$($Comprehensive problem$)$ Over the past few years, Microsoft founder Bill Gates' net worth has fluctuated between $$20$ billion and $$130$ billion In early $2006,$ it 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 $$24$ in $1626$ Now, $387$ years later in $2013,$ Bill Gates wants to buy the island from the "current natives $"$ How much would Bill have to pay for Manhattan if the "current natives" want a $6$ percent annual return on the original $$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 $10$ years. How much would Bill have to invest today at $10$ percent compounded annually in order to purchase Seattle in $10$ years? purchase in $10$ 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 retirement fund beginning a year from today, running his retirement fund to $$0$ at the end of $40$ 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 $6$ 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 $10$ percent compounded annually in order to purchase Seattle for $$50$

c$.$ If Bill Gates only wants to invest half his net worth today, $$13$ billion, in order to buy Seattle for $$50$ billion in $10$ years, the annual rate of re years is $\%($Round to two decimal places$)$

d$.$ If Bill invests his $$20$ billion fortune in safe investments with an expected annual rate of return of $7$ percent and he also wants to make $40$ equal annual withdrawals from this retirement fund beginning a from today, the amount of his annual withdrawal in this case would be $$$ billion per year $($Round to two decimal places$)$