During 16 weeks at the height of an influenza​ outbreak, the rate at which the number of cases of infection changed could be approximated by  Upper I prime left parenthesis t right parenthesisI′(t) =−6.56t+141.6​, where I is the total number of infected people and t is time measured in weeks. Answer parts ​a) through ​d).
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Part 1
​a) Estimate​ I(t), the total number of people who have contracted influenza by time t. Assume that ​I(0)equals=1419

To find \( I(t) \), the total number of people infected by time \( t \), we need to integrate the given rate of change function \( I'(t) = -6.56t + 141.6 \).

### Step-by-Step Solution:

1. **Given:** 
   \[
   I'(t) = -6.56t + 141.6
   \]
   and \( I(0) = 1419 \).

2. **Integrate \( I'(t) \) to find \( I(t) \):**

   The integral of \( I'(t) \) will give us \( I(t) \):
   \[
   I(t) = \int (-6.56t + 141.6) \, dt
   \]

3. **Compute the integral:**
   \[
   I(t) = \int (-6.56t) \, dt + \int 141.6 \, dt
   \]
   \[
   I(t) = -6.56 \cdot \frac{t^2}{2} + 141.6t + C
   \]
   Simplify:
   \[
   I(t) = -3.28t^2 + 141.6t + C
   \]

4. **Determine the constant \( C \) using \( I(0) = 1419 \):**

   Substitute \( t = 0 \) and \( I(0) = 1419 \):
   \[
   I(0) = -3.28(0)^2 + 141.6(0) + C = 1419
   \]
   So, \( C = 1419 \).

5. **Final equation for \( I(t) \):**
   \[
   I(t) = -3.28t^2 + 141.6t + 1419
   \]

This is the formula for the total number of people who have contracted influenza by time \( t \) in weeks.

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Do you want more details on any step or have any questions?

### Related Questions:
1. What is the estimated number of people infected at \( t = 10 \) weeks?
2. When will the number of infected people start decreasing?
3. How many people will have contracted influenza at the end of 16 weeks?
4. What is the maximum number of people infected at any given time?
5. Can you explain how the rate of infection changes over time?

### Tip:
Always verify your solution by differentiating \( I(t) \) to see if you get back \( I'(t) \). This confirms the integration is correct.

During 16 weeks at the height of an influenza outbreak, the rate at which the number of cases of infection changed could be approximated by I′(t) = −6.56t + 141.6, 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) = 1419.

Solve a calculus-based problem involving the rate of infection during an influenza outbreak, where the infection rate is modeled by I′(t) = −6.56t + 141.6. Learn how to estimate the total number of infected people, I(t), using integration and given initial conditions. Suitable for Grades 11-12 students.

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