The equation is this An virus is spreading through a school according to the function 𝑁(𝑡) = 2(3^t) +___, where 𝑁 is the number of people infected and 𝑡 is the time in days.

   a.

b.

2.

Determine the average rate of change between 𝑑𝑎𝑦 1 and 𝑑𝑎𝑦 2 in exact form. [2]

Given the function \( N(t) = 2(3^t) + C \), where \( N \) represents the number of people infected and \( t \) is the time in days, let's determine the average rate of change between day 1 and day 2.

### Average Rate of Change

The average rate of change of a function \( N(t) \) over an interval \([t_1, t_2]\) is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{N(t_2) - N(t_1)}{t_2 - t_1}
\]
For the interval from day 1 to day 2, we have:
- \( t_1 = 1 \)
- \( t_2 = 2 \)

The function is \( N(t) = 2(3^t) + C \).

First, calculate \( N(1) \) and \( N(2) \):

\[
N(1) = 2(3^1) + C = 2 \cdot 3 + C = 6 + C
\]

\[
N(2) = 2(3^2) + C = 2 \cdot 9 + C = 18 + C
\]

Now, substitute these values into the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{(18 + C) - (6 + C)}{2 - 1} = \frac{18 + C - 6 - C}{1} = \frac{12}{1} = 12
\]

### Answer:
The exact form of the average rate of change between day 1 and day 2 is **12**. This is the constant difference in the number of infected people per day between the two days.

Would you like any further details or explanations? Here are some related questions:

1. How would the average rate of change be affected if \( C \) were negative or positive?
2. What does this rate of change suggest about the speed of the virus spread between days 1 and 2?
3. How would the average rate of change differ if the interval was between day 0 and day 2?
4. If the function had an additional term (e.g., \( +5t \)), how would that affect the average rate of change?
5. Could the average rate of change be zero for any interval? Under what circumstances?

**Tip:** Always remember that the constant \( C \) in the function does not affect the rate of change; it only shifts the function vertically.

Learn how to calculate the average rate of change in exponential growth using a specific function. Explore the impact of initial conditions on the rate of spread of an infection in a school setting.

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