An electric kettle has two coils. When one of these is switched on, the water in the kettle boils in 6 minutes. When the other coil is switched on, the water boils in 3 minutes. If the two coils are connected in series, find the time taken to boil the water in the kettle.

Options

A. 3 minutes
B. 6 minutes
C. 2 minutes
D. 9 minutes

Answer

To solve the problem step by step, we need to analyze the heating effect of the two coils in the electric kettle and how they behave when connected in series.

### Step 1: Understand the heating effect of each coil
When one coil (let's call it Coil 1) is used alone, it boils the water in 6 minutes. We can denote the resistance of Coil 1 as \( R_1 \) and the time taken as \( T_1 = 6 \) minutes.

When the other coil (Coil 2) is used alone, it boils the water in 3 minutes. We denote the resistance of Coil 2 as \( R_2 \) and the time taken as \( T_2 = 3 \) minutes.

### Step 2: Write the heat produced by each coil
The heat produced by a coil can be expressed using the formula:
\[ H = \frac{V^2}{R} \times T \]
where \( H \) is the heat required to boil the water, \( V \) is the voltage, \( R \) is the resistance, and \( T \) is the time.

For Coil 1:
\[ H = \frac{V^2}{R_1} \times T_1 \]
For Coil 2:
\[ H = \frac{V^2}{R_2} \times T_2 \]

### Step 3: Set up equations for each coil
From the above equations, we can express the resistances in terms of heat and time:
\[ R_1 = \frac{V^2 \times T_1}{H} \]
\[ R_2 = \frac{V^2 \times T_2}{H} \]

### Step 4: Find the equivalent resistance when coils are in series
When the coils are connected in series, the equivalent resistance \( R_{eq} \) is given by:
\[ R_{eq} = R_1 + R_2 \]
Substituting the expressions for \( R_1 \) and \( R_2 \):
\[ R_{eq} = \frac{V^2 \times T_1}{H} + \frac{V^2 \times T_2}{H} \]
\[ R_{eq} = \frac{V^2}{H} (T_1 + T_2) \]

### Step 5: Calculate the total time taken to boil the water
The total time \( T \) taken to boil the water with both coils in series can be expressed as:
\[ H = \frac{V^2}{R_{eq}} \times T \]
Equating the two expressions for \( H \):
\[ \frac{V^2}{R_{eq}} \times T = \frac{V^2}{H} (T_1 + T_2) \]
Cancelling \( V^2 \) from both sides and substituting \( R_{eq} \):
\[ T = \frac{H}{\frac{V^2}{H} (T_1 + T_2)} \]
This simplifies to:
\[ T = T_1 + T_2 \]
Substituting the values \( T_1 = 6 \) minutes and \( T_2 = 3 \) minutes:
\[ T = 6 + 3 = 9 \text{ minutes} \]

### Final Answer
The time taken to boil the water in the kettle when both coils are connected in series is **9 minutes**.

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