To find the maximum concentration of the drug, we are tasked with finding the maximum value of $C(t) = 8t e^{-0.2t}$. This involves calculus:

1. Find the derivative $C'(t)$: Using the product rule: $C'(t) = \frac{d}{dt}(8t) \cdot e^{-0.2t} + 8t \cdot \frac{d}{dt}(e^{-0.2t})$ $C'(t) = 8e^{-0.2t} - 1.6t \cdot 8e^{-0.2t}$ Factor out $8e^{-0.2t}$: $C'(t) = 8e^{-0.2t}(1 - 0.2t)$

2. Set $C'(t) = 0$: For $8e^{-0.2t}(1 - 0.2t) = 0$, note that $e^{-0.2t} \neq 0$, so: $1 - 0.2t = 0 \implies t = 5$

3. Verify it's a maximum: The second derivative or the behavior of $C'(t)$ around $t = 5$ confirms this is a maximum.

4. Find $C(5)$: Substitute $t = 5$ into $C(t)$: $C(5) = 8 \cdot 5 \cdot e^{-0.2 \cdot 5}$ $C(5) = 40 \cdot e^{-1}$ Since $e^{-1} \approx 0.3679$: $C(5) \approx 40 \cdot 0.3679 \approx 14.7$

The maximum concentration is approximately 14.7.

Would you like further clarification? Here are 5 related questions to explore:

1. How do we confirm if a critical point is a maximum or minimum?
2. What does the derivative $C'(t)$ represent in this context?
3. How does the exponential decay affect the concentration over time?
4. What are some practical uses of such models in pharmacology?
5. How can technology (like graphing tools) simplify finding maxima?

Tip: When dealing with exponential functions, logarithmic properties and approximations are often helpful for calculations.