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The image shows exercises focused on creating mathematical models, tables, and graphs to represent different scenarios. I'll address each problem, guide you through solving them, and assist with identifying similarities and differences.

### Problem 3:
- **Scenario**: You save money in a bank account, earning 3% interest at the end of each month. You start with $50.
- **Objective**: Use a **table**, a **graph**, and an **equation** to model the amount of money in the account after $ m $ months.

#### 1. Equation:
- Since you're earning 3% interest monthly, the situation describes compound interest. The formula for compound interest is:
  \[
  A = P(1 + r)^m
  \]
  where:
  - $ A $ is the amount after $ m $ months,
  - $ P $ is the principal ($50),
  - $ r $ is the interest rate (0.03),
  - $ m $ is the number of months.

  So the equation becomes:
  \[
  A = 50(1 + 0.03)^m = 50(1.03)^m
  \]

#### 2. Table:
Create a table that shows the balance over time. For example:

| Months (m) | Balance (A) |
|------------|-------------|
| 0          | 50          |
| 1          | 51.50       |
| 2          | 53.05       |
| 3          | 54.64       |
| ...        | ...         |

#### 3. Graph:
Plot the values from the table on a graph where the x-axis represents the months ($ m $) and the y-axis represents the balance ($ A $). The graph will show an exponential growth curve.

---

### Problem 4:
- **Scenario**: You're draining a 1,500-gallon swimming pool, and 3% of the water drains out each minute.
- **Objective**: Use a **table**, a **graph**, and an **equation** to model the amount of water in the pool after $ t $ minutes.

#### 1. Equation:
This situation describes exponential decay because 3% of the water is lost each minute. The formula for exponential decay is:
\[
A = P(1 - r)^t
\]
where:
- $ A $ is the amount of water left after $ t $ minutes,
- $ P $ is the initial amount (1,500 gallons),
- $ r $ is the decay rate (0.03),
- $ t $ is the time in minutes.

So, the equation becomes:
\[
A = 1500(1 - 0.03)^t = 1500(0.97)^t
\]

#### 2. Table:
Create a table similar to the one in Problem 3, showing the amount of water remaining over time.

| Minutes (t) | Gallons (A) |
|-------------|-------------|
| 0           | 1500        |
| 1           | 1455        |
| 2           | 1411.35     |
| 3           | 1369.01     |
| ...         | ...         |

#### 3. Graph:
Plot the values from the table on a graph. The x-axis represents the time in minutes ($ t $) and the y-axis represents the gallons of water ($ A $). This graph will show an exponential decay curve.

---

### Problem 5:
- **Compare Problems 1 and 3 (which refers to the bank scenario)**:
  
#### Similarities:
- Both involve exponential functions.
- Both scenarios feature a percentage change (3%) per unit of time (either months or minutes).
- They both require modeling with a table, graph, and equation.

#### Differences:
- **Problem 3** is about exponential growth (bank account balance increases), while **Problem 4** is about exponential decay (water in the pool decreases).
- **Problem 3** has an initial value of $50, while **Problem 4** starts with 1,500 gallons.
- Time increments are different: one uses months, the other minutes.

---

Would you like to see more details on how to complete the tables and graphs?

### Follow-up Questions:
1. Can you calculate the balance after 6 months in Problem 3?
2. How much water remains after 10 minutes in Problem 4?
3. What are the key features of exponential growth vs. decay graphs?
4. How would a change in the interest rate affect the equation in Problem 3?
5. What happens to the pool's water if the percentage drained changes to 5%?

### Tip:
Remember, in exponential functions, small changes in the rate can lead to big differences over time!

I'm more sophisticated than my little sister, so I save my money in a bank account that pays me 3% interest on the money in the account at the end of each month. I started the account with $50 that I got for my birthday. At the end of the summer, I decided to drain the 1,500-gallon swimming pool and noticed 3% of the water was draining out each minute. Use a table, a graph, and an equation to create a mathematical model of both scenarios.

Learn how to model exponential growth with a 3% interest bank account and exponential decay with a 1,500-gallon swimming pool that loses 3% water per minute. This guide covers table, graph, and equation representations for both scenarios.

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