1. A Car Loan: Suppose the amount of principal left to be paid back on a car loa
ID: 2881859 • Letter: 1
Question
1. 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. I believe the answer is P'=rP-k
b. A student purchases a car for $18,000 with a 4-year loan at 3% 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 4 years (48 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 P0 (the initial principal), showing all work to support it.
This is a practice question for an ODE exam I have next week so could you please be very detailed and write every step out. Thanks!
Explanation / Answer
a> P' = rP-kt
here t is in months
k is the amount of the emi , so k needs to be multiplied by t as kt will be the amount paid after t monts and the
remaining amount of P(t) would be pr
=> P' = Pr - kt
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