The concentration of a drug in an organ at any time t (in seconds) is given by the function below where C(t) is measured in grams per cubic centimeter (g/cm3). (Round your answers to 3 decimal places.)
C(t) = {(0.4 t - 24\(1 - e^(-t/60)\) text( if ) 0<=t<=20, 24 e^(-t/60) - 16 e^(-(t-20)/60) text( if ) t>20)


(b) What is the concentration of the drug in the organ after 36 sec?



The concentration of the drug is given as a piecewise function:

\[
C(t) = 
\begin{cases} 
0.4t - 24(1 - e^{-t/60}) & \text{if } 0 \leq t \leq 20 \\
24e^{-t/60} - 16e^{-(t-20)/60} & \text{if } t > 20 
\end{cases}
\]

To find the concentration at \( t = 36 \) seconds, we note that \( t = 36 \) is in the second piece of the piecewise function (since \( t > 20 \)).

The relevant formula is:
\[
C(t) = 24e^{-t/60} - 16e^{-(t-20)/60}
\]

Substituting \( t = 36 \) into the equation:

\[
C(36) = 24e^{-36/60} - 16e^{-(36-20)/60}
\]

Now we will calculate this step by step.

The concentration of the drug in the organ after 36 seconds is approximately \( 0.917 \, \text{g/cm}^3 \).

Would you like more details on the steps, or have any further questions?

Here are 5 related questions to consider:
1. What would the concentration be after 50 seconds?
2. How would you derive the formula for the maximum concentration?
3. What is the rate of change of concentration at \( t = 36 \) seconds?
4. How does the concentration behave as \( t \to \infty \)?
5. How would the function change if the decay rate of the drug were faster?

**Tip:** When solving piecewise functions, always ensure the correct interval is used before applying the formula.

The concentration of a drug in an organ at any time t (in seconds) is given by the function below where C(t) is measured in grams per cubic centimeter (g/cm^3). (Round your answers to 3 decimal places.) C(t) = {(0.4t - 24(1 - e^(-t/60)) if 0 <= t <= 20, 24e^(-t/60) - 16e^(-(t-20)/60) if t > 20} (b) What is the concentration of the drug in the organ after 36 sec?

Learn how to evaluate the concentration of a drug in an organ using a piecewise function at 36 seconds. This solution involves concepts of exponential decay and calculus, suitable for Grades 11-12 and college-level students.

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