A certain college graduate borrows 5526 dollars to buy a car. The lender charges

Question

A certain college graduate borrows 5526 dollars to buy a car. The lender charges interest at an annual rate of 13%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k dollars per year, determine the payment rate that is required to pay off the loan in 2 years. Also determine how much interest is paid during the 2-year period. Round your answers to two decimal places.

Payment rate = dollars per year

Interest paid = dollars

Explanation / Answer

a)  Let S(t) be the amount due after t years.

Then S(t) satisfies the following DE

dS(t)/dt = 0.13 * S(t) k,

S(0) = 5526

Solving the DE by either using separation of variables

S(t) = 5526 * e 0.13t (k/0.13)(e 0.13t 1) = 0

Thus, when t = 2

k = 3137.74

b) The college graduate pays in total 2k = $6,275.48 after two years, hence he paid $6,275.48 $5526 = 749.48$ in interest.

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