Given, `P prop T ` on increasing the value of T by `1^(@)C`, the value of P increases by `0.4%`, then find initial value of T :-

Options

A. 25
B. 250
C. 350
D. 2500

Answer

To solve the problem step by step, we will follow the relationship given in the question and apply the concepts of proportionality and percentage change.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given that \( P \) is proportional to \( T \). This can be expressed mathematically as:
\[ P = k \cdot T \]
where \( k \) is a constant.

2. **Change in Pressure**:
We know that when the temperature \( T \) increases by \( 1^\circ C \), the pressure \( P \) increases by \( 0.4\% \). This means:
\[ \Delta P = 0.004 \cdot P \]
where \( \Delta P \) is the change in pressure.

3. **Change in Temperature**:
The change in temperature is given as:
\[ \Delta T = 1^\circ C \]

4. **Using the Formula for Relative Change**:
The relative change in pressure can also be expressed as:
\[ \frac{\Delta P}{P} = \frac{\Delta T}{T} \]

5. **Substituting the Known Values**:
We can substitute the values we have into the equation:
\[ \frac{0.004 \cdot P}{P} = \frac{1}{T} \]
Simplifying this gives:
\[ 0.004 = \frac{1}{T} \]

6. **Solving for \( T \)**:
Rearranging the equation to solve for \( T \):
\[ T = \frac{1}{0.004} \]
Calculating this gives:
\[ T = 250 \, \text{degrees Celsius} \]

### Final Answer:
The initial value of \( T \) is \( 250^\circ C \).

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