Let's break down the problem and solve it step by step.

Given Data:

The biological model of toxin concentration over time is given by the formula: $f(x) = b \cdot a^x$ where:

• $f(x)$ is the concentration in ppm at time $x$ (measured in days).
• $b$ and $a$ are constants we need to determine.

The data table provides time $x$ (in days after the first measurement) and corresponding concentrations in ppm:

Part (a) - Determine $a$ and $b$ using regression:

We need to fit the exponential model $f(x) = b \cdot a^x$ to the given data. Taking the natural logarithm of both sides will linearize the equation: $\ln(f(x)) = \ln(b) + x \cdot \ln(a)$ So we can perform a linear regression on $\ln(f(x))$ against $x$.

Let's proceed to the regression analysis.

Part (b) - Time when concentration is under 5 ppm:

We are asked to determine when the concentration drops below 5 ppm. From the model $f(x) = b \cdot a^x$, we set $f(x) = 5$ and solve for $x$: $5 = b \cdot a^x$ $x = \frac{\ln(5) - \ln(b)}{\ln(a)}$

Part (c) - Half-life of the toxin:

The half-life $T_{1/2}$ is the time it takes for the concentration to be halved. So, we need to solve for $T_{1/2}$ from: $\frac{f(0)}{2} = f(T_{1/2}) = b \cdot a^{T_{1/2}}$ Let me calculate these values based on the data provided and get back to you with the results.From the regression analysis, the values of $b$ and $a$ in the model $f(x) = b \cdot a^x$ are:

• $b \approx 276.62$
• $a \approx 0.925$

Next steps:

Now, let's move to part (b) to calculate the time when the concentration drops below 5 ppm, and then determine the half-life for part (c).

I'll calculate these values now.### Results:

• Part (b): The concentration will drop below 5 ppm approximately 51.60 days after the first measurement.
• Part (c): The half-life of the toxin in the lake is approximately 8.91 days.

Would you like more details on any of the calculations? Here are a few related questions:

1. How is the regression model determined for exponential decay?
2. How can the half-life formula be derived from the model?
3. What is the impact of small variations in $a$ on the toxin decay rate?
4. How accurate are the regression predictions compared to the actual data?
5. What are the assumptions behind using an exponential decay model?

Tip: When dealing with exponential decay, small changes in the base $a$ can significantly impact the time predictions for reaching specific concentrations.