A brand new car! Sarah Lee is buying a new car and must borrow money for it. (You must choose a vehicle for her and research how much it costs.) She wants a 6-year car loan and has two choices. She can either get a car loan from NC SECU at 2.0% per year or she can get a loan from the dealer at 1.5% but they charge her an origination fee of 1.0%. (This is a fee that the borrower must pay the lender at the beginning of the loan. For example, for a $30,000 vehicle, the origination fee would be 0.01(30000) = $300 extra just to get the loan.)
We can model the loan assuming that payments are made and interest is compounded continuously.
Let t denote time in years, let x(t) be the balance (the amount remaining to be paid) at time t, let r be the annual interest rate (expressed as a decimal number, of course), and let p be the constant annual payment (continously paid out). The DE is dx/dt = rx − p.
A- How much does Sarah pay per year in each case? (Hint: Remember that she starts with the full amount of the loan and pays it off exactly in 6 years.)
B- Which lender offers a better deal over the entire time of the loan (assuming she does not invest the money she would have paid in the origination fee)?