Math Problem Statement
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?
Solution
The concentration of the drug is given as a piecewise function:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Piecewise Functions
Exponential Decay
Formulas
C(t) = 0.4t - 24(1 - e^(-t/60)) for 0 <= t <= 20
C(t) = 24e^(-t/60) - 16e^(-(t-20)/60) for t > 20
Theorems
Exponential Decay
Piecewise Function Evaluation
Suitable Grade Level
Grades 11-12, College Level
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