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The problem provides exponential functions that represent populations of towns over time. You are tasked with determining two things for each function:
1. **Initial population** (i.e., the population at time \( t = 0 \)).
2. **Annual percentage growth rate**, including whether the population is growing or shrinking.

Let's analyze each function one by one.

---

### (a) \( P(t) = 1000(1.03)^t \)

1. **Initial population**:
   \[
   P(0) = 1000(1.03)^0 = 1000
   \]
   So, the initial population is **1000 people**.

2. **Annual percentage growth rate**:
   The factor \( 1.03 \) represents the growth multiplier per year. To find the percentage growth rate:
   \[
   \text{Growth rate} = (1.03 - 1) \times 100 = 3\%
   \]
   Since the growth rate is positive, the population is **growing** by **3% per year**.

---

### (b) \( P(t) = 900(0.90)^t \)

1. **Initial population**:
   \[
   P(0) = 900(0.90)^0 = 900
   \]
   So, the initial population is **900 people**.

2. **Annual percentage growth rate**:
   The factor \( 0.90 \) represents the decay multiplier per year. To find the percentage change:
   \[
   \text{Growth rate} = (0.90 - 1) \times 100 = -10\%
   \]
   Since the growth rate is negative, the population is **shrinking** by **10% per year**.

---

### (c) \( P(t) = 720 \cdot 3^t \)

1. **Initial population**:
   \[
   P(0) = 720 \cdot 3^0 = 720
   \]
   So, the initial population is **720 people**.

2. **Annual percentage growth rate**:
   The factor \( 3 \) represents the growth multiplier per year. To find the percentage growth rate:
   \[
   \text{Growth rate} = (3 - 1) \times 100 = 200\%
   \]
   Since the growth rate is positive, the population is **growing** by **200% per year**.

---

### Summary:
- (a) Initial population: 1000 people, Growth rate: 3% per year (growing)
- (b) Initial population: 900 people, Growth rate: -10% per year (shrinking)
- (c) Initial population: 720 people, Growth rate: 200% per year (growing)

---

Would you like more details on any part of this? 

Here are 5 related questions for practice:
1. How would you interpret a function like \( P(t) = 500(1.05)^t \)?
2. What happens to the population over time if the growth rate is negative?
3. How do you derive the growth rate from a multiplier like \( 1.02 \)?
4. Can a population growth rate ever exceed 100%? What would that mean?
5. How can you convert a percentage growth rate back into an exponential multiplier?

**Tip:** Exponential functions of population are commonly used to model real-world growth in finance, biology, and economics!

The following exponential functions give the populations of four towns with time t in years. For each of the following exponential functions, state the following:
1. The initial population, i.e., the population in year t = 0. Specify the units.
2. The annual percentage growth rate, i.e., the percentage per year at which the population is changing; also specify whether the population is growing or shrinking. Specify units.
Given functions are:
(a) P(t) = 1000(1.03)^t.
(b) P(t) = 900(0.90)^t.
(c) P(t) = 720 * 3^t.

Understand how to find the initial population and the annual percentage growth rate for various exponential functions. Solve problems like P(t) = 1000(1.03)^t, P(t) = 900(0.90)^t, and P(t) = 720 * 3^t using exponential growth and decay formulas.

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