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PS.53 Brother I.D. Ricks is a faculty member at BYU-Idaho whose grandchildren live in Oklahoma and California. He and his wife would like to visit their grandchildren at least once a year in these states. They currently have one vehicle with well over 100,000 miles on it, so they want to buy a newer vehicle with fewer miles and that gets better gas mileage. They are considering two options: (1) a new subcompact car that would cost $19,500 to purchase or (2) a used sedan that would cost $13,700.

They anticipate that the new subcompact would get 41 miles per gallon (combined highway and around town driving) while the sedan would get 26 miles per gallon. Based on their road tripping history they expect to drive 18,000 miles per year. For the purposes of their analysis they are assuming that gas will cost $2.24 per gallon.

Required:
a. How many miles would the Ricks need to drive before the cost of these two options would be the same?
b. How many years would it take for these two options to cost the same?

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Answer:

Results are below.

Explanation:

First, we need to structure the total cost of each car:

Subcompact= 19,500 + 0.055*x

Sedan= 13,700 + 0.086*x

x= miles driven

Now, we equal both formulas and isolate x:

19,500 + 0.055x = 13,700 + 0.086x

5,800 = 0.031x

187,096.77 = x

The indifference point is 187,096.77 miles.

Prove:

Subcompact= 19,500 + 0.055*187,096.77= $29,790.32

Sedan= 13,700 + 0.086*187,096.77= $29,790.32

Finally, the time required:

Number of years= 187,096.77/18,000= 10.39 years

(a) First, we need to structure the total cost of each car is:  

  • When the Subcompact= 19,500 + 0.055*x
  • Then the Sedan= 13,700 + 0.086*x
  • So that x= miles driven
  • After that Now, we equal both formulas and isolate x:
  • 19,500 + 0.055x = 13,700 + 0.086x
  • 5,800 = 0.031x
  • 187,096.77 = x
  • When The indifference point is 187,096.77 miles.
  • hence Prove that:  
  • When the Subcompact is = 19,500 + 0.055*187,096.77= $29,790.32
  • when the Sedan is = 13,700 + 0.086*187,096.77= $29,790.32

                          so that Finally, the time required:

(b) The Number of years is = 187,096.77/18,000= 10.39 years

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