$($Comprehensive problem$)$ Over the past few years, Microsoft founder Bill Gates' net worth has fluctuated between $$20$ billion and $$130$ billion. In early $2006$ was about $$26$ billion$-$after he reduced his stake in Microsoft from $21$ percent to around $14$ percent by moving billions into his charitable foundation. Let's see what Bill Gates can do with his money in the following problems.

a$.$ Manhattan's native tribe sold Manhattan Island to Peter Minuit for $$24$ in $1626.$ Now, $387$ years later in $2013,$ Bill Gates wants to buy the island from the "current natives." How much would Bill have to pay for Manhattan if the "current natives" want a $5$ percent annual return on the original $$24$ purchase price? b$.$ Bill Gates decides to pass on Manhattan and instead plans to buy the city of Seattle, Washington, for $$50$ billion in $11$ years. How much would Bill have to invest today at $11$ percent compounded annually in order to purchase Seattle in $11$ years?

c$.$ Now assume Bill Gates only wants to invest half his net worth today, $$13$ billion in order to buy Seattle for $$50$ billion in $11$ years. What annual rate of return would he have to earn in order to complete his purchase in $11$ years?

d$.$ Instead of buying and running large cities, Bill Gates is considering quitting the rigors of the business world and retiring to work on his golf game. To fund his retirement, Bill would invest his $$20$ billion fortune in safe investments with an expected annual rate of return of $8$ percent. He also wants to make $50$ equal annual withdrawals from this retirement fund beginning a year from today, running his retirement fund to $$0$ at the end of $50$ years. How much can his annual withdrawal be in this case?

a$.$ The amount Bill would have to pay for Manhattan if the "current natives" wanted a $5$ percent annual return on the original $$24$ purchase price after $387$ years is $ billion. $($Round to two decimal places.$)$

b$.$ The amount Bill would have to invest at $11$ percent compounded annually in order to purchase Seattle for $$50$ billion in $11$ years is $ billion. $($Round to two decimal places.$)$

c$.$ If Bill Gates only wants to invest half his net worth today, $$13$ billion, in order to buy Seattle for $$50$ billion in $11$ years, the annual rate of return he would have to earn in order to complete his purchase in $11$ years is $\%.($Round to two decimal places.$)$

d$.$ If Bill invests his $$20$ billion fortune in safe investments with an expected annual rate of return of $8$ percent and he also wants to make $50$ equal annual withdrawals from this retirement fund beginning a year from today, the amount of his annual withdrawal in this case would be $$$ billion per year. $($Round to two decimal places.$)$