Questions: Rachel plans to purchase a new sports car. The dealer requires a 10% down payment on the 45,000 vehicle. Rachel will finance the rest of the cost with a fixed-rate amortized auto loan at 3.5% annual interest with monthly payments over 6 years. Complete the parts below. Do not round any intermediate computations. Round your final answers to the nearest cent if necessary. If necessary, refer to the list of financial formulas. (a) Find the required down payment. (b) Find the amount of the auto loan. (c) Find the monthly payment.

Solution Steps

The down payment is calculated as a percentage of the total cost of the vehicle. Given that the down payment is \(10\%\) of the \(\$45,000\) vehicle, we calculate:

\[ \text{Down Payment} = 0.10 \times 45,000 = 4,500 \]

The amount of the auto loan is the total cost of the vehicle minus the down payment. Therefore:

\[ \text{Auto Loan Amount} = 45,000 - 4,500 = 40,500 \]

To find the monthly payment, we use the formula for the monthly payment on an amortized loan:

\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]

where:

• \(M\) is the monthly payment,
• \(P\) is the principal amount (\$40,500),
• \(r\) is the monthly interest rate (annual rate divided by 12), and
• \(n\) is the total number of payments (months).

Given:

• Annual interest rate = \(3.5\%\), so the monthly interest rate \(r = \frac{3.5}{100 \times 12} = 0.00291667\),
• Loan term = 6 years, so \(n = 6 \times 12 = 72\).

Substitute these values into the formula:

\[ M = \frac{40,500 \times 0.00291667 \times (1 + 0.00291667)^{72}}{(1 + 0.00291667)^{72} - 1} \]

Calculating the above expression gives:

\[ M \approx 622.67 \]

Final Answer