A pendulum clock is 5 sec. Slow at a temperature `30^(@)C` and `10` sec. fast at a temperature of `15^(@)C`, At what temperature does it give the correct time-

Options

A. `18^(@)C`
B. `20^(@)C`
C. `25^(@)C`
D. None of these

Explanation

To solve the problem of determining the temperature at which a pendulum clock gives the correct time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The pendulum clock is 5 seconds slow at 30°C.
- It is 10 seconds fast at 15°C.
- We need to find the temperature \( \theta \) at which the clock is neither slow nor fast (i.e., it gives the correct time).

2. **Setting Up the Equations**:
- We can use the formula for the time error of the pendulum clock based on temperature changes:
\[ \Delta t = \frac{1}{2} \alpha \Delta \theta \cdot t \]
- Here, \( \Delta t \) is the time error, \( \alpha \) is the coefficient of linear expansion, \( \Delta \theta \) is the change in temperature, and \( t \) is the time period.

3. **For the Slow Condition** (5 seconds slow at 30°C):
- The equation becomes:
\[ 5 = \frac{1}{2} \alpha (30 - \theta) \cdot t \]

4. **For the Fast Condition** (10 seconds fast at 15°C):
- The equation becomes:
\[ 10 = \frac{1}{2} \alpha (\theta - 15) \cdot t \]

5. **Eliminating Common Factors**:
- From both equations, we can express \( \alpha t \) in terms of \( \Delta t \):
- From the slow condition:
\[ \alpha t = \frac{10}{30 - \theta} \]
- From the fast condition:
\[ \alpha t = \frac{20}{\theta - 15} \]

6. **Setting the Equations Equal**:
- Since both expressions equal \( \alpha t \), we can set them equal to each other:
\[ \frac{10}{30 - \theta} = \frac{20}{\theta - 15} \]

7. **Cross-Multiplying**:
- Cross-multiply to solve for \( \theta \):
\[ 10(\theta - 15) = 20(30 - \theta) \]

8. **Expanding and Rearranging**:
- Expanding both sides:
\[ 10\theta - 150 = 600 - 20\theta \]
- Rearranging gives:
\[ 10\theta + 20\theta = 600 + 150 \]
\[ 30\theta = 750 \]

9. **Solving for \( \theta \)**:
- Divide both sides by 30:
\[ \theta = \frac{750}{30} = 25°C \]

### Final Answer:
The temperature at which the pendulum clock gives the correct time is **25°C**.