2)Jenny 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.
Jenny 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 Jenny expects to keep the car for 2 years, which model is more economical?
e. What do you suppose is causing this trend?
a. The total cost of owning Model A for any number of years can be calculated using the following formula:
Total Cost of Model A = (Price of Model A + (Average Annual Mileage * (Price of Gas / Mileage per Gallon)) + Annual Insurance Cost) * Number of Years
b. The total cost of owning Model B for any number of years can be calculated using the following formula:
Total Cost of Model B = (Price of Model B + (Average Annual Mileage * (Price of Gas / Mileage per Gallon)) + Annual Insurance Cost) * Number of Years
c. A table of the total cost of owning each model from 1 year to 6 years, in 1 year increments, can be created as follows:
Year Total Cost of Model A Total Cost of Model B
1 Formula for year 1 Formula for year 1
2 Formula for year 2 Formula for year 2
3 Formula for year 3 Formula for year 3
4 Formula for year 4 Formula for year 4
5 Formula for year 5 Formula for year 5
6 Formula for year 6 Formula for year 6
d. To determine which model is more economical if Jenny expects to keep the car for 2 years, calculate the total cost of owning each model for 2 years using the formulas in a and b. Compare the total costs for both models and choose the one with the lower total cost.
e. The trend of Model B being more expensive than Model A could be attributed to several factors such as the higher price of Model B, higher insurance cost, and the lower mileage per gallon of Model B. These factors contribute to the increased overall cost of owning Model B compared to Model A.
a. To find the formula for the total cost of owning Model A for any number of years, we need to consider the cost of the car itself, the cost of gas, and the cost of insurance.
The cost of the car itself is $12,500.
The cost of gas per year can be calculated using the formula: cost of gas = (number of miles driven per year) / (miles per gallon) * (price per gallon). In this case, Jenny drives 36,000 miles per year, and gas costs $3 per gallon. So the formula for the cost of gas is: cost of gas = 36,000 / 25 * 3.
The cost of insurance per year is $300.
Therefore, the formula for the total cost of owning Model A for any number of years is:
Total cost = (Cost of car) + (Number of years) * [(Number of miles driven per year) / (miles per gallon) * (price per gallon)] + (Number of years) * (insurance cost per year)
Plugging in the values, the formula becomes:
Total cost = $12,500 + (Number of years) * [(36,000 / 25) * 3] + (Number of years) * $300
b. Similarly, for Model B, the formula for the total cost of owning Model B for any number of years is:
Total cost = (Cost of car) + (Number of years) * [(Number of miles driven per year) / (miles per gallon) * (price per gallon)] + (Number of years) * (insurance cost per year)
Plugging in the values, the formula becomes:
Total cost = $16,100 + (Number of years) * [(36,000 / 36) * 3] + (Number of years) * $400
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 simply substitute the values for the number of years in the formulas obtained in parts a and b.
Using the formulas:
For Model A:
Number of years | Total cost
1 | (Total cost calculated using the formula)
2 | (Total cost calculated using the formula)
3 | (Total cost calculated using the formula)
4 | (Total cost calculated using the formula)
5 | (Total cost calculated using the formula)
6 | (Total cost calculated using the formula)
For Model B:
Number of years | Total cost
1 | (Total cost calculated using the formula)
2 | (Total cost calculated using the formula)
3 | (Total cost calculated using the formula)
4 | (Total cost calculated using the formula)
5 | (Total cost calculated using the formula)
6 | (Total cost calculated using the formula)
d. To determine which model is more economical if Jenny expects to keep the car for 2 years, we can calculate the total cost for both models using the formulas obtained in parts a and b, and then compare the two values. The model with the lower total cost would be more economical.
Plug in the value of 2 for the number of years in the formulas obtained in parts a and b, and calculate the total cost for both Model A and Model B. Then compare the two values to determine which model is more economical.
e. The trend in this case is that Model B has a higher total cost of ownership compared to Model A. This is likely due to the higher purchase price of Model B ($16,100) compared to Model A ($12,500), as well as the lower fuel efficiency of Model B (36 miles per gallon) compared to Model A (25 miles per gallon). These factors contribute to the overall cost of owning and operating the car, making Model A more economical in this scenario.