3. Congratulations, you just won the lottery! In one optionpresented to you, you will be paid one
million dollars a year for the next 25 years. You can depositthis money in an account that will earn
5% each year.
(a) Let M(t) be the amount of money in the account (measured inmillions of dollars) at time
t (measured in years). Set up a differential equation thatdescribes the rate of change in the
amount of money in the account. Two factors cause the amount togrow – first, you are
depositing one millon dollars per year and second, you areearning 5% interest.
(b) If there is no amount of money in the account when you openit, how much money will you
have in the account after 25 years?
(c) The second option presented to you is to take a lump sum of10 million dollars, which you
will deposit into a similar account. Set up a new initial valueproblem (that is, differential
equation with initial condition) to model this situation.
(d) How much money will you have in the account after 25 yearswith in this case?
(e) Do you prefer the first or second option? Explain yourthinking.
(f) At what time does the amount of money in the account underthe first option overtake the
amount of money in the account under the second option?
