Certainly! Let's walk through the calculation for determining Taylor's monthly car loan payment step-by-step.
Taylor is looking to take a 48-month loan with an annual interest rate of 7.9%, compounded monthly, and the principal amount of the loan is [tex]$19,076.
1. Convert the annual interest rate to a monthly interest rate:
\[
\text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{12} = \frac{7.9\%}{12} = \frac{7.9}{100 \times 12} = 0.006583\overline{3}
\]
2. Identify the terms in the given formula for the monthly payment \( P \):
\[
P = \frac{P_p(i)}{1 - (1 + i)^{-n}}
\]
where:
- \( P_p \) is the principal amount (\$[/tex]19,076)
- [tex]\( i \)[/tex] is the monthly interest rate (0.006583\overline{3})
- [tex]\( n \)[/tex] is the number of monthly payments (48)
3. Calculate the numerator:
[tex]\[
P_p \times i = 19,076 \times 0.006583\overline{3}
\][/tex]
4. Calculate the denominator:
[tex]\[
1 - (1 + i)^{-n} = 1 - (1 + 0.006583\overline{3})^{-48}
\][/tex]
5. Plug these values into the formula and compute:
[tex]\[
P = \frac{19,076 \times 0.006583\overline{3}}{1 - (1 + 0.006583\overline{3})^{-48}}
\][/tex]
6. Numerical Calculation:
After performing these calculations step-by-step:
- Calculating the numerator: [tex]\(19,076 \times 0.006583\overline{3} \approx 125.628892\)[/tex]
- Calculating the denominator: [tex]\(1 - (1.006583\overline{3})^{-48} \approx 0.270222142\)[/tex]
Combining these, the monthly payment [tex]\( P \)[/tex] is approximately:
[tex]\[
P = \frac{125.628892}{0.270222142} \approx 464.81
\][/tex]
The monthly payment for Taylor's loan will be approximately \[tex]$464.81.
Therefore, the correct answer is:
C. Taylor's approximate monthly payment for the loan will be \$[/tex]464.81.
