Question $10$

$($a$)$ The concentration of caffeine in the blood plasma of an adult $t$ hours

after they have consumed a $150$ milligram dose of caffeine can be

modelled by the function: $c(t)=15e^{-0\mathrm{.}3t}-15e^{-0\mathrm{.}6t},$ for $t0,$ where

the concentration, $c(t),$ is measured in milligrams per litre $(\frac{mg}{l}).$

Show that the concentration of caffiene in an adult's blood

plasma reaches its maximum at $t=\frac{10ln2}{3}$ hours.

$($b$)$ On the day Paul's grandson Ollie was born,Paul deposited $5000$

into an account earning $3\%$ per annum compounded annually. On each

birthday after this he deposited $1000$ into the same account, making

his final deposit on Ollie's $17$ th birthday. That is$,$ a total of $18$ deposits

were made. Let $A_n$ be the amount in the account on Ollie's nth birthday

after the deposit $n$ is made. Show that $A_3=8554\mathrm{.}54$

$($c$)$ Find the value of Ollie's account on his $17$ th birthday, just after

the final deposit is made.