4) Each day, 10% of a certain drug dissipates from the system.

Question

A) Classify the function representing this situation as either exponential growth or decay, and identify the growth or decay factor. Then graph the function. (Let a be any number. I suggest 100 mg) 

B) How much of the original amount remains in the system after 9 days? (The answer should be in a percentage)  

C) If a second dose should not be taken if more than 50% of the original amount is in the system, when should the label say it is safe to redose? Design the label and explain your reasoning. 

 

Answer 1

If something dissipates that mean it spreads out or becomes less concentrated and since it is referring to a drug that has a concentrated dose of something that means that when this drug dissipates in the body it becomes less concentrated so this function is an exponential decay function. Because of this the exponential decay function is this 100(0.9)^x where (0.9) is the decay factor because (100-10)->(10-1)->(1-0.1)=0.9

So then the concentration of the drug after 9 days, where x is the number of days, will be modelled by 100(0.9)^9=38.7 so 38.7% of the drug remains after 9 days. If a second dose cannot be administered if 50% of the drug remains then the label warning for redose should say the number of days someone should wait to take a second dose and since it was found that 38.7%if the drug is left after 9 days we must find the number of days someone should wait for redose so lets choose a number less than 9 and see what happens. 100(0.9)^7=47.8% & 100(0.9)^6=53.1% so the label should say "Wait 7 days to take a second dose"

Answer 2

(a) Every day we have 90% of drug left from previous day, so exponential decay, decay factor 0.9

If you start with 100 mg then y = 100 (0.9)^x where x is the number of days

(b) Well if we start with a mg, will be a (0.9)⁹

Percentage left = [ a (0.9)⁹ / a] x 100 = 100 (0.9)⁹ = 38.7 %

(c) When do we have 50% left, well

if we start with 100 mg, then if after t days

50 = 100 (0.9)^t

0.5 = 0.9^t

^{ln 0.5 = t ln 0.9}

t = ln 0.5 / ln 0.9 = 6.58

So safe to redose after 7 days