Modeling with Mathematics: Exponential Decay of Ibuprofen

MODELING WITH MATHEMATICS: IBUPROFEN EXPONENTIAL DECAY

Exponential decay refers to the process where a quantity decreases at a rate proportional to its current value.
Commonly encountered in pharmacokinetics, where the concentration of a drug in the bloodstream decreases over time.

Dosage of Medication: 325 milligrams of ibuprofen is initially administered.
Decay Rate: The amount of medication in the bloodstream decreases by approximately 29% each hour.

Objective: Formulate an exponential decay model to represent the amount of ibuprofen remaining in the bloodstream over time.
Model Derivation:
- Let ( y ) denote the amount of ibuprofen in milligrams.
- The initial amount, ( y_0 ), is set at 325 mg.
- The decay factor, which represents the remaining percentage after each hour, is given by:
- Remaining percentage after one hour = 100% - 29% = 71% = 0.71.
- Therefore, the decay model can be expressed as:
- [ y(t) = y_0 imes (0.71)^t ]
- where ( t ) is the time in hours after the initial dose.
- Substituting the initial amount, we have:
- [ y(t) = 325 imes (0.71)^t ]

Objective: Determine the time it takes for the amount of ibuprofen to decay to 100 mg.
Model Equation for Estimation:
- Set up the equation based on the decay model:
- [ 100 = 325 imes (0.71)^t ]
- Solving for ( t ):
- Divide both sides by 325:
  - [ (0.71)^t = \frac{100}{325} ]
  - [ (0.71)^t = 0.3076923 ]
- Take the natural logarithm on both sides:
  - [ \ln((0.71)^t) = \ln(0.3076923) ]
- Apply logarithm properties:
  - [ t \cdot \ln(0.71) = \ln(0.3076923) ]
- Solve for ( t ):
  - [ t = \frac{\ln(0.3076923)}{\ln(0.71)} ]
  - Using a calculator,
  - Calculate ( \ln(0.3076923) \approx -1.179581 )
  - Calculate ( \ln(0.71) \approx -0.342595 )
  - Thus,
  - [ t \approx \frac{-1.179581}{-0.342595} \approx 3.44 \text{ hours} ]
- Final Answer: Rounding to the nearest tenth, it takes approximately 3.4 hours for the ibuprofen level to decrease to 100 mg.