Question

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 $22,000 to purchase or (2) a used sedan that would cost $17,000. They anticipate that the new subcompact would get 36 miles per gallon (combined highway and around town driving) while the sedan would get 25 miles per gallon. Based on their road tripping history they expect to drive 17,000 miles per year. For the purposes of their analysis they are assuming that gas will cost $3.29 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.) How many years would it take for these two options to cost the same? (Display your answer to two decimal places.) Suppose a severe disruption to the petroleum supply resulted in a $1.50 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.)

Answer 1

Step-by-step explanation:

To find out how many miles the Ricks need to drive before the cost of the two options is the same, we can use the break-even point formula:

Break-even point (in miles) = (Fixed cost of the subcompact car - Fixed cost of the sedan) / (Variable cost per mile for the subcompact car - Variable cost per mile for the sedan)

Let's calculate it step by step:

1. Calculate the fixed costs:

• Fixed cost of the new subcompact car: $22,000
• Fixed cost of the used sedan: $17,000

2. Calculate the variable costs per mile:

• Variable cost per mile for the subcompact car: (Cost of gas per gallon) / (Miles per gallon) = $3.29 / 36 = $0.0914 per mile
• Variable cost per mile for the sedan: (Cost of gas per gallon) / (Miles per gallon) = $3.29 / 25 = $0.1316 per mile

3. Plug the values into the formula:

Break-even point (in miles) = ($22,000 - $17,000) / ($0.0914 - $0.1316)

Break-even point (in miles) = $5,000 / (-$0.0402)

Now, calculate the break-even point in miles:

Break-even point (in miles) ≈ -124,378.11 miles

This negative value doesn't make sense in this context. The break-even point should be a positive number, so it means that the sedan option is more cost-effective for the Ricks. In other words, they will never reach a point where the subcompact car becomes cheaper.

To find out how many years it would take for these two options to cost the same, you can use the annual mileage of 17,000 miles and the calculated break-even point in miles (which is approximately 124,378 miles). Divide the break-even point in miles by the annual mileage:

Years to break-even = Break-even point (in miles) / Annual mileage

Years to break-even ≈ 124,378 miles / 17,000 miles ≈ 7.31 years

So, it would take approximately 7.31 years for the two options to cost the same.

Now, let's consider the scenario where there's a $1.50 increase in the price of gasoline. The new cost of gas per gallon would be $3.29 + $1.50 = $4.79.

To find out how many miles it would now take before the cost of these two options would be the same, repeat the calculation with the new variable cost per mile for the subcompact car:

Variable cost per mile for the subcompact car with increased gas price:

= (New cost of gas per gallon) / (Miles per gallon)

= $4.79 / 36

≈ $0.1331 per mile

Now, use the break-even point formula with this new variable cost per mile:

Break-even point (in miles) = ($22,000 - $17,000) / ($0.1331 - $0.1316)

Break-even point (in miles) ≈ $5,000 / $0.0015

Break-even point (in miles) ≈ 3,333.33 miles

With the increased gas price, it would now take approximately 3,333 miles before the cost of these two options would be the same.