To find a formula for the town's population $P$ in terms of $t$, where $t$ represents the number of years since 1920, we can use the information that the population is decreasing linearly. This suggests a linear equation of the form:

$P(t) = mt + b$

Where:

• $P(t)$ is the population at time $t$ (years since 1920),
• $m$ is the rate of change of population (slope),
• $b$ is the population at $t = 0$ (i.e., in 1920).

Step 1: Set up known points

We are given two points in time with corresponding population values:

• In 1920, when $t = 0$, the population was $P(0) = 3100$.
• In 1924, when $t = 4$, the population was $P(4) = 2960$.

Step 2: Find the slope $m$

The slope of a line is given by the formula:

$m = \frac{\Delta P}{\Delta t} = \frac{P(4) - P(0)}{4 - 0}$

Substituting the given values:

$m = \frac{2960 - 3100}{4 - 0} = \frac{-140}{4} = -35$

Thus, the slope $m = -35$, meaning the population decreases by 35 people per year.

Step 3: Write the equation

Now we can substitute the slope $m = -35$ and the initial population $P(0) = 3100$ into the linear equation $P(t) = mt + b$:

$P(t) = -35t + 3100$

Final Answer:

The formula for the population $P$ in terms of $t$, the number of years since 1920, is:

$P(t) = -35t + 3100$

Would you like further details on this solution, or do you have any questions?

Here are 5 related questions to explore further:

1. What would the population be in 1926 according to this formula?
2. In what year will the population reach 2500 if the decline continues at the same rate?
3. How would the formula change if the population decreased exponentially instead of linearly?
4. Can we model the population decrease as a quadratic function instead?
5. What is the significance of the slope $m = -35$ in real-world terms?

Tip: The slope of a linear equation gives the rate of change, so interpreting its value helps understand how quickly the population (or other variables) changes over time.