Mega Millions set a new record in August $2023$ when a single ticket sold in Florida won

$$1\mathrm{.}602$ billion jackpot. Every Mega Millions and Powerball jackpot winner can choose

one of two ways to receive their prize: a lump sum or an annuity. Suppose you are

lucky enough to match all six numbers and you can choose to have your Mega Millions

winnings distributed in one of two different ways:

Option $A-$ $$5$ million today, $$5$ million per year for nineteen years $($years $1-19),$

$$10$ million per year for nineteen years $($years $20-38),$ and $$710$ million in year $39.$

The annual payments are made at the end of each year.

Option B $-$ $$250$ million today.

$($a$)$ What is the present value of Option A assuming an annual discount rate of $4\%?$

Which option should you choose?

$($b$)$ If your annual discount rate is instead $5\%,$ should you choose Option A or Option

B $?$