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Let's analyze the given problem step by step.

### 1. Function Analysis:
The concentration \( c(t) \) of the drug in the bloodstream as a function of time \( t \) is given by:
\[
c(t) = \frac{20t}{t^2 + 2}, \quad t \geq 0
\]

#### a) **Intercepts:**
- **y-intercept** occurs when \( t = 0 \):
  \[
  c(0) = \frac{20(0)}{0^2 + 2} = 0
  \]
  So, the graph passes through the origin \( (0, 0) \).

- **x-intercept** occurs when \( c(t) = 0 \). Since the numerator is zero when \( t = 0 \), the only x-intercept is also at \( t = 0 \).

#### b) **Asymptotes:**
- **Horizontal asymptote**: 
  As \( t \to \infty \), we analyze the behavior of the function:
  \[
  \lim_{t \to \infty} \frac{20t}{t^2 + 2} = \lim_{t \to \infty} \frac{20}{t + \frac{2}{t}} = 0
  \]
  So, there is a horizontal asymptote at \( y = 0 \).

- **Vertical asymptote**: 
  The denominator \( t^2 + 2 \) never equals zero for any real value of \( t \), so there is no vertical asymptote.

#### c) **Critical Points** (maxima/minima):
To find the critical points, we take the derivative of \( c(t) \) and set it equal to zero:
\[
c'(t) = \frac{d}{dt} \left( \frac{20t}{t^2 + 2} \right)
\]
Using the quotient rule:
\[
c'(t) = \frac{(t^2 + 2)(20) - 20t(2t)}{(t^2 + 2)^2} = \frac{20(t^2 + 2 - 2t^2)}{(t^2 + 2)^2} = \frac{20(2 - t^2)}{(t^2 + 2)^2}
\]
Setting the numerator equal to zero to find the critical points:
\[
20(2 - t^2) = 0 \quad \Rightarrow \quad t^2 = 2 \quad \Rightarrow \quad t = \sqrt{2}
\]
At \( t = \sqrt{2} \), we have a maximum.

#### d) **Behavior**:
- As \( t \) increases from 0 to \( \sqrt{2} \), the concentration increases.
- After \( t = \sqrt{2} \), the concentration decreases towards zero as time progresses.

### 2. Practical Interpretation:
The graph suggests that after the drug is injected, its concentration in the bloodstream initially increases, reaching a peak concentration at around \( t = \sqrt{2} \) minutes (approximately 1.41 minutes). After this point, the drug's concentration starts to decrease over time as it is metabolized or eliminated from the bloodstream.

### Sketch of the graph:
- The graph starts at \( (0,0) \) and increases to a peak at \( t = \sqrt{2} \).
- Then, the graph decreases and approaches \( y = 0 \) as \( t \) tends to infinity.

Would you like to see a graph sketch of the function, or do you have any further questions?

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Here are 5 related questions to think about:
1. How would the graph change if the constant 20 in the numerator was increased or decreased?
2. What happens to the concentration if the denominator included a higher degree term like \( t^4 \)?
3. How does the concentration's peak change if the formula for \( c(t) \) changes to \( \frac{a t}{t^2 + b} \)?
4. Can you find the exact time when the drug's concentration falls to half its peak value?
5. What does the behavior of the function as \( t \to \infty \) imply about the long-term effects of the drug?

**Tip:** When interpreting the practical meaning of a graph, always relate the function's behavior (like intercepts and asymptotes) to the real-world context of the problem.

After a drug is injected into a patient’s bloodstream, the concentration c of the drug in the bloodstream t minutes after the injection is given by c(t) = (20t) / (t^2 + 2), t >= 0. Sketch the graph of c(t) and include the intercepts and the asymptotes. What does the graph imply in practical terms?

Explore how to graph the concentration of a drug in a patient's bloodstream over time using calculus. The problem involves calculating intercepts, asymptotes, and finding critical points through derivatives, providing key insights into how the drug's concentration changes. Suitable for students in Grades 11-12.

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