A serving of a popular cola soft drink has 46 mg of caffeine. Suppose the half-life for caffeine remaining in he body of a typical adult is 6 h. How do you write an exponential decay function that gives the amount of caffeine in the body after t hours?

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Answer 1 · 2017-11-27T19:35:59

See below.

We need an equation of the form:

#y(t)=ae^kt#

Where #a# is the initial amount, #k# is the growth/decay constant and #t# is the time, in this case hours.

To find the the constant #k#, we need to know the initial amount #a#

#a=46mg#

We know the half life is 6 hours, so after 6 hours the amount #y# will be #23mg#

So:

#y(t)=ae^kt#

#23=46e^6k#

Solve for #k#:

#23/46=e^(6k)#

Taking natural logs of both sides:

#ln(23/46)=6klnecolor(white)(88)# ( #lne=1# )

#k=1/6ln(23/46)#

So our equation is:

#y(t)=46e^(1/6ln(23/46)t)#

This could also be written:

#y(t)=46e^(t/6ln(23/46))#

TEST:

#t = 6#

#y(t)=46e^(1/6ln(23/46)6)=23#

Graph of #y(t)=46e^(1/6ln(23/46)t)#

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