A cylindrical container with diameter of base 42 cm contains sufficient water to submerge a rectangular solid of iron with dimensions `22 cm xx 14 cm xx 10.5 cm.` Find the rise in level of the water when the solid is submerged.

Answer

To solve the problem of finding the rise in water level when a rectangular solid is submerged in a cylindrical container, we can follow these steps:

### Step 1: Understand the dimensions of the cylindrical container and the rectangular solid
- The diameter of the cylindrical container is given as 42 cm.
- The dimensions of the rectangular solid are given as:
- Length (l) = 22 cm
- Breadth (b) = 14 cm
- Height (h) = 10.5 cm

### Step 2: Calculate the radius of the cylindrical container
- The radius (r) of the cylindrical container can be calculated using the formula:
\[ r = \frac{\text{Diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm} \]

### Step 3: Calculate the volume of the rectangular solid
- The volume (V) of the rectangular solid can be calculated using the formula:
\[ V = l \times b \times h \]
- Substituting the values:
\[ V = 22 \text{ cm} \times 14 \text{ cm} \times 10.5 \text{ cm} \]
- Calculating this:
\[ V = 22 \times 14 = 308 \text{ cm}^2 \]
\[ V = 308 \times 10.5 = 3234 \text{ cm}^3 \]

### Step 4: Set up the equation for the rise in water level
- The volume of water displaced by the submerged rectangular solid is equal to the volume of the solid.
- The volume of water displaced can also be expressed in terms of the rise in water level (h) in the cylindrical container:
\[ \text{Volume of water displaced} = \pi r^2 h \]
- Therefore, we can set up the equation:
\[ \pi r^2 h = 3234 \text{ cm}^3 \]

### Step 5: Substitute the radius and solve for h
- Substitute \( r = 21 \text{ cm} \) into the equation:
\[ \pi (21)^2 h = 3234 \]
- Calculate \( \pi (21)^2 \):
\[ \pi (21)^2 = \pi \times 441 \approx \frac{22}{7} \times 441 \]
\[ \approx 1386 \text{ cm}^2 \]
- Now substitute this value back into the equation:
\[ 1386 h = 3234 \]
- Solve for h:
\[ h = \frac{3234}{1386} \approx 2.33 \text{ cm} \]

### Final Answer
The rise in the level of the water when the solid is submerged is approximately **2.33 cm**.

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