A electric tea kettle has two heating coils. When first coil of resistance `R_1` is switched on, the kettle begins to boil tea in 6 minutes. When second coil of resistance `R_2` is switched on, the boiling begins in 8 minutes. The value of `R_1 // R_2` is

Options

A. `(7)/(3)`
B. `(3)/(7)`
C. `(3)/(4)`
D. `(4)/(3)`

Explanation

To solve the problem, we need to find the value of \( R_1 // R_2 \), which represents the equivalent resistance when the two resistances \( R_1 \) and \( R_2 \) are connected in parallel.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have two heating coils with resistances \( R_1 \) and \( R_2 \). The first coil boils the tea in 6 minutes, and the second coil boils it in 8 minutes. The heat supplied by both coils is the same since they are boiling the same amount of tea.

2. **Use the Heat Formula**:
The heat supplied by a coil can be expressed as:
\[ Q = \frac{V^2}{R} \cdot t \]
where \( V \) is the voltage, \( R \) is the resistance, and \( t \) is the time.

3. **Set Up the Equations**:
For the first coil:
\[ Q = \frac{V^2}{R_1} \cdot T_1 \]
For the second coil:
\[ Q = \frac{V^2}{R_2} \cdot T_2 \]
Since both coils provide the same heat \( Q \), we can equate the two expressions:
\[ \frac{V^2}{R_1} \cdot T_1 = \frac{V^2}{R_2} \cdot T_2 \]

4. **Cancel Common Terms**:
Since \( V^2 \) is common in both equations, we can cancel it out:
\[ \frac{T_1}{R_1} = \frac{T_2}{R_2} \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ \frac{R_1}{R_2} = \frac{T_1}{T_2} \]

6. **Substitute the Values**:
We know \( T_1 = 6 \) minutes and \( T_2 = 8 \) minutes:
\[ \frac{R_1}{R_2} = \frac{6}{8} = \frac{3}{4} \]

7. **Find the Equivalent Resistance**:
The equivalent resistance \( R_1 // R_2 \) for two resistors in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Using the ratio \( \frac{R_1}{R_2} = \frac{3}{4} \), we can express \( R_1 \) as \( 3k \) and \( R_2 \) as \( 4k \) for some constant \( k \).

8. **Calculate \( R_1 // R_2 \)**:
Substitute \( R_1 \) and \( R_2 \) into the parallel resistance formula:
\[ \frac{1}{R_{eq}} = \frac{1}{3k} + \frac{1}{4k} = \frac{4 + 3}{12k} = \frac{7}{12k} \]
Therefore, the equivalent resistance \( R_{eq} \) is:
\[ R_{eq} = \frac{12k}{7} \]

### Final Answer:
The equivalent resistance \( R_1 // R_2 \) is \( \frac{12k}{7} \) where \( k \) is a constant that can be determined if the actual resistances are known.