A Dr. Prescribes 200 milligrams of a drug that decays by 14%/hr. To the nearest minute, what is the half life of the drug?
The formula for exponential decay is:
A = A0 * e^(-kt)
where A is the amount of the drug remaining after time t, A0 is the initial amount of the drug, k is the decay constant, and e is the mathematical constant approximately equal to 2.71828.
The half-life of the drug is the time it takes for half of the initial amount of the drug to decay. So, we can set A = 0.5A0 and solve for t:
0.5A0 = A0 * e^(-kt)
0.5 = e^(-kt)
ln(0.5) = -kt
t = ln(0.5) / (-k)
We are given that the drug decays by 14% per hour, so k = 0.14. Substituting this into the equation above, we get:
t = ln(0.5) / (-0.14) ≈ 4.95 hours
To convert this to minutes, we multiply by 60:
t ≈ 4.95 * 60 ≈ 297 minutes
Therefore, to the nearest minute, the half-life of the drug is 297 minutes.