A cylindrical cirstern whose diameter is 21 cm is party filled with water. If a rectangular block of iron 14 cm in length , 10.5 cm in breadth and 11 cm in thickness is wholly immersed in water, by how many centimetres will the water level rise ?

Answer

To solve the problem of how much the water level will rise in a cylindrical cistern when a rectangular block is immersed, we will follow these steps:

### Step 1: Calculate the Volume of the Rectangular Block
The volume \( V \) of a rectangular block can be calculated using the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given:
- Length = 14 cm
- Breadth = 10.5 cm
- Height (thickness) = 11 cm

Substituting the values:
\[ V = 14 \, \text{cm} \times 10.5 \, \text{cm} \times 11 \, \text{cm} \]
Calculating this:
\[ V = 14 \times 10.5 = 147 \, \text{cm}^2 \]
\[ V = 147 \times 11 = 1617 \, \text{cm}^3 \]

### Step 2: Calculate the Radius of the Cylindrical Cistern
The diameter of the cylindrical cistern is given as 21 cm. Therefore, the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{21 \, \text{cm}}{2} = 10.5 \, \text{cm} \]

### Step 3: Set Up the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where \( h \) is the height of the water level rise we want to find.

### Step 4: Equate the Volumes
Since the volume of water displaced by the rectangular block is equal to the volume of water that rises in the cylindrical cistern:
\[ \pi r^2 h = 1617 \, \text{cm}^3 \]
Substituting \( r = 10.5 \, \text{cm} \):
\[ \pi (10.5)^2 h = 1617 \]
Calculating \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
Thus, the equation becomes:
\[ \pi \times 110.25 \times h = 1617 \]

### Step 5: Solve for \( h \)
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} \times 110.25 \times h = 1617 \]
To isolate \( h \):
\[ h = \frac{1617 \times 7}{22 \times 110.25} \]

Calculating the right side:
\[ h = \frac{11319}{2425.5} \]
Calculating \( 22 \times 110.25 = 2425.5 \):
\[ h = \frac{11319}{2425.5} \approx 4.67 \, \text{cm} \]

### Final Answer
The water level will rise by approximately **4.67 cm**.

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