To determine Antonio's total lifetime cost for his loans, we must calculate the total repayment amount for both his Junior and Senior year loans, considering the respective interest rates and the compounding effect over a 10-year period with monthly payments.
### Junior Year Loan Calculation
1. Loan amount for Junior year: \[tex]$5,894
2. Interest rate: 6.9% per year, compounded monthly.
3. Monthly interest rate: \(\frac{6.9}{12} = 0.575\%\) or \(0.00575\) in decimal form.
4. Number of payments: \(10\) years \(\times 12\) months/year = 120 payments.
Using the loan amortization formula for calculating monthly payment:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
where:
- \( P \) = loan amount = \$[/tex]5,894
- [tex]\( r \)[/tex] = monthly interest rate = 0.00575
- [tex]\( n \)[/tex] = number of payments = 120
[tex]\[ M_{\text{junior}} = 5894 \times \frac{0.00575(1 + 0.00575)^{120}}{(1 + 0.00575)^{120} - 1} \][/tex]
The monthly payment for the Junior year loan is approximately \[tex]$81.76.
To find the total payment over the life of the loan:
\[ \text{Total Junior Payment} = M_{\text{junior}} \times n = 81.76 \times 120 = \$[/tex]8,175.71 \]
### Senior Year Loan Calculation
1. Loan amount for Senior year: \[tex]$5,258
2. Interest rate: 7.5% per year, compounded monthly.
3. Monthly interest rate: \(\frac{7.5}{12} = 0.625\%\) or \(0.00625\) in decimal form.
4. Number of payments: \(10\) years \(\times 12\) months/year = 120 payments.
Using the loan amortization formula for calculating monthly payment:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
where:
- \( P \) = loan amount = \$[/tex]5,258
- [tex]\( r \)[/tex] = monthly interest rate = 0.00625
- [tex]\( n \)[/tex] = number of payments = 120
[tex]\[ M_{\text{senior}} = 5258 \times \frac{0.00625(1 + 0.00625)^{120}}{(1 + 0.00625)^{120} - 1} \][/tex]
The monthly payment for the Senior year loan is approximately \[tex]$74.90.
To find the total payment over the life of the loan:
\[ \text{Total Senior Payment} = M_{\text{senior}} \times n = 74.90 \times 120 = \$[/tex]7,489.61 \]
### Total Lifetime Cost
To calculate Antonio's total lifetime cost, we sum the total payments for the Junior and Senior year loans:
[tex]\[ \text{Total Lifetime Cost} = \text{Total Junior Payment} + \text{Total Senior Payment} \][/tex]
[tex]\[ \text{Total Lifetime Cost} = 8,175.71 + 7,489.61 = \$15,665.32 \][/tex]
Thus, the total lifetime cost of Antonio's loans, once rounded to the nearest cent, is [tex]\( \$15,665.32 \)[/tex].
Answer: The correct option is not listed among the choices, but the total lifetime cost is \$15,665.32.
