To solve the problem, we need to determine the resistance of a wire at a different temperature given its resistance at a known temperature. The resistance of a conductor typically increases with an increase in temperature.
### Step-by-Step Solution:
1. **Identify the Given Values**:
- Initial resistance \( R_1 = 20 \, \Omega \) at initial temperature \( T_1 = 15^\circ C \).
- New temperature \( T_2 = 30^\circ C \).
2. **Understand the Relationship Between Resistance and Temperature**:
The resistance of a conductor can be expressed as:
\[
R = R_0(1 + \alpha(T - T_0))
\]
where:
- \( R_0 \) is the resistance at the reference temperature \( T_0 \),
- \( \alpha \) is the temperature coefficient of resistance,
- \( T \) is the new temperature.
3. **Calculate the Change in Temperature**:
\[
\Delta T = T_2 - T_1 = 30^\circ C - 15^\circ C = 15^\circ C
\]
4. **Assume a Positive Temperature Coefficient**:
For most metallic conductors, the temperature coefficient \( \alpha \) is positive, meaning resistance increases with temperature.
5. **Estimate the New Resistance**:
Since we do not have the exact value of \( \alpha \), we can infer that the resistance at \( T_2 \) will be greater than \( R_1 \). Thus, we can say:
\[
R_2 > R_1 = 20 \, \Omega
\]
6. **Choose the Possible Values**:
We are looking for the possible value of resistance at \( 30^\circ C \). The options provided must be evaluated to find one that is greater than \( 20 \, \Omega \).
7. **Evaluate the Options**:
- If the options are \( 20 \, \Omega, 21 \, \Omega, 22.5 \, \Omega, 19 \, \Omega \):
- \( 20 \, \Omega \) is not valid as it does not increase.
- \( 19 \, \Omega \) is also not valid as it decreases.
- \( 21 \, \Omega \) and \( 22.5 \, \Omega \) are valid since they are greater than \( 20 \, \Omega \).
- However, \( 22.5 \, \Omega \) is the only option that represents a reasonable increase in resistance.
8. **Conclusion**:
The possible value of the resistance at \( 30^\circ C \) is:
\[
R_2 = 22.5 \, \Omega
\]
