Follow the steps above and find [tex]\( c \)[/tex], the total of the payments, and the monthly payment. Choose the right answers.

Sally Seair buys a sailboat. The price, including tax, is [tex]\$5,275.00[/tex]. She finances the boat over 36 months after making a [tex]\$500[/tex] down payment. The true annual interest rate is [tex]15\%[/tex]. What are Sally's monthly payments (principal plus interest)?

To the nearest penny,
[tex]\[ c = \ \$\][/tex]

Total of payments = amount financed + [tex]\( c \)[/tex] = \ \[tex]$\]

Total of payments / number of payments = monthly payment = \ \$[/tex]\]



Answer :

Let's break down Sally Seair's sailboat financing step by step:

1. Purchase Price and Down Payment:
- The price of the sailboat, including tax, is \[tex]$5,275. - The down payment Sally makes is \$[/tex]500.

2. Amount Financed:
- The amount Sally finances is the purchase price minus the down payment.
- [tex]\(\$5,275 - \$500 = \$4,775\)[/tex]

3. Annual Interest Rate and Loan Term:
- The true annual interest rate is 15%.
- The loan term is 36 months.

4. Monthly Interest Rate:
- The annual interest rate is 15%, so the monthly interest rate is [tex]\(\frac{15\%}{12} = 1.25\%\)[/tex].

5. Monthly Payment Calculation:
- We need to use the formula for the monthly payment of an installment loan:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment,
- [tex]\( P \)[/tex] is the loan amount (\[tex]$4,775), - \( r \) is the monthly interest rate (0.0125), - \( n \) is the number of payments (36). 6. Calculating Monthly Payment: - Using the given values, we find the monthly payment to be approximately \$[/tex]165.53.

7. Total of Payments:
- The total amount paid over the course of the loan can be calculated by multiplying the monthly payment by the number of payments:
[tex]\[ c = M \times n \][/tex]
- Substituting the values, we get:
[tex]\[ c = \$165.53 \times 36 \approx \$5,958.97 \][/tex]

8. Total of Payments and Amount Financed:
- The total of payments consists of the down payment and [tex]\( c \)[/tex]:
[tex]\[ \text{Total of payments} = \$500 + c \][/tex]
- Substituting [tex]\( c \)[/tex], we get:
[tex]\[ \text{Total of payments} = \$500 + \$5,958.97 = \$6,458.97 \][/tex]

Summarizing the solutions:

- To the nearest penny, [tex]\( c = \$5,958.97 \)[/tex].
- Total of payments = \[tex]$4,775 (amount financed) + \$[/tex]5,958.97 = \[tex]$10,733.97. - Monthly payment = \$[/tex]165.53.