Lynette bought a new car for \( \$ 34,000 \). She paid a \( 20 \% \) down payment and financed the remaining balance for 48 months with an APR of \( 4.3 \% \). Assuming she made monthly payments, determine the total cost of Lynette's car. Round your answer to the nearest cent, if necessary.

To determine the total cost of Lynette's car, we need to follow these steps: 1. **Calculate the down payment.** 2. **Determine the financed amount.** 3. **Calculate the monthly payment using the loan formula.** 4. **Calculate the total cost of the car including the down payment and total payments made.** ### Step 1: Calculate the Down Payment The down payment is \( 20\% \) of the car's price. \[ \text{Down Payment} = 0.20 \times 34000 \] ### Step 2: Determine the Financed Amount The financed amount is the total price minus the down payment. \[ \text{Financed Amount} = 34000 - \text{Down Payment} \] ### Step 3: Calculate the Monthly Payment The formula for the monthly payment \( M \) on an amortizing loan is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( P \) is the loan amount (financed amount), - \( r \) is the monthly interest rate (annual rate divided by 12), - \( n \) is the number of payments (months). First, we need to calculate \( r \): \[ r = \frac{4.3\%}{12} = \frac{0.043}{12} \] Now we can calculate the monthly payment. ### Step 4: Calculate the Total Cost of the Car The total cost of the car is the sum of the down payment and the total of all monthly payments. \[ \text{Total Cost} = \text{Down Payment} + (M \times n) \] Now, let's perform the calculations step by step. 1. Calculate the down payment. 2. Calculate the financed amount. 3. Calculate the monthly payment. 4. Calculate the total cost. Let's compute these values. Calculate the value by following steps: - step0: Calculate: \(0.2\times 34000\) - step1: Multiply the numbers: \(6800\) Calculate or simplify the expression \( 34000 - (0.20 * 34000) \). Calculate the value by following steps: - step0: Calculate: \(34000-\left(0.2\times 34000\right)\) - step1: Multiply the numbers: \(34000-6800\) - step2: Subtract the numbers: \(27200\) Calculate or simplify the expression \( 0.043 / 12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.043}{12}\) - step1: Convert the expressions: \(\frac{\frac{43}{1000}}{12}\) - step2: Multiply by the reciprocal: \(\frac{43}{1000}\times \frac{1}{12}\) - step3: Multiply the fractions: \(\frac{43}{1000\times 12}\) - step4: Multiply: \(\frac{43}{12000}\) Calculate or simplify the expression \( 27200 * (\frac{43}{12000} * (1 + \frac{43}{12000})^{48}) / ((1 + \frac{43}{12000})^{48} - 1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{27200\left(\frac{43}{12000}\left(1+\frac{43}{12000}\right)^{48}\right)}{\left(\left(1+\frac{43}{12000}\right)^{48}-1\right)}\) - step1: Remove the parentheses: \(\frac{27200\times \frac{43}{12000}\left(1+\frac{43}{12000}\right)^{48}}{\left(1+\frac{43}{12000}\right)^{48}-1}\) - step2: Add the numbers: \(\frac{27200\times \frac{43}{12000}\left(\frac{12043}{12000}\right)^{48}}{\left(1+\frac{43}{12000}\right)^{48}-1}\) - step3: Add the numbers: \(\frac{27200\times \frac{43}{12000}\left(\frac{12043}{12000}\right)^{48}}{\left(\frac{12043}{12000}\right)^{48}-1}\) - step4: Multiply: \(\frac{\frac{731\times 12043^{48}}{15\times 2^{239}\times 375^{48}}}{\left(\frac{12043}{12000}\right)^{48}-1}\) - step5: Subtract the numbers: \(\frac{\frac{731\times 12043^{48}}{15\times 2^{239}\times 375^{48}}}{\frac{12043^{48}-12000^{48}}{12000^{48}}}\) - step6: Multiply by the reciprocal: \(\frac{731\times 12043^{48}}{15\times 2^{239}\times 375^{48}}\times \frac{12000^{48}}{12043^{48}-12000^{48}}\) - step7: Rewrite the expression: \(\frac{731\times 12043^{48}}{15\times 2^{239}\times 375^{48}}\times \frac{15^{48}\times 800^{48}}{12043^{48}-12000^{48}}\) - step8: Reduce the numbers: \(\frac{731\times 12043^{48}}{2^{239}\times 375^{48}}\times \frac{15^{47}\times 800^{48}}{12043^{48}-12000^{48}}\) - step9: Rewrite the expression: \(\frac{731\times 12043^{48}}{2^{239}\times 15^{48}\times 25^{48}}\times \frac{15^{47}\times 800^{48}}{12043^{48}-12000^{48}}\) - step10: Reduce the numbers: \(\frac{731\times 12043^{48}}{2^{239}\times 15\times 25^{48}}\times \frac{800^{48}}{12043^{48}-12000^{48}}\) - step11: Rewrite the expression: \(\frac{731\times 12043^{48}}{2^{239}\times 15\times 25^{48}}\times \frac{32^{48}\times 25^{48}}{12043^{48}-12000^{48}}\) - step12: Rewrite the expression: \(\frac{731\times 12043^{48}}{2^{239}\times 15\times 25^{48}}\times \frac{2^{240}\times 25^{48}}{12043^{48}-12000^{48}}\) - step13: Reduce the numbers: \(\frac{731\times 12043^{48}}{15\times 25^{48}}\times \frac{2\times 25^{48}}{12043^{48}-12000^{48}}\) - step14: Rewrite the expression: \(\frac{731\times 12043^{48}}{5\times 3\times 25^{48}}\times \frac{2\times 25^{48}}{12043^{48}-12000^{48}}\) - step15: Rewrite the expression: \(\frac{731\times 12043^{48}}{5\times 3\times 25^{48}}\times \frac{2\times 5^{96}}{12043^{48}-12000^{48}}\) - step16: Reduce the numbers: \(\frac{731\times 12043^{48}}{3\times 25^{48}}\times \frac{2\times 5^{95}}{12043^{48}-12000^{48}}\) - step17: Rewrite the expression: \(\frac{731\times 12043^{48}}{3\times 5^{96}}\times \frac{2\times 5^{95}}{12043^{48}-12000^{48}}\) - step18: Reduce the numbers: \(\frac{731\times 12043^{48}}{3\times 5}\times \frac{2}{12043^{48}-12000^{48}}\) - step19: Multiply the fractions: \(\frac{731\times 12043^{48}\times 2}{15\left(12043^{48}-12000^{48}\right)}\) - step20: Multiply: \(\frac{1462\times 12043^{48}}{15\times 12043^{48}-15\times 12000^{48}}\) Calculate or simplify the expression \( 6800 + (617.808513 * 48) \). Calculate the value by following steps: - step0: Calculate: \(6800+\left(617.808513\times 48\right)\) - step1: Multiply the numbers: \(6800+29654.808624\) - step2: Add the numbers: \(36454.808624\) Let's summarize the calculations step by step: 1. **Down Payment:** \[ \text{Down Payment} = 0.20 \times 34000 = 6800 \] 2. **Financed Amount:** \[ \text{Financed Amount} = 34000 - 6800 = 27200 \] 3. **Monthly Interest Rate:** \[ r = \frac{4.3\%}{12} = \frac{0.043}{12} \approx 0.00358333 \] 4. **Monthly Payment Calculation:** Using the formula for the monthly payment: \[ M = 27200 \frac{0.00358333(1 + 0.00358333)^{48}}{(1 + 0.00358333)^{48} - 1} \approx 617.81 \] 5. **Total Cost of the Car:** \[ \text{Total Cost} = 6800 + (617.81 \times 48) \approx 36454.81 \] Thus, the total cost of Lynette's car, rounded to the nearest cent, is approximately: \[ \boxed{36454.81} \]