The population of a small town in New Hampshire has shown a linear increase between the years 2001 and 2013. The population at the end of 2001 was 3,851 and at the end of 2013 it was 4,727.

Write a linear equation expressing the population of this town, $P$, as a function of $t$, the number of years since 2001.

$P(t)=$ ____

Please fill in the blank by inserting the right side of the equation in slope-intercept form $(mt + b)$.

The problem asks to determine a linear equation $P(t) = mt + b$ that models the population growth of a small town in New Hampshire from 2001 to 2013. We are given the population at the end of 2001 as 3,851 and at the end of 2013 as 4,727. Since the time variable $t$ represents the number of years since 2001, the year 2001 corresponds to $t = 0$, and the year 2013 corresponds to $t = 2013 - 2001 = 12$. We therefore have two points: $(0, 3851)$ and $(12, 4727)$.

Calculating the slope and y-intercept

A linear function is in the form $P(t) = mt + b$, where $m$ is the rate of change (the slope) and $b$ is the initial value (the y-intercept, at $t=0$). We calculate the slope $m$ as the ratio of the change in population to the change in time:
\[m = \frac{P_2 - P_1}{t_2 - t_1} = \frac{4727 - 3851}{12 - 0} = \frac{876}{12} = 73.\]
This indicates that the town's population increases by 73 people per year.

Constructing the linear equation

The y-intercept $b$ represents the value of the function at $t = 0$. Since the population in 2001 (which is $t=0$) is 3,851, we have $b = 3851$. Substituting the values of $m$ and $b$ into the linear equation formula, we obtain $P(t) = 73t + 3851$.

Examples

This type of linear modeling is used in urban planning to predict future infrastructure needs. For example, city planners might use this equation to estimate if the town will need a new elementary school by 2025 (which would be $t = 24$) by calculating $P(24) = 73(24) + 3851 = 1752 + 3851 = 5603$. This helps in making informed decisions about resource allocation and public service growth.