ V = \frac{4}{3} \pi (6)^3 \]
Calculate \( (6)^3 = 216 \):
\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \, \text{cm}^3 \]

4. **Volume of Water Displaced**:
The volume of water displaced by the submerged sphere is equal to the volume of the sphere, which is \( 288 \pi \, \text{cm}^3 \).

5. **Volume of Water in the Cylindrical Vessel**:
The volume \( V \) of water in the cylindrical vessel can also be expressed as:
\[ V = \pi R^2 h \]
where \( R \) is the radius of the cylindrical vessel and \( h \) is the height the water level rose (4.5 cm).

6. **Set the Volumes Equal**:
Since the volume of water displaced is equal to the volume of the sphere, we set the two volumes equal:
\[ \pi R^2 (4.5) = 288 \pi \]

7. **Cancel \( \pi \)**:
We can cancel \( \pi \) from both sides:
\[ R^2 (4.5) = 288 \]

8. **Solve for \( R^2 \)**:
Divide both sides by 4.5:
\[ R^2 = \frac{288}{4.5} \]
Calculate \( \frac{288}{4.5} = 64 \):
\[ R^2 = 64 \]

9. **Find \( R \)**:
Take the square root of both sides:
\[ R = \sqrt{64} = 8 \, \text{cm} \]

### Final Answer:
The radius of the cylindrical vessel is **8 cm**.