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.