Question

If \( R_c = m \times \ln \left(1 + \frac{R_m}{m} \right) \) then \( R_m \) is equal to

• A. \( R_m = \ln \left(1 + \frac{R_c}{m} \right) \)
• B. \( R_m = \ln \left(1 + \frac{R_c}{e} \right) \)
• C. \( R_m = m \left( e^{\frac{R_c}{m}} - 1 \right) \)
• D. Cannot be determined

Answer 1

The Correct Option is C

Given the equation \( R_c = m \times \ln \left(1 + \frac{R_m}{m} \right) \), we need to solve for \( R_m \).
1. Start by dividing both sides by \( m \):
\[ \frac{R_c}{m} = \ln \left(1 + \frac{R_m}{m} \right) \]
2. Exponentiate both sides to remove the natural logarithm:
\[ e^{\frac{R_c}{m}} = 1 + \frac{R_m}{m} \]
3. Subtract 1 from both sides:
\[ e^{\frac{R_c}{m}} - 1 = \frac{R_m}{m} \]
4. Multiply both sides by \( m \) to solve for \( R_m \):
\[ R_m = m \left( e^{\frac{R_c}{m}} - 1 \right) \]
Thus, the correct option is C: \( R_m = m \left( e^{\frac{R_c}{m}} - 1 \right) \).