The electric current in a discharging R-C circuit is given by `i=i_0e^(-t/(RC))` where `i_0` `R `and `C `are constant parameters and `t `is time. Let `i_0=2.00A`, `R=6.00xx10^5 ohm` and `C=0.500 muF`.
a. Find the current at t=0.3 s.
b. Find the rate of change of current at t=0.3 s.
Find approximately the current at t=0.31 s.

Explanation

To solve the problem step by step, we will follow the instructions given in the question and use the formula provided for the electric current in a discharging R-C circuit.

### Given:
- \( i_0 = 2.00 \, \text{A} \)
- \( R = 6.00 \times 10^5 \, \Omega \)
- \( C = 0.500 \, \mu\text{F} = 0.500 \times 10^{-6} \, \text{F} \)

### Step 1: Calculate \( RC \)
First, we need to calculate the product \( RC \):
\[ RC = R \times C = (6.00 \times 10^5 \, \Omega) \times (0.500 \times 10^{-6} \, \text{F}) \]
\[ RC = 6.00 \times 0.500 \times 10^{-1} = 3.00 \times 10^{-1} = 0.3 \, \text{s} \]

### Step 2: Find the current at \( t = 0.3 \, \text{s} \)
Using the formula for current:
\[ i = i_0 e^{-\frac{t}{RC}} \]
Substituting \( t = 0.3 \, \text{s} \):
\[ i = 2.00 \, e^{-\frac{0.3}{0.3}} = 2.00 \, e^{-1} \]
Calculating \( e^{-1} \):
\[ e^{-1} \approx 0.3679 \]
Thus,
\[ i \approx 2.00 \times 0.3679 \approx 0.7358 \, \text{A} \]

### Step 3: Find the rate of change of current at \( t = 0.3 \, \text{s} \)
To find the rate of change of current, we differentiate the current with respect to time:
\[ \frac{di}{dt} = -\frac{i_0}{RC} e^{-\frac{t}{RC}} \]
Substituting \( i_0 = 2.00 \, \text{A} \), \( RC = 0.3 \, \text{s} \), and \( t = 0.3 \, \text{s} \):
\[ \frac{di}{dt} = -\frac{2.00}{0.3} e^{-1} \]
Calculating:
\[ \frac{di}{dt} = -\frac{2.00}{0.3} \times 0.3679 \approx -6.6667 \times 0.3679 \approx -2.4489 \, \text{A/s} \]

### Step 4: Find the current at \( t = 0.31 \, \text{s} \)
Now we calculate the current at \( t = 0.31 \, \text{s} \):
\[ i = i_0 e^{-\frac{t}{RC}} = 2.00 \, e^{-\frac{0.31}{0.3}} \]
Calculating \( e^{-\frac{0.31}{0.3}} \):
\[ e^{-\frac{0.31}{0.3}} \approx e^{-1.0333} \approx 0.3567 \]
Thus,
\[ i \approx 2.00 \times 0.3567 \approx 0.7134 \, \text{A} \]

### Summary of Results:
a. Current at \( t = 0.3 \, \text{s} \): \( \approx 0.7358 \, \text{A} \)

b. Rate of change of current at \( t = 0.3 \, \text{s} \): \( \approx -2.4489 \, \text{A/s} \)

c. Current at \( t = 0.31 \, \text{s} \): \( \approx 0.7134 \, \text{A} \)