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The problem you've uploaded involves an exponential growth model where the initial population is \( P_0 = 6 \) and the growth rate is \( r = 0.35 \). Let's break down the solution step by step:

### Step 1: Formula for Exponential Growth
The general formula for exponential growth is:

\[
P_n = P_0 (1 + r)^n
\]

Where:
- \( P_n \) is the population at time \( n \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate, and
- \( n \) is the number of time periods.

### Step 2: Calculate \( P_1 \)
Using the formula for \( n = 1 \):

\[
P_1 = 6 \times (1 + 0.35)^1 = 6 \times 1.35 = 8.1
\]

### Step 3: Calculate \( P_2 \)
Now, for \( n = 2 \):

\[
P_2 = 6 \times (1 + 0.35)^2 = 6 \times 1.35^2 = 6 \times 1.8225 = 10.935
\]

### Step 4: General Formula for \( P_n \)
The explicit formula for \( P_n \) is:

\[
P_n = 6 \times 1.35^n
\]

### Step 5: Use the Formula to Find \( P_{10} \)
Finally, to find \( P_{10} \), substitute \( n = 10 \) into the formula:

\[
P_{10} = 6 \times 1.35^{10}
\]

Let's calculate this value:

\[
P_{10} = 6 \times 19.5345 = 117.207
\]

### Summary:
- \( P_1 = 8.1 \)
- \( P_2 = 10.935 \)
- \( P_n = 6 \times 1.35^n \)
- \( P_{10} = 117.21 \)

Let me know if you'd like more detailed explanations for any part of the process or if you have questions! 

Here are 5 related questions:
1. How would the formula change if the growth rate were negative (decay)?
2. Can you derive a formula to find the time when the population reaches a specific number?
3. How would the population change if the initial population were 10 instead of 6?
4. How is exponential growth different from linear growth?
5. How long would it take for the population to double with this growth rate?

**Tip**: When working with exponential models, always check if your base is greater than 1 for growth, or less than 1 for decay.

A population grows according to an exponential growth model. The initial population is P_0 = 6, and the growth rate is r = 0.35.

This problem involves solving an exponential growth model where the initial population is P_0 = 6 and the growth rate is r = 0.35. Learn how to calculate P_1, P_2, and find an explicit formula for P_n. The problem covers key exponential growth concepts and is suitable for students in Grades 9-12.

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