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$ billionlong dashafter 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$\text{'}$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$$ Seattle, Washington$,$ for $$$$70$ billion in $10$ years. How much would Bill have to invest today at $9$ percent compounded annually in order to purchase Seattle in $10$years$?$

c$.$Now assume Bill Gates only wants to invest half his net worth$$ today, $$$$13$billion$,$ in order to buy Seattle for $$$$70$ billion in $10$ years. What annual rate of return would he have to earn in order to complete his purchase in $10$years$?$

d$.$Instead of buying and running large$$ cities, Bill Gates is considering quitting the rigors of the business world and retiring to work on his golf game. To fund his$$ retirement, Bill would invest his $$$$20$ billion fortune in safe investments with an expected annual rate of return of $7$ percent. He also wants to make $45$ equal annual withdrawals from this retirement fund beginning a year from$$ today, running his retirement fund to$$ $$0$ at the end of $45$ years. How much can his annual withdrawal be in this$$ case?