The number of people with the flu during an epidemic is a function, , of the number of days, , since the apidemic bega
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

What does mean in this situation? make sense in this situation?

Ask by Curry Vargas. 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 bega
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

What does mean in this situation? make sense in this situation?

Ask by Curry Vargas. in the United States

Jan 02,2025

UpStudy AI · Accepted Answer

Let's analyze the problem step by step based on the given function $$ f(d) = 40 \cdot \left(\frac{5}{4}\right)^{d} $$.

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

To find the number of people who had the flu at the beginning of the epidemic, we need to evaluate $$ f(0) $$:

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

Since any number raised to the power of 0 is 1, we have:

$$
f(0) = 40 \cdot 1 = 40
$$

Thus, **40 people had the flu at the beginning of the epidemic.**

### 2. Each day, the number of infected people grows by a factor of

The growth factor can be determined from the equation. The term $$ \left(\frac{5}{4}\right)^{d} $$ indicates that each day, the number of infected people increases by a factor of $$ \frac{5}{4} $$.

This means that each day, the number of infected people grows by **25%** (since $$ \frac{5}{4} = 1.25 $$).

### 3. What does $$ f(1) $$ mean in this situation? Does $$ f(3.5) $$ make sense in this situation?

- **$$ f(1) $$**: This represents the number of people who had the flu after 1 day since the epidemic began. We can calculate it as follows:

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

So, **50 people had the flu after 1 day.**

- **$$ f(3.5) $$**: This represents the number of people who had the flu after 3.5 days since the epidemic began. While we can calculate this value, it is important to note that in a real-world context, the number of infected people is typically counted in whole numbers. However, mathematically, we can still evaluate it:

$$
f(3.5) = 40 \cdot \left(\frac{5}{4}\right)^{3.5}
$$

Let's calculate $$ f(3.5) $$:

$$
f(3.5) = 40 \cdot \left(\frac{5}{4}\right)^{3.5}
$$

Now, I will calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$40\left(\frac{5}{4}\right)^{3.5}$$
- step1: Convert the expressions:
  $$40\left(\frac{5}{4}\right)^{\frac{7}{2}}$$
- step2: Evaluate the power:
  $$40\times \frac{5^{\frac{7}{2}}}{128}$$
- step3: Multiply:
  $$\frac{5^{\frac{9}{2}}}{16}$$
- step4: Simplify:
  $$\frac{625\sqrt{5}}{16}$$
Calculating $$ f(3.5) $$:

$$
f(3.5) \approx 87.35
$$

This means that approximately **87 people had the flu after 3.5 days.**

### Summary:
1. At the beginning of the epidemic, **40 people had the flu.**
2. Each day, the number of infected people grows by a factor of **1.25 (or 25%).**
3. $$ f(1) $$ means **50 people had the flu after 1 day.** $$ f(3.5) $$ makes sense mathematically, yielding approximately **87 people after 3.5 days**, although it may not represent a whole number of infected individuals in a real-world context.
1. **40 people had the flu at the beginning of the epidemic.** 2. Each day, the number of infected people grows by a factor of **1.25 (25%).** 3. $$ f(1) $$ means **50 people had the flu after 1 day.** $$ f(3.5) $$ indicates approximately **87 people had the flu after 3.5 days.**

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