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.
@aibot
**To determine the payment rate k that is required to pay off the loan in 3 years, we can use the formula for the continuous compound interest:
A = Pe^(rt)
Where:
A = Final amount (loan amount + interest)
P = Principal amount (loan amount)
e = Euler’s number (approximately 2.71828)
r = Interest rate
t = Time period
In this case, the principal amount (P) is $8000, the interest rate (r) is 10% or 0.10, and the time period (t) is 3 years. We need to solve for the payment rate (k) that will result in the loan being paid off in 3 years.
Let’s break down the steps to find the payment rate k:
Step 1: Calculate the final amount (A) after 3 years by using the continuous compound interest formula:
A = 8000 * e^(0.10*3)
Step 2: Set up the equation to find the payment rate k:
8000 = P * e^(0.10*3) * (1 - e^(-kt))/k
Where P is the principal amount, e^(0.10*3) is the compound interest factor after 3 years, and (1 - e^(-kt))/k represents the continuous payment formula.
Step 3: Solve the equation for k.
This equation can be quite complex to solve algebraically. It may require iterative or numerical methods to find an approximate solution.
Step 4: Calculate the amount of interest paid during the 3-year period by subtracting the principal amount from the final amount.
Interest paid = A - P
By performing these calculations, we can determine the payment rate k required to pay off the loan in 3 years and the amount of interest paid during that period.