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To solve for the maximum power transferred in the given circuit, we will follow these steps:

1. **Determine the Thevenin Equivalent Circuit**: We need to find the Thevenin equivalent voltage (\(V_{th}\)) and Thevenin equivalent resistance (\(R_{th}\)) as seen from the load (the 17-ohm resistor in this case).

2. **Apply the Maximum Power Transfer Theorem**: Maximum power is transferred to the load when the load resistance equals the Thevenin equivalent resistance, i.e., \(R_{load} = R_{th}\).

### Step 1: Calculate Thevenin Equivalent Resistance (\(R_{th}\))

To find \(R_{th}\), remove the load (the 17-ohm resistor) and calculate the resistance seen from the open terminals.

- The resistors in series: 
  \[
  R_{15\text{ ohm and }5\text{ ohm}} = 15 \text{ ohm} + 5 \text{ ohm} = 20 \text{ ohm}
  \]
  
  Now, this combination is in parallel with the 8-ohm resistor:
  \[
  R_{\text{parallel with 8 ohm}} = \frac{1}{\left(\frac{1}{20} + \frac{1}{8}\right)} = 5.71 \text{ ohm}
  \]
  
- Adding the 22-ohm resistor in series:
  \[
  R_{th} = 5.71 \text{ ohm} + 22 \text{ ohm} = 27.71 \text{ ohm}
  \]

### Step 2: Calculate Thevenin Voltage (\(V_{th}\))

We will calculate \(V_{th}\), the open-circuit voltage across the 17-ohm resistor. This involves voltage division across the resistors in the circuit.

- The total resistance in the loop is:
  \[
  R_{\text{total}} = 27.71 \text{ ohm}
  \]
  
- The current through the circuit due to the 20V source is:
  \[
  I_{\text{total}} = \frac{20V}{27.71\text{ ohm}} = 0.722A
  \]
  
- The voltage across the 22-ohm resistor (which is in series with the load) can be found using:
  \[
  V_{22} = 0.722A \times 22 \text{ ohm} = 15.88V
  \]
  
- The voltage across the parallel combination of the 5-ohm and 8-ohm resistors is:
  \[
  V_{th} = 20V - V_{22} = 20V - 15.88V = 4.12V
  \]

### Step 3: Maximum Power Transfer

According to the maximum power transfer theorem, the maximum power is transferred to the load when:
\[
R_{load} = R_{th} = 27.71 \text{ ohm}
\]

The maximum power transferred is given by:
\[
P_{\text{max}} = \frac{V_{th}^2}{4R_{th}} = \frac{(4.12V)^2}{4 \times 27.71 \text{ ohm}} = 0.153W
\]

So, the **maximum power transferred** is approximately **0.153 W**.

---

Would you like more details on any step or have further questions?

Here are 5 related questions for deeper understanding:
1. How is the Thevenin resistance calculated in complex circuits?
2. What is the significance of the maximum power transfer theorem?
3. How can we calculate Thevenin voltage in different configurations?
4. What happens if the load resistance is not equal to the Thevenin resistance?
5. How does current division differ from voltage division in circuit analysis?

**Tip**: When analyzing circuits, always double-check if resistors are in series or parallel to avoid calculation errors.

Find the maximum power transferred for the given circuit with a 20V source, resistances of 15 ohms, 5 ohms, 8 ohms, 22 ohms, and a load resistor of 17 ohms.

Learn how to calculate the maximum power transferred in a circuit with resistors of 15 ohms, 5 ohms, 8 ohms, 22 ohms, and a load resistor of 17 ohms, using Thevenin's theorem and the maximum power transfer theorem.

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