Quote:
Originally Posted by Spuds
Let Y be cost, X be distance, Pr be price of regular, and Pp be price of Premium. Assume mpg is a constant.
For the truck:
Yt = (Pr/12)X
For the car:
Yc = (Pp/20)X
What we want to know is what prices cause the rate of cost for the car and truck to be equal. Thus:
Y't = Y'c
(The derivative of Y is Y', and we want to set both equal to each other)
Therefore:
Pr/12 = Pp/20 --> Pr/Pp = 12/20 = 3/5 = 0.6
So when the ratio of regular price to premium price is 3:5 then it is a wash. If the ratio is closer (eg 4:5), then it will cost less per mile to drive the car with premium gas. If the ratio is farther (2:5) then it will be more cost efficient to drive the truck. If your mpg changes, that changes the crossover ratio.
TLDR: If premium is $5.60/gal, then regular would have to be $3.36/gal or less to make driving the truck more cost efficient, assuming your mpg numbers are accurate.
Also, imma be real embarrassed if I did that wrong...
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I'd like to revisit this because I finally got around to doing it for myself and think it's important to confess how much trouble I had with it. One of the most important concepts I learned in school was unit agreement. I rarely got it right until I learned to keep those pesky units in the mix. Spoiler alert - we agree.
Our units are dollars, gallons, and miles.
price per gallon of premium
price per gallon of regular
miles per gallon of premium (car)
miles per gallon of regular (truck)
$
p/gal
$
r/gal
mi
p/gal
mi
r/gal
$/gal(gal/mi)=$/mi
we break even when
$
p/mi
p=$
r/mi
r
so, as you also concluded,
$
p(mi
r)/mi
p=$
r