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Let's break down the problem based on the information provided.

### Given:
- You want a total resistance of \( R_{total} = 3.08 \, \Omega \).
- One known resistor is \( R_1 = 12.25 \, \Omega \).
- We need to find the value of the unknown resistance \( R_x \), and determine whether the connection should be in series or parallel.

### Part (a) - Finding the unknown resistance \( R_x \):

#### Case 1: Resistors in **series**:
- For resistors connected in series, the total resistance is the sum of the individual resistances:
  \[
  R_{total} = R_1 + R_x
  \]
  So,
  \[
  3.08 = 12.25 + R_x
  \]
  This results in:
  \[
  R_x = 3.08 - 12.25 = -9.17 \, \Omega
  \]
  Since resistance cannot be negative, the series connection is not possible.

#### Case 2: Resistors in **parallel**:
- For resistors connected in parallel, the total resistance is given by the formula:
  \[
  \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_x}
  \]
  Substituting the values we have:
  \[
  \frac{1}{3.08} = \frac{1}{12.25} + \frac{1}{R_x}
  \]
  Rearranging to find \( R_x \):
  \[
  \frac{1}{R_x} = \frac{1}{3.08} - \frac{1}{12.25}
  \]
  Using a calculator:
  \[
  \frac{1}{R_x} \approx 0.3247 - 0.0816 = 0.2431
  \]
  So,
  \[
  R_x \approx \frac{1}{0.2431} \approx 4.11 \, \Omega
  \]

Thus, the unknown resistance is \( R_x \approx 4.11 \, \Omega \), and the resistors should be connected in **parallel**.

### Final Answers:
- (a) \( R_x = 4.11 \, \Omega \)
- (b) The connection should be in **parallel**.

Would you like further clarification or more details on any step?

Here are 5 questions to explore further:
1. How do resistors behave differently in series vs. parallel connections?
2. How does the total resistance formula change for more than two resistors in parallel?
3. Can a negative resistance have any real-world physical meaning?
4. What are some practical applications of parallel resistor configurations?
5. How would the calculation change if you added more resistors to the circuit?

**Tip**: When calculating parallel resistances, always remember that the total resistance is lower than the smallest individual resistance.

A total resistance of 3.08 Ω is to be produced by connecting an unknown resistance to a 12.25 Ω resistance. (a) What must be the value of the unknown resistance, and (b) Should it be connected in series or in parallel?

Solve a problem to find the unknown resistance in a circuit with a 12.25 Ω resistor to achieve a total resistance of 3.08 Ω. Learn whether the connection should be in series or parallel and use key formulas for resistors.

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