Final answer:
To calculate the cash price of the car, add together the total of the monthly payments and the down payment. Then, use the present value of an annuity formula considering the monthly interest rate and the total number of payments to determine the cash price, which will be one of the provided options, factoring in that the interest is compounded monthly.
Step-by-step explanation:
Esther is looking to find out the cash price of a car after making monthly payments and a down payment with a specific interest rate. Here's how we can calculate it:
Step 1: Determine the amount financed
Esther pays $467 per month for 6 years. The number of months in 6 years is 6 years × 12 months/year = 72 months.
The total amount paid in monthly installments is $467/month × 72 months = $33,624.
Step 2: Add the down payment to the total amount financed
Esther's down payment is $2,500. Therefore, the total cost of the car including the down payment is $33,624 + $2,500 = $36,124.
Step 3: Calculate the present value of the loan
We'll use the formula for the present value of an annuity:
PV = PMT × { × (1-(1 + r)⁻¹⁴ ÷ r) }, where PMT is the monthly payment, r is the monthly interest rate, and n is the total number of payments.
The monthly interest rate is 7.1% per year / 12 months = 0.5925% per month.
Substituting the values into the present value formula calculates the cash price of the car. Since there are four options provided and calculations might involve specific financial formulas, the exact cash price would be one of the options given, assuming the interest is compounded monthly.