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The question provides the concentration of a drug \( C(t) \) in an organ at any time \( t \) (in seconds) and asks for the concentration at specific time points. The formula is piecewise, with two different cases:

\[
C(t) = \begin{cases}
0.25t - 15(1 - e^{-t/60}) & \text{if } 0 \leq t \leq 20 \\
15e^{-t/60} - 10e^{-(t - 20)/60} & \text{if } t > 20
\end{cases}
\]
where \( C(t) \) is measured in mg/cm³.

### (a) Initial concentration (at \( t = 0 \)):
From the formula:
\[
C(0) = 0.25(0) - 15(1 - e^{0}) = 0 - 15(1 - 1) = 0 \, \text{mg/cm}^3
\]
So, the initial concentration is 0 mg/cm³, which is correct as marked.

### (b) Concentration at \( t = 10 \) seconds:
Since \( 10 \leq 20 \), we use the first part of the function:
\[
C(10) = 0.25(10) - 15(1 - e^{-10/60}) 
\]
\[
C(10) = 2.5 - 15(1 - e^{-1/6})
\]
Calculate \( e^{-1/6} \approx 0.84648 \):
\[
C(10) = 2.5 - 15(1 - 0.84648) = 2.5 - 15(0.15352) 
\]
\[
C(10) \approx 2.5 - 2.3028 = 0.1972 \, \text{mg/cm}^3
\]
Rounded to three decimal places, \( C(10) \approx 0.197 \, \text{mg/cm}^3 \).

Your answer of 0.19 is close but should have been rounded to 0.197.

### (c) Concentration at \( t = 35 \) seconds:
Since \( t = 35 > 20 \), we use the second part of the function:
\[
C(35) = 15e^{-35/60} - 10e^{-(35 - 20)/60} 
\]
\[
C(35) = 15e^{-35/60} - 10e^{-15/60}
\]
Approximate the values of the exponentials:
\[
e^{-35/60} \approx 0.58566, \quad e^{-15/60} \approx 0.77979
\]
Now substitute:
\[
C(35) = 15(0.58566) - 10(0.77979) 
\]
\[
C(35) \approx 8.7849 - 7.7979 = 0.987 \, \text{mg/cm}^3
\]
Rounded to three decimal places, \( C(35) \approx 0.987 \, \text{mg/cm}^3 \).

Your answer of 0.58 is incorrect. The correct concentration should be approximately 0.987 mg/cm³.

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Would you like further details or have any other questions?

Here are five related questions:
1. How does the concentration change as \( t \to \infty \)?
2. How would you find the time at which the concentration is maximum?
3. Can the concentration ever become negative based on this formula?
4. What happens to the concentration when \( t \) is slightly greater than 20?
5. How does the rate of change of the concentration behave at \( t = 0 \)?

**Tip**: When solving for values involving exponentials, ensure that you round correctly at each step, especially when asked to provide answers to specific decimal places.

The concentration of a drug in an organ at any time t (in seconds) is given by the function C(t) with different expressions for 0 ≤ t ≤ 20 and t > 20. The task is to calculate the drug concentration at specific time points (initial, at 10 seconds, and at 35 seconds).

Learn how to calculate drug concentration in an organ over time using piecewise exponential functions. This problem involves algebraic manipulations and properties of exponential decay. Suitable for undergraduate math and science students.

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