Assume that you just won the state lottery. Your prize can be taken either in the form of $$$$65,000$ at the end of each of the next $30$ years$($that is$,$$$1,950,000$ over $30$years$)$ or as a single amount of $$$$1,049,000$ paid immediately.$$

a$.$ If you expect to earn $9\%$ annually on your investments over the next $30$years$,$ ignoring taxes and other$$ considerations, which alternative should you$$ take? Why?

b$.$ Would your decision in part a change if you could earn $11\%$ rather than $9\%$ on your investments over the next $30$years$?$ Why?

c$.$ At approximately what interest rate would you be indifferent between the two$$ options?