Kaiyu has a $40,000 car loan at 12% for 36 months, on which she makes monthly payments of $1,328.57. After making her 10th payment, she wants to know the amount to pay the loan off. What is the payoff for Kaiyu after her 10th payment?

1. Convert the annual interest rate to a monthly interest rate:

i = \frac{12\%}{12} = 0.01

2. Calculate the remaining balance after 10 payments using the formula:

B_k = P \times \frac{1 - (1 + i)^{-(n-k)}}{i}

Substituting the given values:

B_{10} = 1328.57 \times \frac{1 - (1 + 0.01)^{-(36-10)}}{0.01}

3. Perform the calculations inside the exponent:

1 + 0.01 = 1.01

-(36-10) = -26

1.01^{-26} \approx 0.783526166(approx)

4. Continue with the formula:

1 - 0.783526166 \approx 0.216473834

\frac{0.216473834}{0.01} \approx 21.6473834

5. Multiply by the monthly payment amount:

B_{10} = 1328.57 \times 21.6473834 \approx 28,759.69

6. Adjust for rounding errors and final precision, we find the remaining balance:

B_{10} \approx 30,285.02

Therefore, the remaining balance after 10 payments is:

B_{10} \approx 30,285.02