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Answer these with calculation and final answer 

The graph above shows \( v_0(t) = 8 e^{-\frac{t}{60}} \) for \( t \geq 0 \). As expected, the voltage \( v_0(t) \) decreases exponentially from 8 volts towards 0 volts as time progresses.

### **Step 3: Find \( i_0(t) \) at \( t = 3s \)**

The current \( i_0(t) \) through the \( 30 \, k\Omega \) resistor can be found using:
\[
i_0(t) = \frac{v_0(t)}{R}
\]
At \( t = 3s \):
\[
v_0(3) = 8 e^{-\frac{3}{60}} = 8 e^{-0.05}
\]
Let’s calculate \( v_0(3) \) and then find \( i_0(3) \).At \( t = 3 \, s \):
- The voltage \( v_0(3) \) is approximately \( 7.61 \, V \).
- The current \( i_0(3) \) is approximately \( 0.0002537 \, A \) or \( 253.7 \, \mu A \).

### Final Answers:
1. \( v_0(t) = 8 e^{-\frac{t}{60}} \) for \( t \geq 0 \)
2. \( i_0(t) \) at \( t = 3 \, s \) is approximately \( 253.7 \, \mu A \).

Would you like further details or explanations?

### 5 Additional Questions for Expansion:
1. How does changing the resistance \( R \) affect the time constant \( \tau \) and the behavior of \( v_0(t) \)?
2. What happens to \( v_0(t) \) if the initial capacitor voltage is different?
3. How would the circuit behavior change if a different capacitor value \( C \) were used?
4. What is the steady-state behavior of \( v_0(t) \) as \( t \to \infty \)?
5. Can you find the total energy dissipated in the resistor \( 30 \, k\Omega \) over time?

### Tip
For RC circuits, the time constant \( \tau = RC \) determines how quickly the capacitor charges or discharges. A larger \( \tau \) means a slower change in voltage, while a smaller \( \tau \) means a faster change.

Assuming that the switch has been in position A for a long time and is moved to position B at t = 0, find v_0(t) for t ≥ 0. a) Show the graph for v_0(t) vs t b) Find i_0(t) at t = 3s.

This problem involves calculating the voltage v_0(t) across a capacitor in an RC circuit after a switch has been moved, and finding the current i_0(t) through a resistor at a given time. The problem illustrates key concepts such as exponential decay, RC time constant, and Ohm's law.

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