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: \( R_m = m \left( e^{\frac{R_c}{m}} - 1 \right) \).

Answer 2

Rather than solving the equation from scratch, we can check which option satisfies the original relation \(R_{c}=m\ln\left(1+\frac{R_{m}}{m}\right)\) when substituted back in.

1. Option A \(R_{m}=\ln\left(1+\frac{R_{c}}{m}\right)\): Substituting back does not simplify to \(R_c\) in general, so this is incorrect.
2. Option B \(R_{m}=\ln\left(1+\frac{R_{c}}{e}\right)\): This mixes \(e\) into the formula in a way that does not cancel correctly on substitution, so it is incorrect.
3. Option C \(R_{m}=m\left(e^{\frac{R_{c}}{m}}-1\right)\): Substituting, \(1+\frac{R_m}{m}=1+\left(e^{\frac{R_c}{m}}-1\right)=e^{\frac{R_c}{m}}\), so \(m\ln\left(1+\frac{R_m}{m}\right)=m\ln\left(e^{\frac{R_c}{m}}\right)=m\times\frac{R_c}{m}=R_c\). This is an exact match.
4. Option D (Cannot be determined): Ruled out since the equation can be solved uniquely, as shown above.

Only Option C reproduces the original equation exactly when substituted back.

Hence, the correct answer is Option C: \(R_{m}=m\left(e^{\frac{R_{c}}{m}}-1\right)\).