# A cup of coffee contains approximately 100 mg of caffiene. When we drink coffee, the caffiene is absorbed in the bloodstream and is eventually metabolised by the body. Every 5 hrs the amount of...

A cup of coffee contains approximately 100 mg of caffiene. When we drink coffee, the caffiene is absorbed in the bloodstream and is eventually metabolised by the body. Every 5 hrs the amount of caffiene in the bloodstream is reduced by 50%. How many hours does it take fo the amount of caffiene to be reduced to 10 mg?

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Since the caffeine is reduced by 50% every 5 hours, it may be modelled by an exponential half-life function.

`A(t)=A_0(1/2)^{t/k}` where `A(t)` is the amount after `t` hours, `A_0` is the initial amount, and `k` is the time for the amount to be reduced by half. Upon substitution we get

`10=100(1/2)^{t/5}` divide by 100 and simplify

`1/10=(1/2)^{t/5}` now to solve, take logarithms

`log(1/10)=log(1/2)^{t/5}` use power rule on right side

`log(1/10)=t/5log(1/2)` isolate t

`5log(1/10)/log(1/2)=t` simplify

`t={5log10}/{log2} approx 16.61`

**It will take approximately 16.61 hours to reduce the caffeine to 10 mg.**