Suppose the amount of principal left to be paid back on a car loan is given by P(t) and suppose a payment of k dollars is made on the loan each month. Then the rate at which the remaining principal P(t) changes with respect to time (in exist/month) is the net result of the monthly payment of k dollars (which is a positive constant) and the interest which is proportional to the remaining principal (here the proportionality constant is the monthly continuous interest rate r). a. Write a differential equation to represent this situation using the function and constants named above. b. A student purchases a car for exist12,000 with a 3-year loan at 9% annual interest, compounded continuously. Calculate the monthly interest rate r and then use this value to rewrite the DE from part a. Then solve this differential equation for P(t). Be sure to determine the value of C and state your model for P(t) clearly. c. Use your model for P(t) to find the monthly payment k so that the loan in paid off (P(t) = 0) in 3 years (36 months). Give your answer rounded to the nearest cent. d. Derive the general payment formula, solving for k in terms of r, n (the number of months), and P_0 (the initial principal), showing all work to support it