PS$\mathrm{.}53$ 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 $$18,500$ to purchase or $(2)$ a used sedan that would cost $$13,200.$

They anticipate that the new subcompact would get $34$ miles per gallon $($combined highway and around town driving$)$ while the sedan would get $23$ miles per gallon. Based on their road tripping history they expect to drive $10,000$ miles per year. For the purposes of their analysis they are assuming that gas will cost $$3\mathrm{.}12$ per gallon.

Question: How many miles would the Ricks need to drive before the cost of these two options would be the same? $($Display your answer to the nearest whole number.$)$

$($Hint: The challenge with this problem is finding the variable costs. First determine the variable costs per mile, in terms of gasoline costs. From there you can easily use the break$-$even formula to find the break$-$even point in terms of miles$$and then do some simple math to get the break$-$even point in years.$)$

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Incorrect

Your Answer: $113928$

Incorrect

Your Answer: $113927$

How many years would it take for these two options to cost the same? $($Display your answer to two decimal places.$)$

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Suppose a severe disruption to the petroleum supply resulted in a $$1\mathrm{.}75$ increase to the price of gasoline. How many miles would it now take before the cost of these two options would be the same? $($Display your answer to the nearest whole number.$)$

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