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The concentration of the drug \( C(t) \) in an organ is given by the following piecewise function:

\[
C(t) = 
\begin{cases} 
0.3t - 18(1 - e^{-t/60}), & 0 \leq t \leq 20 \\
18e^{-t/60} - 12e^{-(t - 20)/60}, & t > 20 
\end{cases}
\]
where \( C(t) \) is measured in mg/cm³ and \( t \) is time in seconds.

### Part (a): Initial concentration \( t = 0 \)
The problem states that \( C(0) = 0 \), which was correctly answered.

### Part (b): Concentration at \( t = 5 \) seconds
For \( t = 5 \), we use the first part of the piecewise function, as \( 0 \leq 5 \leq 20 \):

\[
C(5) = 0.3(5) - 18\left(1 - e^{-5/60}\right)
\]
Simplifying:
\[
C(5) = 1.5 - 18\left(1 - e^{-1/12}\right)
\]
Using \( e^{-1/12} \approx 0.920\), we get:
\[
C(5) \approx 1.5 - 18(1 - 0.920) = 1.5 - 18(0.080) = 1.5 - 1.44 = 0.060 \, \text{mg/cm}^3
\]
This answer is **correct**. It seems that the system marked it incorrect due to rounding or input error.

### Part (c): Concentration at \( t = 29 \) seconds
For \( t = 29 \), we use the second part of the piecewise function, as \( t > 20 \):

\[
C(29) = 18e^{-29/60} - 12e^{-(29 - 20)/60}
\]
Simplifying:
\[
C(29) = 18e^{-29/60} - 12e^{-9/60}
\]
Using \( e^{-29/60} \approx 0.625 \) and \( e^{-9/60} \approx 0.866 \):
\[
C(29) \approx 18(0.625) - 12(0.866) = 11.25 - 10.392 = 0.858 \, \text{mg/cm}^3
\]

So the correct concentration at \( t = 29 \) is approximately **0.858 mg/cm³**, not 3.778.

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Would you like more details on these calculations or have any questions? Here are five related questions to explore this concept further:

1. How does the drug concentration change as \( t \) approaches 20 seconds?
2. What is the rate of change of drug concentration at \( t = 15 \) seconds?
3. What would the concentration be at \( t = 50 \) seconds?
4. How would the concentration function change if the exponential decay rate was faster?
5. Can you determine the time when the drug concentration reaches half its maximum value?

**Tip:** Exponential decay processes, like drug absorption, are often modeled using functions involving \( e^{-t} \), which represents how quickly something decreases over time.

The concentration of a drug in an organ at any time t (in seconds) is given by: 
C(t) = { 0.3t - 18(1 - e^(-t/60)) for 0 <= t <= 20, 
18e^(-t/60) - 12e^(-(t - 20)/60) for t > 20 }, where C(t) is measured in mg/cm³. 
(a) What is the initial concentration of the drug in the organ? 
(b) What is the concentration of the drug after 5 sec? 
(c) What is the concentration of the drug after 29 sec?

Explore how to calculate drug concentration in an organ at different time intervals using a piecewise function involving exponential decay. The problem involves solving for initial concentration, concentration after 5 seconds, and concentration after 29 seconds with detailed calculations.

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