Question
A car rental company has two pricing plans: Plan A: $40 per day with no mileage fee (represented by y = 40x ,
where x is the number of days) Plan B: $30 per day plus $0.15 per mile driven
Derive the equation for Plan B, explaining each component. Then, analyze how the equations differ and what
this means in terms of cost for customers. In what situations would each plan be more economical?
where x is the number of days) Plan B: $30 per day plus $0.15 per mile driven
Derive the equation for Plan B, explaining each component. Then, analyze how the equations differ and what
this means in terms of cost for customers. In what situations would each plan be more economical?
Answers
GPT-5 mini
Let y be the total cost in dollars, x the number of days, and m the total miles driven.
Equation for Plan B:
y = 30x + 0.15m
Explanation of components:
- 30x: the flat daily charge, $30 per day multiplied by number of days x.
- 0.15m: the mileage charge, $0.15 per mile times the number of miles m.
- y: total cost in dollars.
How the equations differ:
- Plan A: y = 40x. Cost depends only on days (no mileage charge).
- Plan B: y = 30x + 0.15m. Lower daily rate but adds a per-mile fee, so cost depends on both days and miles.
When each plan is more economical
Set Plan A = Plan B to find the break-even: 40x = 30x + 0.15m
=> 10x = 0.15m
=> m = (10 / 0.15) x ≈ 66.67 x
Interpretation:
- If total miles m < 66.67x (i.e., average miles per day < 66.67), then Plan B is cheaper.
- If m > 66.67x (average miles per day > 66.67), then Plan A is cheaper.
- If m = 66.67x, the two plans cost the same.
Quick examples
- 3 days, 100 miles: Plan A = $120; Plan B = 30(3)+0.15(100)=90+15= $105 → Plan B cheaper.
- 3 days, 250 miles: Plan A = $120; Plan B = 90+37.50= $127.50 → Plan A cheaper.
Practical note: if you expect low mileage (commuting around town or mostly parked), Plan B usually saves money. If you expect long driving (road trips or high daily mileage), Plan A (flat $40/day) is likely better because it effectively gives “free” miles.
Equation for Plan B:
y = 30x + 0.15m
Explanation of components:
- 30x: the flat daily charge, $30 per day multiplied by number of days x.
- 0.15m: the mileage charge, $0.15 per mile times the number of miles m.
- y: total cost in dollars.
How the equations differ:
- Plan A: y = 40x. Cost depends only on days (no mileage charge).
- Plan B: y = 30x + 0.15m. Lower daily rate but adds a per-mile fee, so cost depends on both days and miles.
When each plan is more economical
Set Plan A = Plan B to find the break-even: 40x = 30x + 0.15m
=> 10x = 0.15m
=> m = (10 / 0.15) x ≈ 66.67 x
Interpretation:
- If total miles m < 66.67x (i.e., average miles per day < 66.67), then Plan B is cheaper.
- If m > 66.67x (average miles per day > 66.67), then Plan A is cheaper.
- If m = 66.67x, the two plans cost the same.
Quick examples
- 3 days, 100 miles: Plan A = $120; Plan B = 30(3)+0.15(100)=90+15= $105 → Plan B cheaper.
- 3 days, 250 miles: Plan A = $120; Plan B = 90+37.50= $127.50 → Plan A cheaper.
Practical note: if you expect low mileage (commuting around town or mostly parked), Plan B usually saves money. If you expect long driving (road trips or high daily mileage), Plan A (flat $40/day) is likely better because it effectively gives “free” miles.