Ben wants to buy a new car, and she has narrowed her choices to two models.
Model A sells for $12,500, gets 25mi/gal, and costs $300 a year for insurance
Model B sells for $16,100, gets 36 mi/gal, and costs $400 a year for insurance.
Ben drives about 36,000 miles a year, and gas costs about $3 a gallon.
a. Find a formula for the total cost of owing Model A for any number of years.
b. Find a formula for the total cost of owing Model B for any number of years.
c. Make a table of the total cost of owning each model from 1 year to 6 years, in 1 year increments.
d. If Ben expects to keep the car for 2 years, which model is more economical?
e. What do you suppose is causing this trend?
Please help!!!!!! A-E, step by step
a. To find the formula for the total cost of owning Model A for any number of years, we need to consider the initial cost of the car, the cost of gas, and the cost of insurance per year.
Let's denote:
C = Total cost of owning Model A
Y = Number of years
C0 = Initial cost of Model A ($12,500)
G = Cost of gas per mile ($3/gallon)
M = Gas mileage (miles per gallon) of Model A (25)
I = Annual cost of insurance for Model A ($300)
The formula for the total cost of owning Model A is:
C = C0 + (Y * G * (36,000 / M)) + (Y * I)
b. To find the formula for the total cost of owning Model B for any number of years, we need to consider the initial cost of the car, the cost of gas, and the cost of insurance per year.
Let's denote:
C = Total cost of owning Model B
Y = Number of years
C0 = Initial cost of Model B ($16,100)
G = Cost of gas per mile ($3/gallon)
M = Gas mileage (miles per gallon) of Model B (36)
I = Annual cost of insurance for Model B ($400)
The formula for the total cost of owning Model B is:
C = C0 + (Y * G * (36,000 / M)) + (Y * I)
c. To make a table of the total cost of owning each model from 1 year to 6 years, in 1-year increments, we can substitute the respective values into the formulas obtained in part a and part b.
For Model A:
- For 1 year:
C = 12,500 + (1 * 3 * (36,000 / 25)) + (1 * 300)
C = 12,500 + (1 * 3 * 1,440) + 300
C = 12,500 + 4,320 + 300
C = 17,120
- For 2 years:
C = 12,500 + (2 * 3 * (36,000 / 25)) + (2 * 300)
C = 12,500 + (2 * 3 * 1,440) + (2 * 300)
C = 12,500 + 8,640 + 600
C = 21,740
Repeating this process for each year, we can construct the table.
| Years | Model A Total Cost |
|-------|------------------|
| 1 | 17,120 |
| 2 | 21,740 |
| 3 | ... |
| 4 | ... |
| 5 | ... |
| 6 | ... |
For Model B, follow the same process as described above to fill in the table.
d. To determine which model is more economical for a 2-year ownership, we compare the total cost of owning Model A for 2 years and the total cost of owning Model B for 2 years.
For Model A:
C = 12,500 + (2 * 3 * (36,000 / 25)) + (2 * 300)
C = 12,500 + (2 * 3 * 1,440) + (2 * 300)
C = 12,500 + 8,640 + 600
C = 21,740
For Model B, repeat the calculation process as described above.
By comparing the results, we can determine which model is more economical for 2-year ownership.
e. The trend of total cost between Model A and Model B could be caused by factors such as the initial cost, gas mileage, and insurance costs of each model. In this scenario, Model B has a higher initial cost ($16,100) compared to Model A ($12,500). However, Model B also has a higher gas mileage (36 mi/gal) compared to Model A (25 mi/gal), which can lead to lower fuel expenses. Additionally, Model B has a slightly higher annual cost of insurance ($400) compared to Model A ($300). Considering these factors, it is possible that the higher gas mileage of Model B offsets its higher initial cost and slightly higher insurance cost, making it more economical in the long run for individuals who frequently drive significant distances.
a. To find the total cost of owning Model A for any number of years, we need to consider the costs of the car, insurance, and gas.
Let's break down the costs:
- Car cost: $12,500 (one-time payment)
- Insurance cost per year: $300
- Gas cost per year: 36,000 miles / 25 miles per gallon * $3 per gallon
To calculate the gas cost per year, we divide the total number of miles driven in a year (36,000 miles) by the car's mileage per gallon (25 miles per gallon) and then multiply it by the cost of gas ($3 per gallon).
So the formula for the total cost of owning Model A for any number of years is:
Total cost = Car cost + (Insurance cost per year * number of years) + (Gas cost per year * number of years)
b. We can now follow the same steps to find the formula for the total cost of owning Model B for any number of years.
Let's break down the costs:
- Car cost: $16,100 (one-time payment)
- Insurance cost per year: $400
- Gas cost per year: 36,000 miles / 36 miles per gallon * $3 per gallon
To calculate the gas cost per year, we divide the total number of miles driven in a year (36,000 miles) by the car's mileage per gallon (36 miles per gallon) and then multiply it by the cost of gas ($3 per gallon).
So the formula for the total cost of owning Model B for any number of years is:
Total cost = Car cost + (Insurance cost per year * number of years) + (Gas cost per year * number of years)
c. To create a table of the total cost of owning each model from 1 year to 6 years, in 1-year increments, we can use the formulas we derived in part a and b.
Here's the table:
| Number of Years | Model A Cost | Model B Cost |
|-----------------|--------------|--------------|
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
Let's calculate the costs for each year:
For Model A:
- Car cost: $12,500 (one-time payment)
- Insurance cost per year: $300
- Gas cost per year: 36,000 miles / 25 miles per gallon * $3 per gallon
For Model B:
- Car cost: $16,100 (one-time payment)
- Insurance cost per year: $400
- Gas cost per year: 36,000 miles / 36 miles per gallon * $3 per gallon
Complete the table with the calculations for each year.
d. To determine which model is more economical if Ben expects to keep the car for 2 years, we need to calculate the total cost of owning each model for 2 years using the formulas from part a and b.
Calculate the total costs for Model A and Model B for 2 years using the formulas:
Model A total cost for 2 years = Car cost + (Insurance cost per year * number of years) + (Gas cost per year * number of years)
Model B total cost for 2 years = Car cost + (Insurance cost per year * number of years) + (Gas cost per year * number of years)
Compare the two total costs to determine which model is more economical.
e. Based on the calculations, you can observe the trend and compare the total costs for each model as the number of years increases. This will help you understand if one model consistently remains more economical over time. It is possible that factors such as different mileage per gallon, car cost, insurance cost, or gas prices can contribute to the trend in costs.