$6.$ A young person with no initial capital invests $\backslash ($ k $\backslash )$ dollars per year at an annual rate of return $\backslash ($ r $\backslash ).$ Assume that investments are made continuously and that the return is compounded continuously.

a$.$ Determine the sum $\backslash ($ S$($t$)\backslash )$ accumulated at any time $\backslash ($ t $\backslash ).$

b$.$ If $\backslash ($ r$=7\mathrm{.}5\backslash \%\backslash ),$ determine $\backslash ($ k $\backslash )$ so that $\backslash (\backslash$$ $1\backslash )$ million will be available for retirement in $40$ years.

c$.$ If $\backslash ($ k$=\backslash$$ $2000/\backslash )$ year, determine the return rate $\backslash ($ r $\backslash )$ that must be obtained to have $\backslash (\backslash$$ $1\backslash )$ million available in $40$ years.