Electric current in a wire is time rate of flow of charge. The charge in coulombs that passes through a wire after `t` seconds is given by the function `q(t)=t^(2)-2t^(2)+5t+2`. Determine the average current (in coulmbe per second) during the first two seconds.

Options

A. `4A`
B. `5A`
C. `6A`
D. `12A`

Explanation

To determine the average current during the first two seconds, we will follow these steps:

### Step 1: Identify the function for charge
The charge \( q(t) \) that passes through the wire after \( t \) seconds is given by:
\[ q(t) = t^3 - 2t^2 + 5t + 2 \]

### Step 2: Calculate the charge at \( t = 0 \) seconds
Substituting \( t = 0 \) into the function:
\[ q(0) = (0)^3 - 2(0)^2 + 5(0) + 2 = 2 \text{ coulombs} \]

### Step 3: Calculate the charge at \( t = 2 \) seconds
Substituting \( t = 2 \) into the function:
\[ q(2) = (2)^3 - 2(2)^2 + 5(2) + 2 \]
Calculating each term:
\[ = 8 - 8 + 10 + 2 = 12 \text{ coulombs} \]

### Step 4: Calculate the change in charge (\( \Delta q \))
The change in charge from \( t = 0 \) to \( t = 2 \) seconds is:
\[ \Delta q = q(2) - q(0) = 12 \text{ coulombs} - 2 \text{ coulombs} = 10 \text{ coulombs} \]

### Step 5: Calculate the change in time (\( \Delta t \))
The change in time from \( t = 0 \) to \( t = 2 \) seconds is:
\[ \Delta t = 2 \text{ seconds} - 0 \text{ seconds} = 2 \text{ seconds} \]

### Step 6: Calculate the average current (\( I_{avg} \))
The average current is given by the formula:
\[ I_{avg} = \frac{\Delta q}{\Delta t} \]
Substituting the values we calculated:
\[ I_{avg} = \frac{10 \text{ coulombs}}{2 \text{ seconds}} = 5 \text{ coulombs per second} = 5 \text{ amperes} \]

### Final Answer
The average current during the first two seconds is:
\[ \boxed{5 \text{ A}} \]