An electric kettle has two heating coils. When one coil is used, water in the kettle boils in 5 minutes, while when second coil is used, same water boils in 10 minutes. If the two coils, connected in parallel are used simultaneously, the same water will boil in time

Options

A. 3 min 20 sec
B. 5 min
C. 7 min 30sec
D. 2 min 30sec

Answer

To solve the problem, we need to determine the time it takes for the water to boil when both heating coils are used simultaneously. Let's break it down step by step.

### Step 1: Understanding the Heating Time for Each Coil
- When the first coil (R1) is used, the water boils in 5 minutes (T1 = 5 minutes).
- When the second coil (R2) is used, the water boils in 10 minutes (T2 = 10 minutes).

### Step 2: Calculate the Power of Each Coil
The power (P) of each coil can be expressed in terms of the time taken to boil the water. The heat produced (H) can be given by the formula:
\[ H = P \times t \]
Where \( P \) is the power and \( t \) is the time.

Since the same amount of heat is required to boil the water, we can express the power of each coil as:
- For coil 1:
\[ H = P_1 \times T_1 \]
- For coil 2:
\[ H = P_2 \times T_2 \]

### Step 3: Relate Power to Resistance
The power can also be expressed in terms of voltage (V) and resistance (R):
\[ P = \frac{V^2}{R} \]
Thus, we can write:
- 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 4: Set Up the Equations
Since both coils produce the same amount of heat:
\[ \frac{V^2}{R_1} \times T_1 = \frac{V^2}{R_2} \times T_2 \]

### Step 5: Calculate the Equivalent Resistance
When the coils are connected in parallel, the equivalent resistance (R_eq) can be calculated as:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 6: Calculate the Total Time
The total power when both coils are used together is:
\[ P_{total} = P_1 + P_2 = \frac{V^2}{R_1} + \frac{V^2}{R_2} \]
The total heat required to boil the water remains the same, and we can express it as:
\[ H = P_{total} \times t \]
Where \( t \) is the time taken to boil the water with both coils.

### Step 7: Combine the Equations
Using the relationship of heat produced:
\[ H = \frac{V^2}{R_{eq}} \times t \]
We can equate the two expressions for heat:
\[ \frac{V^2}{R_{eq}} \times t = \frac{V^2}{R_1} \times T_1 \]
This simplifies to:
\[ t = T_1 \times \frac{R_{eq}}{R_1} \]

### Step 8: Substitute the Values
From the earlier steps, we know:
- \( T_1 = 5 \) minutes
- \( T_2 = 10 \) minutes

Now we can calculate:
\[ \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} \]
\[ \frac{1}{T} = \frac{1}{5} + \frac{1}{10} \]
\[ \frac{1}{T} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \]

### Step 9: Solve for Time
Now, solving for \( T \):
\[ T = \frac{10}{3} \text{ minutes} \]
This can be converted to minutes and seconds:
\[ T = 3 \text{ minutes and } 20 \text{ seconds} \]

### Final Answer
The time it takes for the water to boil when both coils are used simultaneously is **3 minutes and 20 seconds**.

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