During 20 weeks at the height of an influenza​ outbreak, the rate at which the number of cases of infection changed could be approximated by l'=- 6.26t+142.6 where I is the total number of infected people and t is time measured in weeks. estimate l(t), the total number of people who have contracted influenza by time t. Assume that l(0)=1418

We are given the rate of change of the total number of influenza cases over time, expressed as a function of time \( t \):

\[
\frac{dI}{dt} = I'(t) = -6.26t + 142.6
\]

We are also told that \( I(0) = 1418 \), meaning that at \( t = 0 \), 1418 people had already contracted influenza. We need to find the function \( I(t) \), which represents the total number of people infected by time \( t \).

### Step 1: Integrate \( I'(t) \)
To find \( I(t) \), we integrate the given rate of change function with respect to \( t \):

\[
I(t) = \int (-6.26t + 142.6) \, dt
\]

### Step 2: Perform the integration
The integral of \( -6.26t + 142.6 \) is:

\[
I(t) = \int (-6.26t) \, dt + \int 142.6 \, dt
\]

This gives us:

\[
I(t) = -6.26 \cdot \frac{t^2}{2} + 142.6t + C
\]

Simplifying:

\[
I(t) = -3.13t^2 + 142.6t + C
\]

### Step 3: Use initial condition to find \( C \)
We know that \( I(0) = 1418 \), so we substitute \( t = 0 \) into the equation to solve for \( C \):

\[
I(0) = -3.13(0)^2 + 142.6(0) + C = 1418
\]

This simplifies to:

\[
C = 1418
\]

### Step 4: Final equation for \( I(t) \)
Now, substituting \( C = 1418 \) into the equation for \( I(t) \), we get:

\[
I(t) = -3.13t^2 + 142.6t + 1418
\]

Thus, the total number of people who have contracted influenza by time \( t \) is given by:

\[
I(t) = -3.13t^2 + 142.6t + 1418
\]

### Do you have any questions, or would you like more details on any part of the process?

---

Here are 5 related questions to further your understanding:

1. How would the total number of cases change if the rate of infection were a different function of \( t \)?
2. What does the negative quadratic term in \( I(t) \) tell you about the trend in the number of cases over time?
3. How would the solution change if the initial condition were different, say \( I(0) = 2000 \)?
4. What is the significance of the constant term \( 1418 \) in the context of this outbreak?
5. Can you find the maximum number of cases over the 20-week period?

**Tip:** When solving problems involving differential equations, always check if you are given initial conditions, as they are essential for finding the particular solution!

During 20 weeks at the height of an influenza outbreak, the rate at which the number of cases of infection changed could be approximated by I' = -6.26t + 142.6, where I is the total number of infected people and t is time measured in weeks. Estimate I(t), the total number of people who have contracted influenza by time t. Assume that I(0) = 1418.

Learn how to estimate the total number of people infected during an influenza outbreak over 20 weeks. Using differential equations and integration, solve I(t) given the rate of infection change and initial conditions. Suitable for high school and college students.

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