To find the solution to the problem, we need to follow these steps:
### Given Data:
- Principal (P): [tex]$18,000
- Annual Percentage Rate (APR): 4%
- Term: 4 years
### Part a) Calculation of Monthly Payment (PMT)
First, let's summarize the information provided:
N : $[/tex]\square[tex]$ months \\
$[/tex]I \%[tex]$: $[/tex]\square[tex]$ \\
P.V: $[/tex]S[tex]$ $[/tex]\square[tex]$ \\
PMT: $[/tex]\[tex]$[/tex] [tex]$\square$[/tex] \\
F.V: 0 \\
[tex]$P / Y$[/tex]: 12 \\
[tex]$C / Y$[/tex]: 12 \\
1. Convert the term in years to months:
[tex]\[ N = \text{Term in years} \times 12 \][/tex]
[tex]\[ N = 4 \times 12 = 48 \text{ months} \][/tex]
So,
[tex]\[ N = 48 \text{ months} \][/tex]
2. Calculate the monthly interest rate:
[tex]\[ \text{APR} = 4\% = 0.04 \][/tex]
[tex]\[ \text{Monthly Interest Rate} = \frac{\text{APR}}{12} = \frac{0.04}{12} = 0.003333.\ldots \][/tex]
So,
[tex]\[ I = 0.003333.\ldots \][/tex]
3. Use the loan payment formula to calculate the monthly payment. The formula for the monthly payment (PMT) is:
[tex]\[ \text{PMT} = P.\text{V} \times \left(\frac{I}{1 - (1 + I)^{-N}}\right) \][/tex]
Substitute the known values into the formula:
[tex]\[ PMT = 18000 \times \left(\frac{0.003333.\ldots}{1 - (1 + 0.003333.\ldots)^{-48}}\right) \][/tex]
This calculation results in:
[tex]\[ PMT = \$406.42 \][/tex]
So the monthly payment is:
[tex]\[ \boxed{\$406.42} \][/tex]
### Part b) Calculation of Total Amount Paid Over the Life of the Loan
To find the total amount paid over the life of the loan, we multiply the monthly payment by the total number of payments:
[tex]\[ A = \text{PMT} \times N \][/tex]
[tex]\[ A = 406.42 \times 48 \][/tex]
[tex]\[ A = \$19508.3 \][/tex]
So the total amount paid over the life of the loan is:
[tex]\[ \boxed{\$19508.30} \][/tex]
### Part c) Calculation of Total Interest Paid Over the Life of the Loan
The total interest paid is the total amount paid over the life of the loan minus the principal:
[tex]\[ I = A - P \][/tex]
[tex]\[ I = 19508.3 - 18000 \][/tex]
[tex]\[ I = \$1508.3 \][/tex]
So the total interest paid is:
[tex]\[ \boxed{\$1508.30} \][/tex]
In summary:
a) The monthly payment is [tex]\(\$406.42\)[/tex].
b) The total amount paid over the life of the loan is [tex]\(\$19508.30\)[/tex].
c) The total interest paid over the life of the loan is [tex]\(\$1508.30\)[/tex].
