A metal cube of side 11 cm is completely submerged in water contained in a cylindrical vessel with diameter 28 cm. Find the rise in the level of water.

Answer

To find the rise in the level of water when a metal cube is submerged in a cylindrical vessel, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of the Cube:**
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
Here, the side of the cube is 11 cm.
\[ V = 11^3 = 11 \times 11 \times 11 = 1331 \, \text{cm}^3 \]

2. **Determine the Radius of the Cylinder:**
The diameter of the cylindrical vessel is given as 28 cm. The radius \( R \) is half of the diameter:
\[ R = \frac{\text{diameter}}{2} = \frac{28}{2} = 14 \, \text{cm} \]

3. **Set Up the Volume of Water Displaced:**
When the cube is submerged, it displaces a volume of water equal to its own volume. The volume of water displaced can also be expressed in terms of the rise in water level \( h \) in the cylindrical vessel:
\[ \text{Volume of water displaced} = \pi R^2 h \]
Substituting the radius:
\[ \text{Volume of water displaced} = \pi (14^2) h = \pi (196) h \]

4. **Equate the Two Volumes:**
Since the volume of water displaced is equal to the volume of the cube, we can set up the equation:
\[ 196\pi h = 1331 \]

5. **Solve for \( h \):**
Rearranging the equation to find \( h \):
\[ h = \frac{1331}{196\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ h = \frac{1331}{196 \times \frac{22}{7}} = \frac{1331 \times 7}{196 \times 22} \]
Simplifying:
\[ h = \frac{9317}{4312} \]
Now, we can simplify this fraction:
\[ h = \frac{121}{56} \, \text{cm} \]

6. **Convert to Mixed Fraction:**
To convert \( \frac{121}{56} \) to a mixed fraction:
\[ 121 \div 56 = 2 \quad \text{(remainder 9)} \]
Thus, \( h = 2 \frac{9}{56} \, \text{cm} \).

### Final Answer:
The rise in the level of water is \( 2 \frac{9}{56} \, \text{cm} \).