Select the correct answer. $\newline$ Taylor wants to purchase a car with an auto loan. He can get a $48$ -month loan from hls bank that is compounded monthy at an annual interest rate of $7.9 \%$ . $\newline$ Suppose Taylor needs to obtaln a loan for $\$ 19,076$ to purchase the car. $\newline$ Use the formula for the sum of a fillte geomettic sertes to determine TEyor's approximate monthly payment. $\newline$ $P=\frac{P_{1}(0)}{1-(1+0)}$ $\newline$ A. Taylor's approxlmate monthly payment for the loan wall be SA $58$ . $35$ . $\newline$ B. Taylor's approximate monthy payment for the loan will be S $464$ . $81$ . $\newline$ C. Tayor's approxdmate monthly payment for the loan will be $\$ 546.50$ . $\newline$ D. Taylor's approxlmate monthy payment for the loan will be S $4132$ .

Full solution

Q. Select the correct answer.

\newline

Taylor wants to purchase a car with an auto loan. He can get a

48

-month loan from hls bank that is compounded monthy at an annual interest rate of

7.9 \%

\newline

Suppose Taylor needs to obtaln a loan for

\$ 19,076

to purchase the car.

\newline

Use the formula for the sum of a fillte geomettic sertes to determine TEyor's approximate monthly payment.

\newline

P=\frac{P_{1}(0)}{1-(1+0)}

\newline

A. Taylor's approxlmate monthly payment for the loan wall be SA

58

35

\newline

B. Taylor's approximate monthy payment for the loan will be S

464

81

\newline

C. Tayor's approxdmate monthly payment for the loan will be

\$ 546.50

\newline

D. Taylor's approxlmate monthy payment for the loan will be S

4132

Calculate Monthly Interest Rate: Calculate the monthly interest rate from the annual rate. $\newline$ Annual interest rate = $7.9\%$ $\newline$ Monthly interest rate = $\frac{7.9\%}{12}$ $\newline$ = $0.6583\%$
Convert to Decimal Form: Convert the monthly interest rate into decimal form for calculation. $\newline$ Monthly interest rate (decimal) = $\frac{0.6583}{100}$ $\newline$ = $0.006583$
Use Amortizing Loan Formula: Use the formula for the monthly payment of an amortizing loan, which is different from the sum of a finite geometric series. $\newline$ Formula: $P = \frac{P_1 \times r}{1 - (1 + r)^{-n}}$ $\newline$ Where $P_1 =$ principal amount ( $\$19,076$ ), $r =$ monthly interest rate ( $0.006583$ ), $n =$ total number of payments ( $48$ )
Calculate Monthly Payment: Plug the values into the formula to calculate the monthly payment. $\newline$ $P = \frac{19076 \times 0.006583}{1 - (1 + 0.006583)^{-48}}$ $\newline$ $= \frac{125.585908}{1 - 0.689974}$ $\newline$ $= \frac{125.585908}{0.310026}$ $\newline$ $= \$405.08$