Math Problem Statement
The drug concentration in the blood-stream t hours after injection is given approximately by
C(t) = 5e-0.3t
where C(t) is concentration in milligrams per milliliter.
What is the rate of change of concentration after 1 hour? After 5 hours?
Solution
To find the rate of change of concentration, we need to compute the derivative of with respect to :
The derivative is:
Now, let's evaluate at and :
-
At :
-
At :
Thus, the rate of change of concentration is approximately:
- After 1 hour:
- After 5 hours:
Would you like more details on this calculation? Here are some related questions you might consider:
- How does the concentration change at ?
- What does the negative rate of change indicate about the drug concentration?
- How would the rate of change be affected if the exponent changed?
- Can you graph the concentration over time?
- What is the significance of the constant in the equation?
Tip: Understanding derivatives helps in analyzing how quantities change over time in real-world applications!
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Exponential Functions
Formulas
C(t) = 5e^(-0.3t)
C'(t) = -1.5e^(-0.3t)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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