Answer :
Let's walk through the process step-by-step to determine which Dodge car Suzanne traded in.
1. Calculate the total cost of the new car:
- The list price of the car is \[tex]$23,860. - Sales tax is 8.11% of the list price. - Registration fee is \$[/tex]1,695.
- Documentation fee is \[tex]$228. First, calculate the sales tax: \[ \text{Sales Tax} = \$[/tex]23,860 \times 0.0811 = \[tex]$1,935.146 \] Now, calculate the total cost: \[ \text{Total Cost} = \$[/tex]23,860 + \[tex]$1,935.146 + \$[/tex]1,695 + \[tex]$228 = \$[/tex]27,718.146
\]
Rounding, we get:
[tex]\[ \text{Total Cost} \approx \$27,718 \][/tex]
2. Determine the trade-in value:
- She receives 85% of the listed trade-in price for her Dodge car.
We'll consider each possible car and calculate the trade-in value:
- For the 2004 Intrepid (\[tex]$8,285): \[ \text{Trade-in Value} = \$[/tex]8,285 \times 0.85 = \[tex]$7,042.25 \] - For the 2008 Neon (\$[/tex]7,440):
[tex]\[ \text{Trade-in Value} = \$7,440 \times 0.85 = \$6,324 \][/tex]
- For the 2005 Viper (\[tex]$7,225): \[ \text{Trade-in Value} = \$[/tex]7,225 \times 0.85 = \[tex]$6,141.25 \] - For the 2007 Dakota (\$[/tex]8,313):
[tex]\[ \text{Trade-in Value} = \$8,313 \times 0.85 = \$7,066.05 \][/tex]
3. Calculate the amount to be financed:
- Subtract the trade-in value from the total cost.
- For the 2004 Intrepid:
[tex]\[ \text{Financing Amount} = \$27,718 - \$7,042.25 = \$20,675.75 \][/tex]
- For the 2008 Neon:
[tex]\[ \text{Financing Amount} = \$27,718 - \$6,324 = \$21,394 \][/tex]
- For the 2005 Viper:
[tex]\[ \text{Financing Amount} = \$27,718 - \$6,141.25 = \$21,576.75 \][/tex]
- For the 2007 Dakota:
[tex]\[ \text{Financing Amount} = \$27,718 - \$7,066.05 = \$20,651.95 \][/tex]
4. Calculating the Monthly Payment using the Annuity Formula:
The formula for calculating the monthly payment is:
[tex]\[ \text{Monthly Payment} = \frac{P \times r}{1 - (1 + r)^{-n}} \][/tex]
Where:
- [tex]\( P \)[/tex] is the financing amount.
- [tex]\( r \)[/tex] is the monthly interest rate.
- [tex]\( n \)[/tex] is the number of total payments (months).
Given:
[tex]\[ r = \frac{0.1162}{12}, \quad n = 5 \times 12 = 60 \text{ months} \][/tex]
We'll use this formula to check which financing amount gives a monthly payment close to \[tex]$455.96: - For the 2004 Intrepid: \[ P = 20,675.75 \] Calculated Monthly Payment: \[ \approx \$[/tex]455.44
\]
- For the 2007 Dakota:
[tex]\[ P = 20,651.95 \][/tex]
Calculated Monthly Payment:
[tex]\[ \approx \$455.44 \][/tex]
Both cars give a monthly payment of around \[tex]$455.44, which is very close to \$[/tex]455.96.
After reconciling all conditions, it turns out:
- The 2004 Intrepid (a)
- 2007 Dakota (d)
Both satisfy Suzanne's requirements. Thus, both are correct.
1. Calculate the total cost of the new car:
- The list price of the car is \[tex]$23,860. - Sales tax is 8.11% of the list price. - Registration fee is \$[/tex]1,695.
- Documentation fee is \[tex]$228. First, calculate the sales tax: \[ \text{Sales Tax} = \$[/tex]23,860 \times 0.0811 = \[tex]$1,935.146 \] Now, calculate the total cost: \[ \text{Total Cost} = \$[/tex]23,860 + \[tex]$1,935.146 + \$[/tex]1,695 + \[tex]$228 = \$[/tex]27,718.146
\]
Rounding, we get:
[tex]\[ \text{Total Cost} \approx \$27,718 \][/tex]
2. Determine the trade-in value:
- She receives 85% of the listed trade-in price for her Dodge car.
We'll consider each possible car and calculate the trade-in value:
- For the 2004 Intrepid (\[tex]$8,285): \[ \text{Trade-in Value} = \$[/tex]8,285 \times 0.85 = \[tex]$7,042.25 \] - For the 2008 Neon (\$[/tex]7,440):
[tex]\[ \text{Trade-in Value} = \$7,440 \times 0.85 = \$6,324 \][/tex]
- For the 2005 Viper (\[tex]$7,225): \[ \text{Trade-in Value} = \$[/tex]7,225 \times 0.85 = \[tex]$6,141.25 \] - For the 2007 Dakota (\$[/tex]8,313):
[tex]\[ \text{Trade-in Value} = \$8,313 \times 0.85 = \$7,066.05 \][/tex]
3. Calculate the amount to be financed:
- Subtract the trade-in value from the total cost.
- For the 2004 Intrepid:
[tex]\[ \text{Financing Amount} = \$27,718 - \$7,042.25 = \$20,675.75 \][/tex]
- For the 2008 Neon:
[tex]\[ \text{Financing Amount} = \$27,718 - \$6,324 = \$21,394 \][/tex]
- For the 2005 Viper:
[tex]\[ \text{Financing Amount} = \$27,718 - \$6,141.25 = \$21,576.75 \][/tex]
- For the 2007 Dakota:
[tex]\[ \text{Financing Amount} = \$27,718 - \$7,066.05 = \$20,651.95 \][/tex]
4. Calculating the Monthly Payment using the Annuity Formula:
The formula for calculating the monthly payment is:
[tex]\[ \text{Monthly Payment} = \frac{P \times r}{1 - (1 + r)^{-n}} \][/tex]
Where:
- [tex]\( P \)[/tex] is the financing amount.
- [tex]\( r \)[/tex] is the monthly interest rate.
- [tex]\( n \)[/tex] is the number of total payments (months).
Given:
[tex]\[ r = \frac{0.1162}{12}, \quad n = 5 \times 12 = 60 \text{ months} \][/tex]
We'll use this formula to check which financing amount gives a monthly payment close to \[tex]$455.96: - For the 2004 Intrepid: \[ P = 20,675.75 \] Calculated Monthly Payment: \[ \approx \$[/tex]455.44
\]
- For the 2007 Dakota:
[tex]\[ P = 20,651.95 \][/tex]
Calculated Monthly Payment:
[tex]\[ \approx \$455.44 \][/tex]
Both cars give a monthly payment of around \[tex]$455.44, which is very close to \$[/tex]455.96.
After reconciling all conditions, it turns out:
- The 2004 Intrepid (a)
- 2007 Dakota (d)
Both satisfy Suzanne's requirements. Thus, both are correct.
