A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period.

Answer:The payment is [tex]k=\$3216.92[/tex].Interest paid during the 3-year period is [tex]I_{T}=\$1651.76[/tex]Step-by-step explanation:HiThe Payment amountWe are going to use the formula below with [tex]VP=\$8000, i=10\%[/tex] and [tex]n=3[/tex].[tex]k=\frac{VP}{[\frac{1-(1+i)^{-n}}{i}] } =\frac{8000}{[\frac{1-(1+0.1)^{-3}}{0.1}] }=3216.92[/tex]Interest paid during the 3 yearFirst period[tex]I_{1}=VP*i=8000*0.1=800[/tex][tex]CS_{1}=k-I_{1}=3216.92-800=2416.92[/tex][tex]Balance_{1}=VP-CS_{1}=8000-2416.92=5583.08[/tex]Second period[tex]I_{2}=B_{1}*i=5583.08*0.1=558.31[/tex][tex]CS_{2}=k-I_{2}=3216.92-558.31=2648.61[/tex][tex]Balance_{2}=VP-CS_{2}=5583.08-2648.61=2924.47[/tex]Third period[tex]I_{3}=B_{2}*i=2924.47*0.1=292.45[/tex]Sum of Interest paid during the 3 years[tex]I_{t}=I_{1}+I_{2}+I_{3}=800+558.31+292.45=1651.76[/tex]