To solve the problem, we need to find the effective resistance when three resistors \( R_1, R_2, \) and \( R_3 \) are connected in parallel. We are given three scenarios with their respective effective resistances when the resistors are arranged in different positions.
### Step 1: Set Up the Equations
From the problem, we have the following three equations based on the effective resistances:
1. When \( R_1, R_2, R_3 \) are in positions A, B, C:
\[
R_{eff1} = R_1 + \frac{1}{\frac{1}{R_2} + \frac{1}{R_3}} = 70 \, \Omega
\]
2. When \( R_2, R_3, R_1 \) are in positions A, B, C:
\[
R_{eff2} = R_2 + \frac{1}{\frac{1}{R_3} + \frac{1}{R_1}} = 35 \, \Omega
\]
3. When \( R_3, R_1, R_2 \) are in positions A, B, C:
\[
R_{eff3} = R_3 + \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} = 42 \, \Omega
\]
### Step 2: Simplify the Equations
We can rewrite the effective resistance equations in terms of \( R_1, R_2, \) and \( R_3 \):
1. From the first equation:
\[
R_1 + \frac{R_2 R_3}{R_2 + R_3} = 70
\]
2. From the second equation:
\[
R_2 + \frac{R_3 R_1}{R_3 + R_1} = 35
\]
3. From the third equation:
\[
R_3 + \frac{R_1 R_2}{R_1 + R_2} = 42
\]
### Step 3: Solve the System of Equations
Now we have a system of three equations with three unknowns \( R_1, R_2, R_3 \). We can solve these equations simultaneously to find the values of \( R_1, R_2, \) and \( R_3 \).
### Step 4: Calculate the Effective Resistance in Parallel
Once we have the values of \( R_1, R_2, \) and \( R_3 \), we can find the effective resistance when they are connected in parallel using the formula:
\[
\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\]
### Step 5: Final Calculation
After calculating \( R_{parallel} \), we will have our final answer.
