A scientist models the concentration of a drug in the bloodstream as C(t) = 50 × e^(-0.2t), where t is in hours. What is the concentration after 4 hours? - Coaching Toolbox

Understanding the Context

This model reflects how drug levels decrease predictably over time after administration—slowing as the body metabolizes and eliminates the substance. In this article, we explore the significance of the formula and, focusing on a key question: What is the drug concentration after 4 hours?

Breaking Down the Model

• C(t) = Drug concentration (mg/L or similar units) at time t
  
• 50 = Initial concentration at t = 0 (maximum level immediately after dosing)
  
• 0.2 = Decay constant, indicating how rapidly the drug is cleared
  
• e^(-0.2t) = Exponential decay factor accounting for natural elimination

Because the function involves e^(-kt), it accurately captures the nonlinear, memory-driven nature of how substances clear from the body—showing rapid initial decline followed by slower change.

Key Insights

Calculating Concentration at t = 4 Hours

To find the concentration after 4 hours, substitute t = 4 into the equation:

C(4) = 50 × e^(-0.2 × 4)
C(4) = 50 × e^(-0.8)

Using an approximate value:
e^(-0.8) ≈ 0.4493 (via calculator or exponential tables)

C(4) ≈ 50 × 0.4493 ≈ 22.465 mg/L

Final Thoughts

Rounded to two decimal places, the concentration after 4 hours is approximately 22.47 mg/L.

Why This Model Matters in Medicine

This precise description of drug kinetics allows clinicians to:

• Predict therapeutic windows and optimal dosing intervals
  
• Minimize toxicity by avoiding excessive accumulation
  
• Personalize treatment based on elimination rates across patient populations

Moreover, such models guide drug development by simulating pharmacokinetics before costly clinical trials.

In summary, the equation C(t) = 50 × e^(-0.2t) is more than a formula—it’s a vital tool for precision medicine. After 4 hours, the bloodstream contains about 22.47 μg/mL of the drug, illustrating how rapidly concentrations fall into steady decline, a key insight in safe and effective drug use.