3 Maximize income A banker from Mathematical Monetization LLC ("Where your money is integral to our success") wants to hire you to work for them. You are offered a lucrative position with an interesting payment structure which allows you to invest your income into the bank throughout the month and to withdraw it at the end of the month. In particular, you get to choose a number p between 0 and 1 that determines your payment. The rules are as follows: 1. Your money at each point in time t throughout the month is M(t) where t is a number measured in months. That is, M(0) is your total money at the beginning of the month and M(1) is your total money at the end of the month which you get to take home. 2. You start with no money at the beginning of each month. In other words, M(0)=0. 3. Your money compounds continuously with the rate 1+p. That is, you are getting interest at a rate of (1+p)M(t) at each moment in time t. 4. You get a flat rate income of (1p)10,000 dollars per month. You are given this money continuously. For example, if you choose p=0, then you will take home 10,000 dollars each month since M(t)= 10,000t resulting in M(1)=10,000. What should you set p to be in order to maximize your income at the end of the month? With that value of p, what is your income? Hint: You may find it helpful to graph M(1) as a function of p. Note: You must fully justify your answer. 3 Maximize income A banker from Mathematical Monetization LLC ("Where your money is integral to our success") wants to hire you to work for them. You are offered a lucrative position with an interesting payment structure which allows you to invest your income into the bank throughout the month and to withdraw it at the end of the month. In particular, you get to choose a number p between 0 and 1 that determines your payment. The rules are as follows: 1. Your money at each point in time t throughout the month is M(t) where t is a number measured in months. That is, M(0) is your total money at the beginning of the month and M(1) is your total money at the end of the month which you get to take home. 2. You start with no money at the beginning of each month. In other words, M(0)=0. 3. Your money compounds continuously with the rate 1+p. That is, you are getting interest at a rate of (1+p)M(t) at each moment in time t. 4. You get a flat rate income of (1p)10,000 dollars per month. You are given this money continuously. For example, if you choose p=0, then you will take home 10,000 dollars each month since M(t)= 10,000t resulting in M(1)=10,000. What should you set p to be in order to maximize your income at the end of the month? With that value of p, what is your income? Hint: You may find it helpful to graph M(1) as a function of p. Note: You must fully justify your