Two heater coils separately take 10 minute and 5 minute to boil certain amount of water . If both the coils are connected in series, the time taken will be

Options

A. 15 min
B. 7.5 min
C. 3.33 mim
D. 2.5 min

Answer

To solve the problem of how long it will take to boil water when two heater coils are connected in series, we can follow these steps:

### Step 1: Understand the heating times of the coils
- The first heater coil takes **10 minutes** to boil the water.
- The second heater coil takes **5 minutes** to boil the same amount of water.

### Step 2: Determine the relationship between time and resistance
- The energy required to heat the water is the same for both coils. The energy can be expressed as:
\[ E = \frac{V^2 T}{R} \]
where \(E\) is energy, \(V\) is voltage, \(T\) is time, and \(R\) is resistance.

### Step 3: Define the resistances of the coils
- Let \(R_1\) be the resistance of the first coil and \(R_2\) be the resistance of the second coil.
- The time taken by each coil can be related to their resistances:
- For the first coil:
\[ E = \frac{V^2 \cdot T_1}{R_1} \quad \text{(where \(T_1 = 10\) minutes)} \]
- For the second coil:
\[ E = \frac{V^2 \cdot T_2}{R_2} \quad \text{(where \(T_2 = 5\) minutes)} \]

### Step 4: Set up the equations for energy
- Since both coils provide the same energy to boil the water, we can set the equations equal to each other:
\[ \frac{V^2 \cdot T_1}{R_1} = \frac{V^2 \cdot T_2}{R_2} \]
- This simplifies to:
\[ \frac{T_1}{R_1} = \frac{T_2}{R_2} \]
- Rearranging gives:
\[ R_1 = \frac{T_1}{T_2} \cdot R_2 \]

### Step 5: Calculate the equivalent resistance when connected in series
- When connected in series, the total resistance \(R_{eq}\) is:
\[ R_{eq} = R_1 + R_2 \]
- Substitute \(R_1\) from the previous step:
\[ R_{eq} = \frac{T_1}{T_2} \cdot R_2 + R_2 = R_2 \left(\frac{T_1}{T_2} + 1\right) \]

### Step 6: Determine the total time taken when both coils are in series
- The time \(T\) taken to boil the water when both coils are connected in series can be expressed as:
\[ E = \frac{V^2 T}{R_{eq}} \]
- Since \(E\) is the same, we can relate it to the individual times:
\[ T = T_1 + T_2 \]
- Substituting the values:
\[ T = 10 \text{ minutes} + 5 \text{ minutes} = 15 \text{ minutes} \]

### Final Answer
The time taken to boil the water when both heater coils are connected in series is **15 minutes**.

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