A Car Loan: 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 $/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 $12,000 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), using a linear integrating factor approach. 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, (the initial principal), showing all work to support it.