. A gray squirrel population was introduced in a certain region 18 years ago. Biologists observe that the
population doubles every 6 years and now the population is 600.
(a) What was the initial squirrel population?
(b) What is the expected squirrel population t years after introduction?
(c) Estimate the expected squirrel population 10 years from now

This problem involves exponential growth. Since the squirrel population doubles every 6 years, we can model the population growth using the exponential growth formula:

\[
P(t) = P_0 \cdot 2^{\frac{t}{T}}
\]

Where:
- \(P(t)\) is the population at time \(t\) (in years).
- \(P_0\) is the initial population (at \(t = 0\)).
- \(T\) is the doubling time (6 years in this case).
- \(t\) is the number of years since the population was introduced.

### Part (a): Initial Squirrel Population

We are given that the current population (after 18 years) is 600. Using the formula:

\[
600 = P_0 \cdot 2^{\frac{18}{6}}
\]

Simplifying:

\[
600 = P_0 \cdot 2^3
\]
\[
600 = P_0 \cdot 8
\]

Solving for \(P_0\):

\[
P_0 = \frac{600}{8} = 75
\]

So, the initial squirrel population was **75**.

### Part (b): Squirrel Population \(t\) Years After Introduction

Now, we can use the general formula for any time \(t\):

\[
P(t) = 75 \cdot 2^{\frac{t}{6}}
\]

This equation gives the population at any time \(t\) years after the introduction.

### Part (c): Expected Squirrel Population 10 Years From Now

To find the expected population 10 years from now (which means 18 + 10 = 28 years after the introduction), we plug \(t = 28\) into the formula:

\[
P(28) = 75 \cdot 2^{\frac{28}{6}} = 75 \cdot 2^{4.67}
\]

Let me calculate that.The expected squirrel population 10 years from now (which is 28 years after introduction) is approximately **1,905 squirrels**.

### Summary:
1. The initial squirrel population was **75**.
2. The squirrel population at any time \( t \) years after introduction is given by \( P(t) = 75 \cdot 2^{\frac{t}{6}} \).
3. The estimated population 10 years from now is approximately **1,905**.

Do you want more details or have any questions?

Here are 5 related questions:
1. How would the model change if the population doubled every 3 years instead of 6 years?
2. What would the population be 20 years from now?
3. How does the exponential growth model compare to a linear growth model in this context?
4. If the population grows by 10% each year instead of doubling every 6 years, how would that affect the equation?
5. Can you determine after how many years the population would reach 10,000?

**Tip:** When analyzing exponential growth, small changes in time can result in significant differences in population size due to the compounding effect.

A gray squirrel population was introduced in a certain region 18 years ago. Biologists observe that the population doubles every 6 years and now the population is 600. (a) What was the initial squirrel population? (b) What is the expected squirrel population t years after introduction? (c) Estimate the expected squirrel population 10 years from now.

Explore how to calculate the initial squirrel population and predict future growth using exponential growth principles. This problem demonstrates population doubling every 6 years, providing key formulas and step-by-step solutions for understanding exponential growth, suitable for Grades 9-12.

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