Bruce added some chlorine to the water in a pool. The chlorine evaporated at a fixed rate every week. The table below shows the amount of chlorine f(n), in ounces, that was left in the pool after n weeks:

n	f(n)
1	54
2	27
3	13.5
4	6.75


Which function best shows the relationship between n and f(n)? (1 point)

f(n) = 108(0.5)n − 1

f(n) = 54(0.5)n

f(n) = 108(0.5)n + 1

f(n) = 54(0.5)n − 1

To find the function that best represents the relationship between \( n \) (the number of weeks) and \( f(n) \) (the amount of chlorine left in the pool in ounces), let's analyze the given data in the table and identify the type of function.

### Step-by-Step Analysis:

1. **Identify the Pattern in Data:**
   - **n = 1, f(n) = 54**
   - **n = 2, f(n) = 27**
   - **n = 3, f(n) = 13.5**
   - **n = 4, f(n) = 6.75**

   Observing the values, we notice that each week, the amount of chlorine is halved. For example:
   - From 54 to 27 (dividing by 2)
   - From 27 to 13.5 (dividing by 2)
   - From 13.5 to 6.75 (dividing by 2)

   This indicates an **exponential decay** with a common ratio of \( 0.5 \).

2. **Identify the Form of the Function:**
   Given that the chlorine evaporates at a fixed rate every week, the relationship can be modeled by an exponential function of the form:
   \[
   f(n) = a \cdot (r)^n
   \]
   where:
   - \( a \) is the initial amount of chlorine,
   - \( r \) is the decay factor (rate of decay per week).

3. **Determine Initial Amount and Decay Factor:**
   - The initial amount of chlorine after 1 week, \( f(1) = 54 \).
   - Given the halving pattern, the decay factor \( r = 0.5 \).

   If \( n = 1 \) corresponds to the first week, then the initial amount before any evaporation (when \( n = 0 \)) can be inferred to be twice \( f(1) \), i.e., \( 108 \) ounces (since \( 54 \) ounces is half of \( 108 \)).

4. **Choose the Correct Function:**
   To match the given choices, convert the correct model:
   
   - Starting from the relationship \( f(n) = 108 \cdot (0.5)^n \), if \( n \) starts from 1, then the model should be adjusted for \( n - 1 \):

   \[
   f(n) = 108 \cdot (0.5)^{n-1}
   \]

5. **Compare with the Given Choices:**
   The option \( f(n) = 108(0.5)^{n-1} \) matches our derived function, but since the choices provided use \( f(n) = 54(0.5)^n \), consider the equivalency:

   - Starting from \( n = 1 \), use the amount at the first week (54 ounces) directly:

   \[
   f(n) = 54 \cdot (0.5)^{n-1} \text{ becomes } f(n) = 54 \cdot (0.5)^n \text{ when considering the start point n = 1}
   \]

   **Thus, the correct function that shows the relationship is:**

   \[
   \boxed{f(n) = 54(0.5)^n}
   \]

### Conclusion:
The function \( f(n) = 54(0.5)^n \) best represents the chlorine decay over weeks, given the data provided.

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the rate of decay in an exponential function?
2. Can you explain how to convert between different forms of exponential functions?
3. What would the chlorine amount be after 5 weeks using the chosen function?
4. How would the function change if the chlorine decreased by 25% instead of 50% each week?
5. How can you derive the initial amount of a substance in a decay problem given subsequent amounts?

### Tip:
When working with exponential functions, always check for a consistent multiplicative pattern in the data, which indicates growth or decay.

Explore how to model the decay of chlorine levels in a pool over weeks using exponential functions. Learn to identify the decay rate and initial amount based on given data points. Suitable for high school students studying exponential functions and decay processes.

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