Part (a): Finding the linear model

Given the population data for the years 2003 and 2015:

• Population in 2003: 108,434
• Population in 2015: 103,922

We are asked to model the population as a linear function of time, where $x$ is the number of years since 2000. So, for 2003, $x = 3$, and for 2015, $x = 15$.

The equation of a line is given by:

$y = mx + b$

We need to find the slope $m$ and the intercept $b$.

Step 1: Calculate the slope $m$

The slope is the rate of change of population per year and is calculated as:

$m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{103922 - 108434}}{{15 - 3}} = \frac{{-4512}}{{12}} = -376$

So, the slope $m = -376$.

Step 2: Find the intercept $b$

To find $b$, we use one of the points (e.g., $(3, 108434)$) and the equation $y = mx + b$.

$108434 = -376(3) + b$

$108434 = -1128 + b$

$b = 108434 + 1128 = 109562$

So, the linear model is:

$y = -376x + 109562$

Part (b): Estimating the population in 2028

For 2028, the value of $x$ is $2028 - 2000 = 28$.

Using the linear model $y = -376x + 109562$, we substitute $x = 28$:

$y = -376(28) + 109562$

$y = -10528 + 109562$

$y = 99034$

So, the estimated population in 2028 is 99,034 people.

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Do you want more details or have any questions? Here are 5 related questions to explore:

1. How would the population change if the decline rate increased to -400 per year?
2. Can you find the population model if the decline were exponential instead of linear?
3. What would be the population in 2050 according to the current model?
4. How long will it take for the population to drop below 90,000?
5. How would changes in birth rate affect the linear model?

Tip: When estimating future values using a linear model, remember that real-world factors may cause deviations from the projected trend.