3. Once again, suppose you take out a loan of P dollars, with monthly interest rate...

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3. Once again, suppose you take out a loan of P dollars, with monthly interest rate r. Suppose also that your bank gives you the option to choose what you pay to principal each month. You expect that your income will increase at a monthly rate of s, from a combination of cost-of-living adjustments and seniority pay. You want to pay off your bank loan by making monthly payments (interest plus principal) that increase in line with your overall income. That is, if you pay $M in interest and principal one month, the next month you'll pay $M (1+ s) in principal and interest. (a) Set up a differential equation for P(n), the amount you owen months after your loan is taken out. Your equation should include M, s, r, P, P', and n. The differential equation in (a) is not separable, so we haven't given you a way to solve it from scratch. (If you continue learning about differential equations, you'll probably learn how. Alternately, if you get used to computer algebra systems, solving the DE is the part you can leave to a computer - coming up with the DE in the first place is where a human is most needed.) Using sums is pretty tricky here, so both methods we've talked about so far (differential equations and geometric sums) come up short. So, as a third option for understanding this loan, let's use a spreadsheet to investigate one particular case. 1/4 1/3 Suppose P750,000, r = and s= 100 You want to pay off your loan in 300 months. 100 (b) First we'll want to find M, the total payment to the bank in the first month. We'll set up a spreadsheet to keep track of things. A template is below. An arrow indicates that the contents of a cell have been copy-pasted down the row. A1-D1 are descriptions of the contents of their columns. You'll be guessing different values of M, and storing them in cell E2. You'll want to know how much debt is left after 300 months, so to save on scrolling, P(300) is displayed in cell F2. B C n Total monthly payment Payment to principal A 1 2 0 3 1 4 2 5 3 n/a n/a D E F P(n) M P(300) 750 000 i. What should you write in cells B3, C3, and D3? For this part, you do not have to justify your answer. ii. You want column D to compute up to P(300). If you copy-paste row 3 down columns B, C, and D, what cell will P(300) be computed in? (To save on scrolling, write "=X" into cell F2, where X is the cell where P(300) is computed.) For this part, you do not have to justify your answer. iii. Play around with different values of M to make P(300) as close as possible to 0. To within one cent, what is the best value of M? For this part, you do not have to justify your answer. iv. Explain why M < 3556.58 makes intuitive sense. (c) Use the spreadsheet to compute the amount of interest you pay this way. (d) Do you pay more or less interest this way than with the type of mortgage in Question 2? 3. Once again, suppose you take out a loan of P dollars, with monthly interest rate r. Suppose also that your bank gives you the option to choose what you pay to principal each month. You expect that your income will increase at a monthly rate of s, from a combination of cost-of-living adjustments and seniority pay. You want to pay off your bank loan by making monthly payments (interest plus principal) that increase in line with your overall income. That is, if you pay $M in interest and principal one month, the next month you'll pay $M (1+ s) in principal and interest. (a) Set up a differential equation for P(n), the amount you owen months after your loan is taken out. Your equation should include M, s, r, P, P', and n. The differential equation in (a) is not separable, so we haven't given you a way to solve it from scratch. (If you continue learning about differential equations, you'll probably learn how. Alternately, if you get used to computer algebra systems, solving the DE is the part you can leave to a computer - coming up with the DE in the first place is where a human is most needed.) Using sums is pretty tricky here, so both methods we've talked about so far (differential equations and geometric sums) come up short. So, as a third option for understanding this loan, let's use a spreadsheet to investigate one particular case. 1/4 1/3 Suppose P750,000, r = and s= 100 You want to pay off your loan in 300 months. 100 (b) First we'll want to find M, the total payment to the bank in the first month. We'll set up a spreadsheet to keep track of things. A template is below. An arrow indicates that the contents of a cell have been copy-pasted down the row. A1-D1 are descriptions of the contents of their columns. You'll be guessing different values of M, and storing them in cell E2. You'll want to know how much debt is left after 300 months, so to save on scrolling, P(300) is displayed in cell F2. B C n Total monthly payment Payment to principal A 1 2 0 3 1 4 2 5 3 n/a n/a D E F P(n) M P(300) 750 000 i. What should you write in cells B3, C3, and D3? For this part, you do not have to justify your answer. ii. You want column D to compute up to P(300). If you copy-paste row 3 down columns B, C, and D, what cell will P(300) be computed in? (To save on scrolling, write "=X" into cell F2, where X is the cell where P(300) is computed.) For this part, you do not have to justify your answer. iii. Play around with different values of M to make P(300) as close as possible to 0. To within one cent, what is the best value of M? For this part, you do not have to justify your answer. iv. Explain why M < 3556.58 makes intuitive sense. (c) Use the spreadsheet to compute the amount of interest you pay this way. (d) Do you pay more or less interest this way than with the type of mortgage in Question 2?