A certain college graduate borrows 6900 dollars to buy a car. The lender charges interest at an annual rate of 18%. 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 7 years. Also determine how much interest is paid during the 7-year period.

a) k = $1734.86 per year

b) $5244.02

Explanation:

Principal ( borrowed ) = $6900

Annual interest rate = 18%  ( compounded continuously )

Borrower makes a continuous payment at a constant rate : $k per year

a) Determine payment rate required to payoff loan ( in 7 years )

let loan at time ( t ) = x

x = $6900 , at t = 0

rate of increase of loan amount = 0.18x

rate of decrease of loan amount = k

∴ net change of loan x

= dx/dt = 0.18x - k  

hence ; dt = dx / ( 0.18x - k )

k = $1734.86 per year

attached below is a part of the solution

b) Determine how much interest is paid during 7-year period

Interest paid in 7 years

= Amount paid - principal amount

= ( 1734.86 * 7 ) - 6900

= $5244.02