https://file.aiquickdraw.com/mathbot/file/1727398980336fruoh5lo.jpg

Let's first break down the image and address each part clearly:

### Example #4:
The population prediction formula is given as:

\[
P = 200,000 \cdot e^{(0.03y)}
\]

where:
- \( P \) is the population,
- \( y \) is the number of years after the year 2020.

In this example, we are asked to predict the population for the year **2030**. 

Since 2030 is **10 years after 2020**, \( y = 10 \).

The formula becomes:

\[
P = 200,000 \cdot e^{(0.03 \cdot 10)}
\]

Using the value of \( e \approx 2.71828 \):

\[
P = 200,000 \cdot (2.71828)^{0.3}
\]

\[
P \approx 200,000 \cdot 1.34986 = 269,971.70
\]

Thus, the predicted population for the year 2030 is approximately **269,971**.

### Next Section: Exponential Function

The **natural exponential function** is described by:

\[
f(x) = e^x
\]

This is the basic form of an exponential function with the base \( e \), where \( e \approx 2.71828 \), and it appears frequently in growth and decay problems, such as population prediction or bacterial growth.

### Problem 1: Bacterial Growth

For a bacterial culture that **doubles every hour**, starting with **100 bacteria**, the table represents how the number of bacteria grows over time.

| t (hours) | No. of Bacteria |
|:---------:|:---------------:|
|     0     |      100        |
|     1     |      200        |
|     2     |      400        |
|     3     |      800        |
|     4     |     1600        |

- **Exponential model:** The number of bacteria after \( t \) hours can be modeled by:

\[
P(t) = 100 \cdot 2^t
\]

- **After 6 hours:** 

\[
P(6) = 100 \cdot 2^6 = 100 \cdot 64 = 6400
\]

Thus, there will be **6,400 bacteria** after 6 hours.

### Problem 2: Radioactive Decay (Half-Life)

This problem involves a substance with a **half-life of 12 hours** and an initial quantity of **100 grams**. Over time, the substance decays to half of its previous quantity every 12 hours.

Let me know if you'd like to proceed with calculations for Problem 2 or clarify any other part! Here are some related questions to explore further:

---

1. How would the population prediction change if the growth rate were 0.04 instead of 0.03?
2. Can you derive a similar exponential decay formula for Problem 2 (radioactive decay)?
3. What would the population be in 2040 using the same model in Example #4?
4. How would you calculate the number of bacteria after 10 hours in Problem 1?
5. How does the half-life concept apply to exponential decay models?

---

**Tip:** In exponential growth models, small changes in the growth rate or time can lead to significant differences in the outcome over long periods.

Example #4: The predicted population of a city is given by P = 200,000e^(0.03y) where y is the number of years after the year 2020. Predict the population for the year 2030.

Learn how to predict population using the exponential growth formula P = 200,000 * e^(0.03y), including the solution for the year 2030, and explore related topics such as bacterial growth and half-life decay models.

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