Assume you are $35$ years old today and are considering your retirement needs. You expect

to retire at age $65($in $30$ years$)$ and you plan to live to age $99.$ You want to buy a house

costing $$300,000$ on your $65$th birthday and your living expenses will be $$30,000$ a year

after that $($starting at the end of year $65$ and continuing through the end of year $99,$ i$.$e$.,$ for

$35$ years$).$ Assume an interest rate of $8\%,$ annual compounding:

$($i$)$ How much will you need to have saved by your retirement date to be able to afford this

retirement plan?

$($ii$)$ Suppose you already have $$50,000$ in savings today. If you can invest money at $8\%$ a

year annual compounding, how much would you need to save at the end of each year for

the next $30$ years to be able to afford this retirement plan?