https://file.aiquickdraw.com/mathbot/file/1728995015621kinb2s9p.jpg

The problem involves calculating the value of an unknown resistance that needs to be connected to a 12.25 Ω resistor to achieve a total resistance of 3.08 Ω. There are two possibilities for connecting resistors: either in series or in parallel. Let’s examine each scenario.

### (a) What must be the value of the unknown resistance?

#### 1. Series Connection:
In a series connection, the total resistance \( R_{total} \) is the sum of the resistances:
\[
R_{total} = R_1 + R_2
\]
where \( R_1 = 12.25 \, \Omega \) and \( R_2 \) is the unknown resistance. Given \( R_{total} = 3.08 \, \Omega \), we can solve for \( R_2 \):
\[
3.08 = 12.25 + R_2
\]
\[
R_2 = 3.08 - 12.25 = -9.17 \, \Omega
\]
A negative resistance value is physically impossible, so the series connection is not viable.

#### 2. Parallel Connection:
In a parallel connection, the total resistance \( R_{total} \) is given by the formula:
\[
\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}
\]
Substituting \( R_{total} = 3.08 \, \Omega \) and \( R_1 = 12.25 \, \Omega \):
\[
\frac{1}{3.08} = \frac{1}{12.25} + \frac{1}{R_2}
\]
First, compute \( \frac{1}{12.25} \):
\[
\frac{1}{12.25} \approx 0.08163
\]
Now solve for \( \frac{1}{R_2} \):
\[
\frac{1}{3.08} = 0.32467
\]
\[
0.32467 = 0.08163 + \frac{1}{R_2}
\]
\[
\frac{1}{R_2} = 0.32467 - 0.08163 = 0.24304
\]
Thus, \( R_2 \) is:
\[
R_2 = \frac{1}{0.24304} \approx 4.115 \, \Omega
\]

### (b) Should the unknown resistance be connected in series or in parallel?
Since the series connection leads to an impossible (negative) resistance, the unknown resistance must be connected in **parallel** to the 12.25 Ω resistor.

### Final Answers:
- (a) The value of the unknown resistance is approximately **4.115 Ω**.
- (b) The resistors should be connected in **parallel**.

Would you like further details or have any other questions?

Here are five related questions to deepen your understanding:
1. How do you calculate total resistance in a series circuit with more than two resistors?
2. How does the total resistance change if additional resistors are added in parallel?
3. What is the physical interpretation of a negative resistance result?
4. Can you explain why parallel circuits result in a lower total resistance?
5. How do resistor values affect current in parallel versus series circuits?

**Tip:** When solving parallel circuits, always remember that the total resistance is less 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?

Find the unknown resistance needed to achieve a total resistance of 3.08 Ω when connected to a 12.25 Ω resistor. The problem explores the difference between series and parallel circuits, using algebraic equations to solve for the unknown resistor. Suitable for high school students in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation