A swimming pool is being filled with water from a hose. At 11:00am, the pool had 500 liters of water. By 11:45am, the pool had 675 liters of water.

a) Determine the rule linking the amount of water in the pool,
𝑤
w, and the number of minutes since 11:00am,
𝑛
n.

To determine the rule linking the amount of water in the pool \( w \) and the number of minutes since 11:00am \( n \), we need to find the rate at which water is being added to the pool and then express \( w \) as a function of \( n \).

### Step 1: Determine the rate of change
From the problem, we know the following:

- At 11:00am (\( n = 0 \) minutes), the pool has 500 liters of water (\( w = 500 \) liters).
- At 11:45am (\( n = 45 \) minutes), the pool has 675 liters of water (\( w = 675 \) liters).

The change in water, \( \Delta w \), over the change in time, \( \Delta n \), gives us the rate at which water is being added to the pool.

\[
\Delta w = 675 \text{ liters} - 500 \text{ liters} = 175 \text{ liters}
\]
\[
\Delta n = 45 \text{ minutes} - 0 \text{ minutes} = 45 \text{ minutes}
\]

Now, calculate the rate of change (rate of water flow):

\[
\text{Rate of flow} = \frac{\Delta w}{\Delta n} = \frac{175 \text{ liters}}{45 \text{ minutes}} \approx 3.89 \text{ liters/minute}
\]

### Step 2: Determine the equation
The amount of water in the pool \( w \) after \( n \) minutes can be expressed as:

\[
w = \text{Initial amount of water} + (\text{Rate of flow}) \times n
\]

Substituting the given values:

\[
w = 500 + 3.89n
\]

Thus, the rule linking the amount of water in the pool \( w \) and the number of minutes since 11:00am \( n \) is:

\[
\boxed{w = 500 + 3.89n}
\]

This equation means that the pool started with 500 liters of water, and every minute, approximately 3.89 liters of water are added to the pool.

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Would you like more details or have any questions?

Here are some related questions:
1. What will be the amount of water in the pool after 90 minutes?
2. At what time will the pool reach 1000 liters of water?
3. If the rate of water flow increases, how would the equation change?
4. How long would it take to fill the pool to 2000 liters starting from 500 liters?
5. What if the pool started with 300 liters at 11:00am? How would the equation change?

**Tip:** When working with linear functions, always identify the initial value and the rate of change; they define the equation of the line.

Explore how to determine the rule linking the amount of water in a swimming pool over time from 11:00am. Understand the concept of linear equations and rate of change through a step-by-step solution. Suitable for Grades 7-9.

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