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The concentration of the drug in the bloodstream is given by the function:

\[
C(t) = \frac{100t}{t^2 + 4}
\]

To find the time at which the concentration is at its maximum, we need to maximize this function. This can be done by finding the critical points of \(C(t)\), which requires us to differentiate \(C(t)\) with respect to \(t\), and then set the derivative equal to zero to solve for \(t\).

### Step 1: Differentiate \(C(t)\)
Using the quotient rule to differentiate \(C(t)\), we get:

\[
C'(t) = \frac{(t^2 + 4) \cdot 100 - 100t \cdot (2t)}{(t^2 + 4)^2}
\]

Simplifying the numerator:

\[
C'(t) = \frac{100(t^2 + 4) - 200t^2}{(t^2 + 4)^2}
\]
\[
C'(t) = \frac{100t^2 + 400 - 200t^2}{(t^2 + 4)^2}
\]
\[
C'(t) = \frac{-100t^2 + 400}{(t^2 + 4)^2}
\]

### Step 2: Set \(C'(t) = 0\)
To find the critical points, we set \(C'(t) = 0\):

\[
-100t^2 + 400 = 0
\]

Solve for \(t^2\):

\[
100t^2 = 400
\]
\[
t^2 = 4
\]

Taking the square root of both sides:

\[
t = 2 \quad \text{or} \quad t = -2
\]

Since time cannot be negative in this context, we discard \(t = -2\). Therefore, \(t = 2\) hours.

### Conclusion:
It takes **2 hours** for the drug to reach its maximum concentration.

Would you like further details on how the quotient rule was applied, or have any other questions?

Here are some related questions to deepen your understanding:

1. What is the significance of critical points in the context of this problem?
2. How do you confirm that \(t = 2\) is indeed a maximum and not a minimum or inflection point?
3. Could this approach be extended to other drug concentration models with different functions?
4. How would the concentration curve look like graphically, and how can we verify the maximum visually?
5. How does the behavior of the function \(C(t)\) change for large values of \(t\)?

**Tip:** Always check the second derivative or analyze the sign of the first derivative around critical points to confirm whether you have found a maximum or a minimum.

The concentration of a drug in the bloodstream C(t) at any time t, in hours, is given by the equation C(t) = 100t / (t^2 + 4). Determine how long it takes for the drug to reach its maximum concentration.

Learn how to find the time when the concentration of a drug in the bloodstream is at its maximum using calculus. The problem involves the function C(t) = 100t / (t^2 + 4) and uses optimization techniques, including differentiation and critical point analysis.

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