23 Jan 2006 - 04:39

another triumph for dimensional analysis

Bernie and I are in the throes of having to buy a new car. This is because our ten-year-old Grand Voyager's transmission finally went out, and will cost $2500 to fix (apparently 96 Grand Voyagers had transmission issues; the guys at the dealer told us we were pretty lucky to have one last 130,000 miles). Given that every member of our family who is old enough to drive (that is, everyone except Ben and the dogs) has put a dent in it, the exterior is fairly rough, and the interior is no picnic, and the transmission is gone; we have pretty much decided that it deserves to rest in peace.

I say 'pretty much decided' because we keep dithering about it. If we don't fix it, what do we replace it with? Our other car is a Honda Civic. We love it; it gets great gas mileage; the Voyager didn't. Bernie is sure that gas is going to go up to 4 dollars per gallon relatively soon, which makes a second Civic (or even a hybrid) an attractive option; but with the boys (3) and the dogs (2) and the neighborhood gang of kids (4) to transport, another ten years of a minivan might be what we need to sign up for. We aren't quite empty-nesters yet.

Bernie asked me to calculate the difference in gas costs over a year between the Honda Civic and the Dodge Grand Caravan, in a world where gas is 4.00 per gallon. I was a little flummoxed; I had the pieces of the puzzle, but couldn't quite see how to put them together.

In the end, I put them together in a completely brain-dead way, using dimensional analysis.

I wanted to figure out the cost per year of driving both a Civic and a Caravan. The pieces I had were: dollars per gallon (4.00 in my brave new imagined world), miles per gallon (35 for the Civic and 25 for the van), and miles per year (13,000, if the new car gets anything like the workout the old one did). I knew that if I put the pieces together so that they came out to dollars per year, I'd likely have my answer.

Dollars needed to end up in the numerator, and years in the denominator; miles and gallons need to cancel. Playing around with the pieces for a few minutes shows that the only way to combine the pieces so that you end up with dollars per year is to write:

Try moving the pieces around yourself. There's no other way it can work. Plus, the answers make sense; the cost per year is $2080 for the van, and $1485 for the Honda (by contrast, the cost per year to operate a Prius, the Toyota hybid vehicle, which gets 50 miles per gallon, is $1040).

Dimensional analysis is a great tool for kids to learn to use to check their work in math and science. Kids will be taught a bit of dimensional analysis in middle school science; unfortunately, it's not emphasized in most math classes and it's not drilled enough in science classes. Kids may be taught to check their units to make sure they come out right (but probably not); they certainly won't be taught that they can use dimensional analysis to try to figure out what to do when they have no clue (which is what I did here).

For other material on dimensional analysis, see:

DrMathOnFractionsAndUnits
UnitConversionsPart2
Dan's dimensional dominoes (manipulatives)
TeachingUnitConversions
DimensionalAnalysis

-- CarolynJohnston - 23 Jan 2006

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Carolyn,

I had to give up my 96 Plymouth Voyager for exactly the same reason!!!

BTW, the dimensional analysis is great!!

-- AnneDwyer - 23 Jan 2006

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Dodge Grand Caravans suck.

The only car to buy, ever, under any circumstances, is a Toyota.

Or a Honda.

-- CatherineJohnson - 23 Jan 2006

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I LOVE dimensional analysis, now that I've learned it (JUST learned it).

Saxon 8/7 has 3 dimensional analysis problems so far that are fantastic; I was planning to post them all.

But here's my question: Saxon hasn't used the term 'dimensional analysis.'

He uses the terms 'unit multiplier' and 'unit conversion.'

Are 'dimensional analysis' and 'unit multipler/unit conversion' exactly the same thing?

And if so, why wouldn't Saxon use the term?

-- CatherineJohnson - 23 Jan 2006

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How do you use dimensional analysis to check your work?

-- CatherineJohnson - 23 Jan 2006

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Dodge grand caravans don't suck. We had ours (it was a Voyager, same thing) for ten years and got 130000 miles out of it, and it was low-trouble.

We also have a Honda Civic and the trouble level with that has been about the same as it was with the Voyager.

-- CarolynJohnston - 23 Jan 2006

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I've always heard that the Civic isn't quite like the Accord. I have no idea where I heard that, but long ago I was warned not to think they were the same car with differences only in size.

I've always had Accords and they will drive for eternity. Friends of mine with Civics (only a couple) seemed to report problems that my Accords (or people with Toyota Corollas and Camrys) never had.

That's my very scientifically-based opinion for ya.'

-- SusanS - 23 Jan 2006

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Catherine

Unit multipliers/conversions are a subset of the toolkit of dimensional analysis.

Here's how dimensional analysis can be used to check your work; if you do all your calculations with units, then at the end your units ought to come out right in your answer. If they don't, you've gone wrong.

Also, at every stage, quantities that you are adding/subtracting ought to have the same units; for example, you can't add something that has units of miles to something that has units of miles/gallon, and if you come to a place where you seem to be having to do that, then you've gone wrong.

-- CarolynJohnston - 23 Jan 2006

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I've always had Accords and they will drive for eternity.

I believe it.

The weird thing is, I get car sick in Hondas.

That happened to me a gazillion years ago, when they used to make the Prelude....and then I discovered it still happens to me today.

Strange.

-- CatherineJohnson - 23 Jan 2006

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Martine is driving the Toyota Camry we bought in....1988, I think.

-- CatherineJohnson - 23 Jan 2006

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Here's how dimensional analysis can be used to check your work; if you do all your calculations with units, then at the end your units ought to come out right in your answer. If they don't, you've gone wrong.

Also, at every stage, quantities that you are adding/subtracting ought to have the same units; for example, you can't add something that has units of miles to something that has units of miles/gallon, and if you come to a place where you seem to be having to do that, then you've gone wrong.

oh, neat

thanks!

-- CatherineJohnson - 23 Jan 2006

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I've gotta post the great Saxon problems.....

-- CatherineJohnson - 23 Jan 2006

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At 130K miles, Hondas/Toyotas/Nissans are just entering early middle age.

-- GoogleMaster - 23 Jan 2006

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At 130K miles, Hondas/Toyotas/Nissans are just entering early middle age.

lol!

that's sure been my experience

-- CatherineJohnson - 23 Jan 2006

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My old chemistry prof called it "factor-label:" You start by figuring out what you want to find out, then lay out all of your ratios and conversion factors, one multiplied by the other (always multiply, invert the relation if you need to), until you can factor out all the unwanted labels (units) in between and leave behind the ones you want. The check is that you shouldn't have any stray units hanging over when you're done.

So your problem becomes:

  dollars   4 dollars   1 gallon   13000 miles
  ------- = --------- * -------- * -----------  
    year     gallon     35 miles      1 year

In the numerator, you've got: dollars-gallons-miles In the denominator: gallons-miles-year Factor out: gallons,miles Leaving: dollars in the numerator, years in the denominator

The "always multiply" trick bypasses figuring out complicated expressions like your first one with fractions in both the numerator and denominator.

-- KtmGuest - 25 Jan 2006

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My old chemistry prof called it "factor-label:"

So did mine. Must be a chemistry thing.

-- KDeRosa - 25 Jan 2006