Question

A particular pain relieving medicine has a decay rate of 22% per hour. A patient was given a dose of the medicine 3 hours ago and there is currently 96 milligrams of the medicine in the patients bloodstream. After the dose was given how long must the patient wait for there to be less than 25% of the original dose of the medicine left in his or her bloodstream?

Answer 1

let when the dose was given, x miligrams of the medicine was present.

the medicine has a decay rate of 22% per hour. Hence it is reducing in compound rate.

so after 3 hours the amount of medicine present will be x(1-22/100)³

As mentioned in the question, that amount is 96 miligrams

so 96=x(78/100)³

or, x=96*100*100*100/(78*78*78)=202.296 miligrams

Now let the patient must wait t hours of time for there to be at most 25% of the original does of the medicine left in his or her bloodstream. Then again by compound rate formula

202.296*25/100=202.296(78/100)^t

or, 0.25=(78/100)^t

or, ln0.25=t*ln0.78

or, t=ln0.25/ln0.78=5.5795~6

so the patient must wait for at least 6 hours for there to be less than 25% of the original dose of the medicine left in his or her bloodstream [answer]