You are trying to decide how much to save for retirement. Assume you plan to save $$5,000$ per year with the first investment made one year from now. You think you can earn $9\%$ per year on your investments and you plan to retire in $42$ years, immediately after making your last $$5,000$ investment.

a$.$ How much will you have in your retirement account on the day you retire? to be$?$

c$.$ If you hope to live for $27$ years in retirement, how much can you withdraw every year in retirement $($starting one year after retirement$)$ so that you will just exhaust your savings with the $27$ th withdrawal $($assume your savings will continue to earn $9\%$ in retirement$)?$

d$.$ If$,$ instead, you decide to withdraw $$200,000$ per year in retirement $($again with the first withdrawal one year after retiring$),$ how many years will it take until you exhaust your savings?

e$.$ Assuming the most you can afford to save is $$1,000$ per year, but you want to retire with $$1,000,000\mathrm{.}00$ in your investment account, how high of a return do you need to earn on your investments?