Final Answer:
(a) The current in the 18.0-Ω resistor is approximately 0.833 A.
(b) The potential difference between points a and b is approximately 15.0 V.
Explaination:
To find the current in the 18.0-Ω resistor, we can use Ohm's Law and Kirchhoff's laws. Let's label the points as follows:
- Point a (starting point)
- Point b (ending point)
Now, let's analyze the circuit step by step:
### Part (a) - Current in the 18.0-Ω resistor:
1. Top Branch:
- Voltage across the top branch battery = 25.0 V (given)
- Resistance in the top branch = 10.0 Ω (given)
- Using Ohm's Law (I = V/R), the current through the top branch is

2. Middle Branch:
- Resistance in the middle branch = 10.0 Ω (given)
- Using Ohm's Law, the current through the middle branch is

3. Bottom Branch:
- Resistance in the bottom branch = 5.00 Ω (given)
- Using Ohm's Law, the current through the bottom branch is

4. Downward Path from b to 18.0-Ω resistor:
- The current through this path is the same as the current through the bottom branch, so let's call it

5. 18.0-Ω Resistor:
- The current through the 18.0-Ω resistor is the difference between the currents coming into and leaving this point.

### Part (b) - Potential difference between points a and b:
The potential difference between two points in a circuit can be found by summing the potential differences along any path between the two points. In this case, we can go along the top branch and then down the right side to get from a to b.
1. Potential difference across the top branch:

2. Potential difference across the 18.0-Ω resistor:

3. Potential difference from b to a (down the right side):

4. Total potential difference between points a and b:

Now, you can substitute the values and calculate the results.