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 C(t)C(t) with respect to tt:

C(t)=5e0.3tC(t) = 5e^{-0.3t}

The derivative C(t)C'(t) is:

C(t)=5(0.3)e0.3t=1.5e0.3tC'(t) = 5 \cdot (-0.3) e^{-0.3t} = -1.5 e^{-0.3t}

Now, let's evaluate C(t)C'(t) at t=1t = 1 and t=5t = 5:

  1. At t=1t = 1: C(1)=1.5e0.311.5e0.31.50.74081.1112C'(1) = -1.5 e^{-0.3 \cdot 1} \approx -1.5 e^{-0.3} \approx -1.5 \cdot 0.7408 \approx -1.1112

  2. At t=5t = 5: C(5)=1.5e0.351.5e1.51.50.22310.3347C'(5) = -1.5 e^{-0.3 \cdot 5} \approx -1.5 e^{-1.5} \approx -1.5 \cdot 0.2231 \approx -0.3347

Thus, the rate of change of concentration is approximately:

  • After 1 hour: 1.11mg/mL/hr-1.11 \, \text{mg/mL/hr}
  • After 5 hours: 0.33mg/mL/hr-0.33 \, \text{mg/mL/hr}

Would you like more details on this calculation? Here are some related questions you might consider:

  1. How does the concentration change at t=0t = 0?
  2. What does the negative rate of change indicate about the drug concentration?
  3. How would the rate of change be affected if the exponent changed?
  4. Can you graph the concentration over time?
  5. What is the significance of the constant 55 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