Provide an example of a real-world scenario that can be modeled by an exponential function.

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**Example: Population Growth of a Bacterial Culture**

**Scenario:**
Imagine a biologist studying the growth of a bacterial colony in a controlled laboratory environment. The biologist starts with a small number of bacteria and observes how the population changes over time.

**Exponential Growth Modeling:**

1. **Initial Population (P₀):** Suppose the initial number of bacteria is 500.

2. **Growth Rate (r):** The bacteria divide at a constant rate, say 5% per hour. This means that every hour, the population increases by 5% of its current size.

3. **Time (t):** The biologist records the population at different hours.

**Exponential Function:**
The population at any time *t* can be modeled by the exponential growth function:

$$ P(t) = P₀ \times e^{rt} $$

Where:
- $$ P(t) $$ = population at time t
- $$ P₀ $$ = initial population (500 bacteria)
- $$ r $$ = growth rate (0.05 per hour)
- $$ t $$ = time in hours
- $$ e $$ = Euler's number (approximately 2.71828)

**Alternatively, since the growth rate is 5% per hour, it can also be modeled using:**

$$ P(t) = P₀ \times (1 + r)^t = 500 \times (1.05)^t $$

**Application:**
- **After 1 hour:** $$ P(1) = 500 \times 1.05 = 525 $$ bacteria
- **After 2 hours:** $$ P(2) = 500 \times (1.05)^2 \approx 551.25 $$ bacteria
- **After 5 hours:** $$ P(5) = 500 \times (1.05)^5 \approx 638.14 $$ bacteria

**Why Exponential Function?**

- **Constant Percentage Growth:** Each hour, the population grows by a fixed percentage (5%), not a fixed number. This characteristic is key to exponential growth.
  
- **Unlimited Growth Potential:** Assuming unlimited resources, the model projects the population can grow indefinitely, which is typical in ideal exponential growth scenarios.

**Real-World Implications:**

- **Biology:** Understanding bacterial growth helps in fields like microbiology, medicine, and environmental science.
  
- **Public Health:** Similar models apply to the spread of diseases, where the number of infected individuals can grow exponentially without intervention.
  
- **Business and Finance:** Companies may use exponential models to project sales, market growth, or investment returns over time.

**Limitations:**

- **Resource Constraints:** In reality, resources (space, nutrients) limit growth, leading to logistic (S-shaped) growth rather than pure exponential growth.

- **Environmental Factors:** Factors like temperature, pH, and presence of antibiotics can affect bacterial growth rates.

**Conclusion:**
The bacterial population growth scenario is a classic example of an exponential function in the real world. It illustrates how populations can increase rapidly over time when growth occurs at a constant percentage rate, making exponential models invaluable tools in various scientific and practical applications.
**Example of Exponential Growth: Bacterial Population** Imagine a biologist studying a bacterial colony. Starting with 500 bacteria, the population grows by 5% each hour. The number of bacteria can be calculated using the formula: $$ P(t) = 500 \times (1.05)^t $$ After 1 hour: 525 bacteria After 2 hours: ≈551 bacteria After 5 hours: ≈638 bacteria This shows how the population increases exponentially over time.

Provide an example of a real-world scenario that can be modeled by an exponential function.

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