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To solve the problem, let's break it into two parts:

### Part A: Finding the formula for the population, \( P(t) \)
Given:
- In 1991 (which is \( t = 1 \) since \( t \) represents years since 1990), the population was 4230.
- In 1997 (which is \( t = 7 \)), the population was 4470.

We assume the population changes linearly, so the equation for population \( P(t) \) is of the form:
\[
P(t) = mt + b
\]
Where:
- \( m \) is the slope, representing the rate of change in population per year.
- \( b \) is the initial population at \( t = 0 \) (i.e., in 1990).

#### Step 1: Find the slope \( m \)
The slope is given by:
\[
m = \frac{P(7) - P(1)}{7 - 1} = \frac{4470 - 4230}{6} = \frac{240}{6} = 40
\]
So, the population increases by 40 moose per year.

#### Step 2: Find the intercept \( b \)
Using the point \( P(1) = 4230 \) (in 1991):
\[
P(1) = m(1) + b = 40(1) + b = 4230
\]
\[
40 + b = 4230 \quad \Rightarrow \quad b = 4230 - 40 = 4190
\]

Thus, the formula for the population is:
\[
P(t) = 40t + 4190
\]

### Part B: Predicting the population in 2008
Since \( t \) represents years since 1990, for the year 2008, we have:
\[
t = 2008 - 1990 = 18
\]
Substituting \( t = 18 \) into the equation for \( P(t) \):
\[
P(18) = 40(18) + 4190 = 720 + 4190 = 4910
\]

Thus, the predicted moose population in 2008 is 4910.

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Would you like further clarification on this? Here are some related questions for practice:

1. What would the population be in 2020 according to this model?
2. How would the equation change if the population decreased over time?
3. What if the population increase was exponential rather than linear?
4. How do you calculate the slope when given two points in a different context (e.g., economics)?
5. If the population in 1993 was measured to be 4320, how would you modify your linear model?

**Tip:** When modeling with a linear equation, always check your slope carefully to ensure it represents the rate of change correctly.

In 1991, the moose population in a park was measured to be 4230. By 1997, the population was measured again to be 4470. If the population continues to change linearly:
A.) Find a formula for the moose population, P, in terms of t, the years since 1990.
B.) What does your model predict the moose population to be in 2008?

Explore how to find the linear population growth model for moose in a park from 1991 to 1997, and predict future populations, such as for the year 2008. This problem involves using slope-intercept form and calculating the rate of change, ideal for Grades 8-10.

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