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.

• 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.