following savings accounts, A and B$.$ For both accounts, interest is calculated and paid daily $($i$.$e$.$

$k=365.$

Account A: Interest is $6\%$ AER. If you withdraw before $180$ days you pay a penalty sum

equivalent to the first $90$ days of interest. If you withdraw on the $181$st day, or later, there is

no penalty.

Account B: Interest is $8\%$ APR. Whenever you withdraw the money, you pay a penalty sum

equivalent to the first $180$ days of interest.

Assume that the first $1,000$ of interest is tax free, and any interest earned over this is taxed at

$20\%.$

$($a$)$ Explain the difference between AER $($annual equivalent rate$)$ and APR $($annual

percentage rate$),$ and express the APR as a function of the AER.

$($b$)$ If you put $20,000$ into account $B,$ show that you would have less than $19,500$ if you

took it all out after exactly $60$ days.

$($c$)$ If you put $20,000$ into account $A,$ calculate how much money you would have if you

took it all out after exactly $181$ days.

Now assume you decide to keep an amount of your money, $x,$ in bank account $A($so that you

also keep amount $20000-x$ in bank account B$).$

$($d$)$ Write down, $G(x),$ your Gross $($pre$-$tax$)$ income you have after exactly $365$ days, as a

function of $x.$

$($e$)$ Write down, $N(x),$ your Net $($post$-$tax$)$ income after exactly $365$ days, as a function of

$x.$

$($f$),$SketchN$(x)$

Now suppose there is a third savings account available, C$,($interest is also calculated and paid

daily$)$:

Account C: Interest is $5\%$ APR. You can take money out of this account whenever you like

without any penalty.

$($g$)$ Suppose you leave the money in one of the three accounts for $d$ days. Carefully

explain in detail why $A$ is the best account iff din$[181,662].$