Answer :
Let's go step-by-step to solve each part of the problem.
### 1. Monthly Loan Payment Calculation
First, we calculate the monthly loan payments for both the non-hybrid and hybrid cars using the given loan payment formula:
[tex]\[ PMT = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \][/tex]
where:
- [tex]\( P \)[/tex] is the loan amount (either \[tex]$41,000 for the non-hybrid or \$[/tex]44,000 for the hybrid).
- [tex]\( r \)[/tex] is the annual interest rate (4.68%), converted to a decimal (0.0468).
- [tex]\( n \)[/tex] is the number of payments per year (12 since monthly payments).
- [tex]\( t \)[/tex] is the term of the loan in years (5 years).
Given:
- [tex]\( P_\text{non-hybrid} = \$41,000 \)[/tex]
- [tex]\( P_\text{hybrid} = \$44,000 \)[/tex]
- Annual interest rate [tex]\( r = 4.68\% = 0.0468 \)[/tex]
- Number of payments per year [tex]\( n = 12 \)[/tex]
- Term of loan [tex]\( t = 5 \)[/tex] years
Firstly, for the non-hybrid car:
Substitute the values into the formula to get the monthly payment:
[tex]\[ PMT_\text{non-hybrid} = \frac{41{,}000 \left(\frac{0.0468}{12}\right)}{1 - \left(1 + \frac{0.0468}{12}\right)^{-12 \times 5}} \][/tex]
This results in:
[tex]\[ PMT_\text{non-hybrid} = \$767.72 \][/tex]
Next, for the hybrid car:
[tex]\[ PMT_\text{hybrid} = \frac{44{,}000 \left(\frac{0.0468}{12}\right)}{1 - \left(1 + \frac{0.0468}{12}\right)^{-12 \times 5}} \][/tex]
This results in:
[tex]\[ PMT_\text{hybrid} = \$823.90 \][/tex]
Therefore:
- The monthly loan payment for the non-hybrid version of the car is [tex]$767.72. - The monthly loan payment for the hybrid version of the car is $[/tex]823.90.
### 2. Annual Fuel Cost Calculation
We will calculate the annual fuel cost for both cars.
Given:
- Annual miles driven [tex]\( = 6,000 \)[/tex]
- Cost of gas per gallon [tex]\( = \$4.12 \)[/tex]
- Miles per gallon for the non-hybrid car [tex]\( = 35 \)[/tex]
- Miles per gallon for the hybrid car [tex]\( = 49 \)[/tex]
For the non-hybrid car:
[tex]\[ \text{Annual fuel cost}_\text{non-hybrid} = \frac{6{,}000 \text{ miles}}{35 \text{ mpg}} \times 4.12 \text{ dollars/gallon} \][/tex]
This results in:
[tex]\[ \text{Annual fuel cost}_\text{non-hybrid} = \$706.29 \][/tex]
For the hybrid car:
[tex]\[ \text{Annual fuel cost}_\text{hybrid} = \frac{6{,}000 \text{ miles}}{49 \text{ mpg}} \times 4.12 \text{ dollars/gallon} \][/tex]
This results in:
[tex]\[ \text{Annual fuel cost}_\text{hybrid} = \$504.49 \][/tex]
Therefore:
- The annual fuel cost for the non-hybrid version of the car is [tex]$706.29. - The annual fuel cost for the hybrid version of the car is $[/tex]504.49.
### 3. Annual Cost of Ownership
Now, we determine the total annual cost of ownership for each car, which includes the monthly loan payments and fuel costs.
For the non-hybrid car:
[tex]\[ \text{Annual cost}_\text{non-hybrid} = (\text{Monthly payment}_\text{non-hybrid} \times 12) + \text{Annual fuel cost}_\text{non-hybrid} \][/tex]
[tex]\[ \text{Annual cost}_\text{non-hybrid} = (767.72 \times 12) + 706.29 = 9352.64 + 706.29 = 9918.97 \][/tex]
For the hybrid car:
[tex]\[ \text{Annual cost}_\text{hybrid} = (\text{Monthly payment}_\text{hybrid} \times 12) + \text{Annual fuel cost}_\text{hybrid} \][/tex]
[tex]\[ \text{Annual cost}_\text{hybrid} = (823.90 \times 12) + 504.49 = 9886.80 + 504.49 = 10391.28 \][/tex]
As a result:
- The annual cost of ownership for the non-hybrid version of the car is [tex]$9918.97. - The annual cost of ownership for the hybrid version of the car is $[/tex]10391.28.
### Conclusion
We compare the total annual costs to determine which car is more economical.
Comparing the annual costs:
[tex]\[ 9918.97 < 10391.28 \][/tex]
This shows:
- The non-hybrid version of the car is more economical over the course of one year.
Let's find the cost difference between the two cars:
[tex]\[ \text{Cost difference} = 10391.28 - 9918.97 = 472.30 \][/tex]
Therefore:
- The annual cost difference between the two cars is $472.30.
### 1. Monthly Loan Payment Calculation
First, we calculate the monthly loan payments for both the non-hybrid and hybrid cars using the given loan payment formula:
[tex]\[ PMT = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \][/tex]
where:
- [tex]\( P \)[/tex] is the loan amount (either \[tex]$41,000 for the non-hybrid or \$[/tex]44,000 for the hybrid).
- [tex]\( r \)[/tex] is the annual interest rate (4.68%), converted to a decimal (0.0468).
- [tex]\( n \)[/tex] is the number of payments per year (12 since monthly payments).
- [tex]\( t \)[/tex] is the term of the loan in years (5 years).
Given:
- [tex]\( P_\text{non-hybrid} = \$41,000 \)[/tex]
- [tex]\( P_\text{hybrid} = \$44,000 \)[/tex]
- Annual interest rate [tex]\( r = 4.68\% = 0.0468 \)[/tex]
- Number of payments per year [tex]\( n = 12 \)[/tex]
- Term of loan [tex]\( t = 5 \)[/tex] years
Firstly, for the non-hybrid car:
Substitute the values into the formula to get the monthly payment:
[tex]\[ PMT_\text{non-hybrid} = \frac{41{,}000 \left(\frac{0.0468}{12}\right)}{1 - \left(1 + \frac{0.0468}{12}\right)^{-12 \times 5}} \][/tex]
This results in:
[tex]\[ PMT_\text{non-hybrid} = \$767.72 \][/tex]
Next, for the hybrid car:
[tex]\[ PMT_\text{hybrid} = \frac{44{,}000 \left(\frac{0.0468}{12}\right)}{1 - \left(1 + \frac{0.0468}{12}\right)^{-12 \times 5}} \][/tex]
This results in:
[tex]\[ PMT_\text{hybrid} = \$823.90 \][/tex]
Therefore:
- The monthly loan payment for the non-hybrid version of the car is [tex]$767.72. - The monthly loan payment for the hybrid version of the car is $[/tex]823.90.
### 2. Annual Fuel Cost Calculation
We will calculate the annual fuel cost for both cars.
Given:
- Annual miles driven [tex]\( = 6,000 \)[/tex]
- Cost of gas per gallon [tex]\( = \$4.12 \)[/tex]
- Miles per gallon for the non-hybrid car [tex]\( = 35 \)[/tex]
- Miles per gallon for the hybrid car [tex]\( = 49 \)[/tex]
For the non-hybrid car:
[tex]\[ \text{Annual fuel cost}_\text{non-hybrid} = \frac{6{,}000 \text{ miles}}{35 \text{ mpg}} \times 4.12 \text{ dollars/gallon} \][/tex]
This results in:
[tex]\[ \text{Annual fuel cost}_\text{non-hybrid} = \$706.29 \][/tex]
For the hybrid car:
[tex]\[ \text{Annual fuel cost}_\text{hybrid} = \frac{6{,}000 \text{ miles}}{49 \text{ mpg}} \times 4.12 \text{ dollars/gallon} \][/tex]
This results in:
[tex]\[ \text{Annual fuel cost}_\text{hybrid} = \$504.49 \][/tex]
Therefore:
- The annual fuel cost for the non-hybrid version of the car is [tex]$706.29. - The annual fuel cost for the hybrid version of the car is $[/tex]504.49.
### 3. Annual Cost of Ownership
Now, we determine the total annual cost of ownership for each car, which includes the monthly loan payments and fuel costs.
For the non-hybrid car:
[tex]\[ \text{Annual cost}_\text{non-hybrid} = (\text{Monthly payment}_\text{non-hybrid} \times 12) + \text{Annual fuel cost}_\text{non-hybrid} \][/tex]
[tex]\[ \text{Annual cost}_\text{non-hybrid} = (767.72 \times 12) + 706.29 = 9352.64 + 706.29 = 9918.97 \][/tex]
For the hybrid car:
[tex]\[ \text{Annual cost}_\text{hybrid} = (\text{Monthly payment}_\text{hybrid} \times 12) + \text{Annual fuel cost}_\text{hybrid} \][/tex]
[tex]\[ \text{Annual cost}_\text{hybrid} = (823.90 \times 12) + 504.49 = 9886.80 + 504.49 = 10391.28 \][/tex]
As a result:
- The annual cost of ownership for the non-hybrid version of the car is [tex]$9918.97. - The annual cost of ownership for the hybrid version of the car is $[/tex]10391.28.
### Conclusion
We compare the total annual costs to determine which car is more economical.
Comparing the annual costs:
[tex]\[ 9918.97 < 10391.28 \][/tex]
This shows:
- The non-hybrid version of the car is more economical over the course of one year.
Let's find the cost difference between the two cars:
[tex]\[ \text{Cost difference} = 10391.28 - 9918.97 = 472.30 \][/tex]
Therefore:
- The annual cost difference between the two cars is $472.30.
