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The task involves determining the constants \(a\) and \(b\) in the exponential decay model of the form:

\[
f(x) = b \cdot a^x
\]

where \(f(x)\) is the concentration of a toxic substance in parts per million (ppm) after \(x\) days. The table provides the concentration at various time points.

### Step-by-Step Approach to Regression:

1. **Logarithmic Transformation**:
   Since the function is exponential, it’s helpful to linearize it by taking the natural logarithm of both sides:
   \[
   \ln(f(x)) = \ln(b \cdot a^x) = \ln(b) + x \cdot \ln(a)
   \]
   This transformation changes the problem into a linear regression problem where:
   \[
   y = \ln(f(x)), \quad \text{slope} = \ln(a), \quad \text{intercept} = \ln(b)
   \]

2. **Apply Linear Regression**:
   You can now apply linear regression to the transformed data \( (x, \ln(f(x))) \), where:
   - \( x \) is the time (in days).
   - \( \ln(f(x)) \) is the natural logarithm of the concentrations.

   By fitting a linear model \( \ln(f(x)) = mx + c \), you can extract:
   - \( m = \ln(a) \), so \( a = e^m \)
   - \( c = \ln(b) \), so \( b = e^c \)

   From this, you will get the values of \( a \) and \( b \).

### Using the Data:

- Time \(x = 0, 4, 8, 12, 16, 20, 24, 28\)
- Concentration \(f(x) = 278, 206, 138, 110, 81, 58, 49, 32\)

### Tools for Regression:
- You can use any statistical software (like Excel, Python, or a graphing calculator) to perform the regression analysis on the given data.

Once the regression is done, you will have the values of \( a \) and \( b \) that best fit the data.

---

**Expanding Questions:**
1. What would happen to the concentration if the value of \( a \) were greater than 1 instead of less than 1?
2. How would you use the model to predict the concentration at 32 days?
3. Could this model be used indefinitely, or would there be a point where the concentration stops decreasing?
4. What is the physical interpretation of the value of \( b \) in this context?
5. How would the model change if the decay were not exponential but followed a different form, such as a logarithmic decay?

**Tip:** Exponential decay models are common in biology and environmental science because many processes decrease at a constant proportional rate over time, such as the breakdown of toxins.

En biolog måler jævnligt koncentrationen af et bestemt giftstof i en forurenet sø. Resultaterne fremgår af nedenstående tabel. Bestem de konstante værdier a og b ved hjælp af regression.

Learn how to determine the constants in an exponential decay model for toxic substance concentration using regression analysis. This problem involves transforming the data to a linear form to find values of a and b, suitable for Grade 10-12 students.

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