Math Problem Statement
Solution
The question provides the concentration of a drug in an organ at any time (in seconds) and asks for the concentration at specific time points. The formula is piecewise, with two different 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$$. Your answer of 0.58 is incorrect. The correct concentration should be approximately 0.987 mg/cm³. --- Would you like further details or have any other questions? Here are five related questions: 1. How does the concentration change as $$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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Exponential decay
Algebra
Formulas
C(t) = 0.25t - 15(1 - e^{-t/60}) for 0 ≤ t ≤ 20
C(t) = 15e^{-t/60} - 10e^{-(t - 20)/60} for t > 20
Theorems
Properties of exponential decay
Suitable Grade Level
Undergraduate (Math/Science)
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