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Unformatted text preview: MATH 150 Applications of Linear Equations Cheung Man Wai Tutorial 4 1 Personal Finance Example 1 Planning for retirement You have decided that you will need $ 50,000 each year to live on after you retire, and that you should plan on living 30 years after your retirement. Assuming that your retirement account will earn 5% interest while you are taking out $ 50,000 each year, how much money must be in the retirement account when you retire? For simplicity, take 1,000 dollars as our unit. According to the model, we have P = 0 . 05 P- 50 . By using integrating factor, [ e . 05 t P ] =- 50 e . 05 t . So, e . 05 t P ( t ) = 1000 e . 05 t + C, P ( t ) = 1000 e . 05 t + C If we let P (0) = P denote the balance at time you retire, then we can evaluate the constant C , C = P- 1000 . Thus, the solution is P ( t ) = 1000 + ( P- 1000) e . 05 t . Since you want to have $ 50,000 each year until you die 30 years after retiring, you will want P (30) ≥ 0. If you spend your last cent the day you die, you will want 0 = P (30) = P ( t ) = 1000 + ( P- 1000) e 1 . 5 . Solving for P , getting P = 1000(1- e- 1 . 5 ) = 776 . 8698 . 1 So you will need to have saved $ 776,870. Example 2 Saving for retirement After some though, you have decide that you should put a fixed percentage ρ of your salary into your retirement account. The question is, what value of ρ will achieve our goal? Assume your current salary is $35,000 pre year and your salary will grow at 4% per year. This leads to the differential equation S = 0 . 04 S , where S ( t ) is your annual salary in thousands of dollars. So, S ( t ) = 35 e . 04 t ....
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