You are saving money from working, and are planning to invest it to get the best...

 

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You are saving money from working, and are planning to invest it to get the best return. Let us suppose that starting today, you save 1,000 dollars per month. You know of a unique investment plan that operates on continously compounding interest with an impressive monthly rate of 5%, meaning that if you put P dollars in, and you withdraw from it k months later, you will get back Pe.sk dollars. However, due to financial restrictions and tax laws, you can only invest in this plan once, and you can only withdraw from the plan once. Since saving money is hard, you will stop saving after investing to pursue some hobbies and vacations. You have a friend who is willing to offer you a loan of an additional 2,000 dollars to add to this plan when you put in money, on the condition that you will repay him at 3% monthy interest once you withdraw the money, so if you withdraw after s months you must pay him back 2,000 (1.03) dollars. Let t represent when (in months, starting from now) you invest in the plan, using money from yourself and your friend, and let s represent how many months you leave the money in the plan before withdrawing it, paying off your loan, and keeping the remainder. (a) Let f(t, s) be the amount of money you end up keeping after withdrawing from the invest- ment plan and paying off the loan. Provide f(t, s). (b) Compute Vf(t, s). (c) Suppose that due to upcoming personal concerns, you need to complete the process of investing and withdrawing the money 24 months from now. Give a constraint of the form g(t, s) K that represents this (note that you do not need to further specify t≥ 0 and 820, although they must also be true!). (d) Determine values of t and s for which f(t, s) is maximized subject to the constraint given in part (c) (including that s 20 and 20). For this part only, you may use a graphing calculator or online source such as Wolfram Alpha to find solutions, and you should provide them rounded to two decimal places. You should still clearly indicate exactly what you put into the calculator or online source. You are saving money from working, and are planning to invest it to get the best return. Let us suppose that starting today, you save 1,000 dollars per month. You know of a unique investment plan that operates on continously compounding interest with an impressive monthly rate of 5%, meaning that if you put P dollars in, and you withdraw from it k months later, you will get back Pe.sk dollars. However, due to financial restrictions and tax laws, you can only invest in this plan once, and you can only withdraw from the plan once. Since saving money is hard, you will stop saving after investing to pursue some hobbies and vacations. You have a friend who is willing to offer you a loan of an additional 2,000 dollars to add to this plan when you put in money, on the condition that you will repay him at 3% monthy interest once you withdraw the money, so if you withdraw after s months you must pay him back 2,000 (1.03) dollars. Let t represent when (in months, starting from now) you invest in the plan, using money from yourself and your friend, and let s represent how many months you leave the money in the plan before withdrawing it, paying off your loan, and keeping the remainder. (a) Let f(t, s) be the amount of money you end up keeping after withdrawing from the invest- ment plan and paying off the loan. Provide f(t, s). (b) Compute Vf(t, s). (c) Suppose that due to upcoming personal concerns, you need to complete the process of investing and withdrawing the money 24 months from now. Give a constraint of the form g(t, s) K that represents this (note that you do not need to further specify t≥ 0 and 820, although they must also be true!). (d) Determine values of t and s for which f(t, s) is maximized subject to the constraint given in part (c) (including that s 20 and 20). For this part only, you may use a graphing calculator or online source such as Wolfram Alpha to find solutions, and you should provide them rounded to two decimal places. You should still clearly indicate exactly what you put into the calculator or online source.