There is a leak proof cylinder of length 1m,made of a metal that has very low coefficient of expansion is floating vertically in water at `0^(@)C` such that its height above the water surface is 20cm. If the temperature of water is increased to `4^(@)C`, the height of the cylinder above the water surface becomes 21 cm. The density of water at `T = 4^(@)C` , relative to the density at `T = 0^(@)C` is approximately

Options

A. 1.03
B. 1.04
C. 1.26
D. 1.01

Answer

To solve the problem, we need to analyze the buoyancy of the cylinder in two different scenarios: when the water is at 0°C and when it is at 4°C.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The cylinder has a total length of 1 m (100 cm).
- At 0°C, the height of the cylinder above the water surface is 20 cm. This means that 80 cm of the cylinder is submerged in water.
- When the temperature of the water is increased to 4°C, the height above the water surface increases to 21 cm, meaning 79 cm of the cylinder is submerged.

2. **Defining Variables**:
- Let \( \rho_0 \) be the density of water at 0°C.
- Let \( \rho_4 \) be the density of water at 4°C.
- The area of the cylinder's cross-section is \( A \).

3. **Calculating the Buoyant Force**:
- The buoyant force acting on the cylinder is equal to the weight of the water displaced by the submerged part of the cylinder.
- At 0°C:
\[ \text{Buoyant Force} = \rho_0 \cdot A \cdot (80 \text{ cm}) \cdot g \]
- At 4°C:
\[ \text{Buoyant Force} = \rho_4 \cdot A \cdot (79 \text{ cm}) \cdot g \]

4. **Setting Up the Equation**:
- Since the weight of the cylinder remains constant, we can equate the buoyant forces at both temperatures:
\[ \rho_0 \cdot A \cdot 0.8 \cdot g = \rho_4 \cdot A \cdot 0.79 \cdot g \]
- The area \( A \) and gravitational acceleration \( g \) cancel out:
\[ \rho_0 \cdot 0.8 = \rho_4 \cdot 0.79 \]

5. **Finding the Ratio of Densities**:
- Rearranging the equation gives:
\[ \frac{\rho_4}{\rho_0} = \frac{0.8}{0.79} \]
- Calculating this ratio:
\[ \frac{\rho_4}{\rho_0} \approx 1.013 \]

6. **Final Answer**:
- The relative density of water at 4°C compared to that at 0°C is approximately 1.01.

### Conclusion:
The density of water at \( T = 4°C \) relative to the density at \( T = 0°C \) is approximately **1.01**.