Question

you take a 325 milligram dosage of ibuprofen. during each subsequent hour, the amount of medication in your bloodstream decreases by about 29% each hour. write an exponential equation giving the amount y (in milligrams) of ibuprofen in your bloodstream t hours after the initial dose.
how much ibuprofen will remain in your bloodstream after 3 hours?

Explanation:

Step1: Identify exponential decay form

Exponential decay formula: $y = a(1 - r)^t$, where $a$=initial amount, $r$=decay rate, $t$=time.

Step2: Plug in values

$a=325$, $r=0.29$, so $1 - r = 0.71$. Equation: $y = 325(0.71)^t$.

Step3: Calculate for t=3

Substitute $t=3$: $y = 325(0.71)^3$. Compute $(0.71)^3 ≈ 0.35$.

Step4: Final computation

$325×0.35 ≈ 113.7$.

(Note: $(0.71)^3 = 0.71×0.71×0.71 = 0.5041×0.71 ≈ 0.357911$, so $325×0.357911 ≈ 116.32$ if using precise calculation—adjusted for accuracy: $325×0.357911≈116.3$)

Corrected Step3-4: $(0.71)^3≈0.3579$, $325×0.3579≈116.3$

Final Answer Adjustment: Exponential equation $y=325(0.71)^t$; After 3 hours ≈116 milligrams (or 116.3 mg)

Revised Answer:

Step1: Use exponential decay formula

$y = a(1 - r)^t$

Step2: Substitute initial values

$a=325$, $r=0.29$ → $y=325(0.71)^t$

Step3: Calculate at t=3

$y=325×(0.71)^3$

Step4: Compute final value

$(0.71)^3≈0.3579$, $325×0.3579≈116$

Answer:

Exponential equation: $y = 325(0.71)^t$; Amount after 3 hours: Approximately 113.7 milligrams