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Let's work through each of these cases step by step, based on the initial population of 1,750 people.

### (a) The town grows by 80 people a year.
Here, the population increases by a fixed amount each year. This is a linear growth model. The population after \( t \) years can be written as:

\[
P(t) = 1750 + 80t
\]

### (b) The town grows at an annual rate of 8 percent a year.
This describes exponential growth, where the population increases by 8% each year. The formula for exponential growth is:

\[
P(t) = P_0(1 + r)^t
\]

where \( P_0 \) is the initial population (1,750), \( r \) is the growth rate (0.08 for 8%), and \( t \) is the number of years. Thus, the formula is:

\[
P(t) = 1750(1 + 0.08)^t = 1750(1.08)^t
\]

### (c) The town shrinks by 60 people a year.
This is another linear model, but the population decreases by a fixed amount each year. The formula is:

\[
P(t) = 1750 - 60t
\]

### (d) The town shrinks at an annual rate of 9 percent a year.
This describes exponential decay, where the population decreases by 9% each year. The formula for exponential decay is:

\[
P(t) = P_0(1 - r)^t
\]

Here, \( r = 0.09 \) for a 9% decline, so the formula becomes:

\[
P(t) = 1750(1 - 0.09)^t = 1750(0.91)^t
\]

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If you'd like further details or clarifications, feel free to ask! 

Here are 5 related questions for further practice:
1. What is the population after 5 years in case (a)?
2. How would the formula change in case (b) if the annual growth rate was 5%?
3. In case (c), after how many years will the population reach 1,000?
4. What is the population after 10 years in case (d)?
5. How does the formula for exponential decay differ from that for exponential growth?

**Tip**: When dealing with growth or decay problems, always pay attention to whether the change is linear (constant addition/subtraction) or exponential (percentage-based).

A town has 1750 people initially. In each of the cases below, find the formula for the population of the town, P, in terms of the number of years, t.
(a) The town grows by 80 people a year.
(b) The town grows at an annual rate of 8 percent a year.
(c) The town shrinks by 60 people a year.
(d) The town shrinks at an annual rate of 9 percent a year.

Learn how to calculate population growth and decay using linear and exponential formulas. The town's initial population is 1,750 people, with scenarios including fixed annual growth, percentage-based growth, fixed annual shrinkage, and percentage-based decay. Suitable for high school students.

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