At `4^(@)C, 0.98` of the volume of a body is immersed in water. The temperature at which the entire body gets immersed in water is (neglect the expansion of the body ) (`gamma_(w) = 3.3 xx10 ^(-4)K^(-1)`):-

Options

A. `40.8^(@)C`
B. `64.6^(@)C`
C. `60.6^(@)C`
D. `58.8^(@)C`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript:

### Step 1: Understand the Problem
At 4°C, 0.98 of the volume of a body is immersed in water. We need to find the temperature at which the entire body gets immersed in water, neglecting the expansion of the body. The volumetric expansion coefficient of water is given as \( \gamma_w = 3.3 \times 10^{-4} \, K^{-1} \).

### Step 2: Determine the Required Volume Increase
Since 0.98 of the volume is currently immersed, we need to find the increase in volume needed for the entire body to be submerged. The fraction of the volume that needs to be submerged is:
\[ 1 - 0.98 = 0.02 \]

### Step 3: Relate Volume Change to Temperature Change
The change in volume due to temperature change can be expressed using the formula for volumetric expansion:
\[ \Delta V = V \cdot \gamma \cdot \Delta T \]
Where:
- \( \Delta V \) is the change in volume,
- \( V \) is the original volume (which we can consider as 1 unit for simplicity),
- \( \gamma \) is the volumetric expansion coefficient of water,
- \( \Delta T \) is the change in temperature.

### Step 4: Set Up the Equation
We can set up the equation based on the volume change required:
\[ 0.02 = 0.98 \cdot \gamma_w \cdot \Delta T \]
Substituting the value of \( \gamma_w \):
\[ 0.02 = 0.98 \cdot (3.3 \times 10^{-4}) \cdot \Delta T \]

### Step 5: Solve for \( \Delta T \)
Rearranging the equation to solve for \( \Delta T \):
\[ \Delta T = \frac{0.02}{0.98 \cdot (3.3 \times 10^{-4})} \]
Calculating \( \Delta T \):
\[ \Delta T \approx \frac{0.02}{0.0003234} \approx 61.8 \, K \]

### Step 6: Calculate Final Temperature
Since the initial temperature is 4°C, the final temperature \( T_f \) is:
\[ T_f = 4 + \Delta T \approx 4 + 61.8 \approx 65.8 \, °C \]

### Step 7: Final Answer
The closest value to our calculated temperature is approximately 64.6°C. Thus, the answer is:
\[ \text{The temperature at which the entire body gets immersed in water is approximately } 64.6°C. \]