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The problem you've shared is asking for a mathematical model to describe how the water drains from a pool. The key points to address are:

1. The initial volume of the pool is 1,500 gallons.
2. The water drains at a rate of 3% per minute, so it's an exponential decay model.

### Step 1: Equation
The amount of water remaining in the pool \( V(t) \), at any time \( t \) (in minutes), can be modeled using an exponential decay function:
\[
V(t) = 1500 \times (0.97)^t
\]
This equation uses 0.97 because 3% is being drained, meaning 97% of the water remains after each minute.

### Step 2: Table
We can now calculate the volume of water remaining in the pool at different times.

| \( t \) (minutes) | \( V(t) \) (gallons) |
|-------------------|----------------------|
| 0                 | 1500                 |
| 1                 | \( 1500 \times 0.97^1 = 1455 \) |
| 2                 | \( 1500 \times 0.97^2 = 1411.35 \) |
| 3                 | \( 1500 \times 0.97^3 = 1368.01 \) |
| 4                 | \( 1500 \times 0.97^4 = 1326.01 \) |
| 5                 | \( 1500 \times 0.97^5 = 1285.22 \) |

### Step 3: Domain
Since the pool will continue to drain over time, the domain for this function is:
\[
t \in [0, \infty)
\]
However, the practical domain would be from \( t = 0 \) until the pool is nearly empty, which would occur after several hours.

### Step 4: Graph
You can graph this equation using software like Desmos. The y-axis represents the gallons of water remaining in the pool, and the x-axis represents time in minutes. The curve will show a steady decline as time increases due to the exponential nature of the decay.

If you have Desmos available, you can input the equation \( V(t) = 1500 \times 0.97^t \) and plot the curve.

---

Would you like further details or clarifications?

Here are 5 related questions to extend your understanding:
1. What would the volume of the pool be after 10 minutes?
2. How long would it take for the pool to reach half its initial volume?
3. Can you describe the significance of the decay factor in an exponential decay function?
4. What other real-life scenarios could be modeled using exponential decay?
5. How does the percentage change in decay affect the steepness of the graph?

**Tip**: When modeling decay processes, always check whether the rate is additive or multiplicative (percentage-based), as this determines the type of function (linear vs exponential).

At the end of the summer, I decided to drain the 1,500-gallon swimming pool. I noticed that it drains faster when there is more water in the pool. I found that 3% was draining out of the pool every minute. Use a table, a graph, and an equation to create a mathematical model of the gallons of water in the pool at t minutes.

Learn how to model the drainage of a 1,500-gallon swimming pool using exponential decay, where 3% of the water drains per minute. This problem covers key mathematical concepts like exponential functions and provides a step-by-step solution, including a table and graph.

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