A standard coil marked 3W is found to have a true resistance of 3.115 W at 300 K. Calculatge the temperature at which marking is correct. Temperature coefficient of resistance of the material of the coil is `4.2 xx 10^(-300C^(-1)`.

Answer

To solve the problem, we need to find the temperature at which the resistance of the coil is correctly marked as 3Ω. We know the true resistance at 300 K is 3.115Ω, and we have the temperature coefficient of resistance (α) given as \(4.2 \times 10^{-3} \, °C^{-1}\).

### Step-by-Step Solution:

1. **Understand the relationship between resistance and temperature**:
The resistance of a material changes with temperature according to the formula:
\[ R' = R(1 + \alpha (T - T_0)) \]
where:
- \(R'\) is the resistance at temperature \(T\),
- \(R\) is the resistance at reference temperature \(T_0\),
- \(\alpha\) is the temperature coefficient of resistance,
- \(T\) is the temperature we want to find,
- \(T_0\) is the reference temperature (300 K in this case).

2. **Identify the known values**:
- \(R' = 3.115 \, \Omega\) (true resistance at 300 K)
- \(R = 3 \, \Omega\) (marked resistance)
- \(T_0 = 300 \, K\)
- \(\alpha = 4.2 \times 10^{-3} \, °C^{-1}\)

3. **Set up the equation**:
Substitute the known values into the resistance formula:
\[ 3.115 = 3(1 + 4.2 \times 10^{-3} (300 - T)) \]

4. **Simplify the equation**:
Divide both sides by 3:
\[ \frac{3.115}{3} = 1 + 4.2 \times 10^{-3} (300 - T) \]
Calculate \(\frac{3.115}{3}\):
\[ 1.03833 = 1 + 4.2 \times 10^{-3} (300 - T) \]

5. **Isolate the term involving temperature**:
Subtract 1 from both sides:
\[ 0.03833 = 4.2 \times 10^{-3} (300 - T) \]

6. **Solve for \(300 - T\)**:
Divide both sides by \(4.2 \times 10^{-3}\):
\[ 300 - T = \frac{0.03833}{4.2 \times 10^{-3}} \approx 9.12 \]

7. **Calculate the temperature \(T\)**:
Rearranging gives:
\[ T = 300 - 9.12 \approx 290.88 \, K \]

### Final Answer:
The temperature at which the marking is correct is approximately \(290.88 \, K\).