Question

A certain college graduate borrows 7277 dollars to buy a car. The lender charges interest at an annual rate of 11%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k dollars per year.
1. Determine the payment rate that is required to pay off the loan in 5 years.
2. Also determine how much interest is paid during the 5-year period?

Answer 1

Answer:

a. $1773.82

b. $1592.1

Step-by-step explanation:

1. If he pays k dollar in the first year, then the amount that he owned without interest is

7277 - k

The amount that he owned including interest of 11% in the 2nd year is

(7277 - k)*1.11 or 7277*1.11 - 1.11k

After 2nd year and paying k then the amount he owned (without interest is)

(7277 - k)*1.11 - k

With interest

[(7277 - k)*1.11 - k]1.11 or $7277*1.11^2 - 1.11^2k - 1.11k$

So after 5 years

$7277*1.11^5 - (1.11^5 + 1.11^4 + 1.11^3 + 1.11^2 +1.11)k$

$12262.17 - 6.91 k$

Since he's dept-free after 5 year then

$12262.17 - 6.91 k = 0$

$k = 12262.17 / 6.91 = 1773.82$ dollar

2. The total amount he would have to pay over 5 years is 5k = 5*1773.82 = 8869.1

So the interest we has to pay over 5 years is the total subtracted by the principal, which is 8869.1 - 7277 = 1592.1 dollar