An electric kettle has two heating coils. When one of the coil is connected to an alternating current source the water in the kettle boils in 10 minutes. When the other coil is used, the same quantity of water takes 15 minutes in boil. How long will it take for the same quantity of water to boil, if the two coils are connected in parallel ?

Question

Accepted Answer

To determine how long it will take for the same quantity of water to boil when the two coils are connected in parallel, we need to understand the concept of power and resistance in electrical circuits.

### Explanation:

1. **Power and Resistance:**
   - The power $ P $ delivered by an electrical device is given by the formula $ P = \frac{V^2}{R} $, where $ V $ is the voltage and $ R $ is the resistance.
   - When two resistors (or coils, in this case) are connected in parallel, the total resistance $ R_{\text{total}} $ is given by $ \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} $.

2. **Boiling Time and Power:**
   - The time taken to boil the water is inversely proportional to the power delivered by the coil. Therefore, if a coil takes less time to boil the water, it delivers more power.
   - Let $ R_1 $ be the resistance of the first coil, which boils water in 10 minutes, and $ R_2 $ be the resistance of the second coil, which boils water in 15 minutes.

3. **Calculating Equivalent Resistance:**
   - Since the time taken is inversely proportional to the power, we can say:
     $$
     \frac{1}{R_1} \propto \frac{1}{10} \quad \text{and} \quad \frac{1}{R_2} \propto \frac{1}{15}
     $$
   - When connected in parallel:
     $$
     \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}
     $$
   - Therefore, the equivalent resistance $ R_{\text{total}} $ is such that the time taken to boil the water is 6 minutes.

4. **Conclusion:**
   - When the two coils are connected in parallel, the combined power is greater than the power of each coil individually, resulting in a shorter boiling time.
   - Thus, the correct answer is **Option 1: 6 minutes**.

### Why Other Options are Incorrect:

- **Option 2: 12 minutes**
  - This would imply that the combined power of the two coils is less effective than the individual power of the first coil, which is incorrect as parallel connection increases the total power.

- **Option 3: 18 minutes**
  - This would suggest that the combined power is even less effective than the second coil alone, which contradicts the principle of parallel circuits increasing total power.

- **Option 4: 24 minutes**
  - This is not feasible as it implies a significant reduction in power, which is not possible with a parallel connection.

By understanding the principles of electrical circuits and the effect of parallel connections on resistance and power, we can confidently conclude that the correct answer is **Option 1: 6 minutes**.