The number of people with the flu during an epidemic is a function, , of the number of days, , since the apidemic began
The equation . defines .

How many people had the flu at the beginning of the epidemic?

Each day, the number of infected people grows by a factor of does mean in this situation?

Does make sense in this situation?

Ask by Gibbs Herrera. in the United States

Jan 02,2025

Question

The number of people with the flu during an epidemic is a function, , of the number of days, , since the apidemic began
The equation . defines .

How many people had the flu at the beginning of the epidemic?

Each day, the number of infected people grows by a factor of does mean in this situation?

Does make sense in this situation?

Ask by Gibbs Herrera. in the United States

Jan 02,2025

UpStudy AI · Accepted Answer

Let's analyze the given function and address each of your questions step by step.

The function provided is:
$$ f(d) = 40 \left(\frac{5}{4}\right)^d $$
where:
- $$ f(d) $$ represents the number of people with the flu.
- $$ d $$ represents the number of days since the epidemic began.

### 1. **How many people had the flu at the beginning of the epidemic?**

To find the number of people infected at the **beginning** of the epidemic, we evaluate the function at $$ d = 0 $$:

$$ f(0) = 40 \left(\frac{5}{4}\right)^0 $$
$$ f(0) = 40 \times 1 $$
$$ f(0) = 40 $$

**Answer:** At the beginning of the epidemic, **40 people** had the flu.

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### 2. **What does $$ f(1) $$ mean in this situation?**

$$ f(1) $$ represents the number of people infected **one day** after the epidemic began. Let's calculate it:

$$ f(1) = 40 \left(\frac{5}{4}\right)^1 $$
$$ f(1) = 40 \times \frac{5}{4} $$
$$ f(1) = 40 \times 1.25 $$
$$ f(1) = 50 $$

This means that **one day after the epidemic began**, the number of infected people increased to **50**.

**Interpretation:** Each day, the number of infected people increases by a factor of $$\frac{5}{4}$$ (or **25%**). So, $$ f(1) $$ shows the first day's increase based on this growth rate.

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### 4. **Does $$ f(3.5) $$ make sense in this situation?**

**Understanding $$ f(3.5) $$:**
- $$ f(3.5) $$ would represent the number of infected people **3.5 days** after the epidemic began.

**Practical Considerations:**
- **Discrete vs. Continuous Time:** 
  - **Discrete Interpretation:** If we consider days as whole units (i.e., you can only have whole days), then 3.5 days doesn't make practical sense because you can't have half a day in real-world counting.
  - **Continuous Interpretation:** If the model allows for fractional days (e.g., considering time continuously, such as hours within a day), then calculating $$ f(3.5) $$ is mathematically valid. It would provide an estimate of the number of infected people at that specific half-day mark.

**Calculation Example:**
$$ f(3.5) = 40 \left(\frac{5}{4}\right)^{3.5} $$
$$ f(3.5) \approx 40 \times 1.25^{3.5} \approx 40 \times 2.3784 \approx 95.136 $$

**Conclusion:** 
- **Mathematically:** $$ f(3.5) $$ is valid and gives an estimate based on the exponential growth model.
- **Practically:** It may not make sense if you can only count whole days, but it can be useful for more granular time-based analyses.

**Answer Summary:**

1. **40 people** were infected at the start of the epidemic.
2. $$ f(1) $$ indicates that each day the number of infected people increases by a factor of **5⁄4** (a 25% daily growth).
4. $$ f(3.5) $$ doesn’t practically make sense if days are counted as whole numbers, since you can’t have half a day in this context.
1. **40 people** had the flu at the beginning of the epidemic. 2. Each day, the number of infected people increases by a factor of $$\frac{5}{4}$$. So, $$ f(1) = 50 $$ people were infected one day after the epidemic began. 4. $$ f(3.5) $$ doesn't make practical sense because you can't have half a day in this context.

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