A View from the Back of the Envelope - approximations and the fun behind them.
posted by Gyan (25 comments total) 1 user marked this as a favorite

Good post, Gyan. I read an interesting article on the Fermi principle in The Sciences magazine. When the first atomic bomb went off in New Mexico, Fermi threw some torn pieces of paper into the air, and from the distance they flew he estimated the power of the explosion. After all the official math was done, he was not far off the mark. Approximation works because, as when rounding off a restaurant bill, the errors tend to fall both directions and cancel one another.
posted by weapons-grade pandemonium at 11:02 AM on October 18, 2005

Interesting stuff.

My sister and I had a "back of the envelope" question this weekend: If there's a penny in my car's change drawer, how many miles will I drive before its extra weight has consumed its value in gasoline? Is it more or less than the expected lifespan of the car?
posted by justkevin at 11:13 AM on October 18, 2005

Cool link, Gyan. Thanks. We all use these ideas regularly (some more adeptly than others). It's interesting to step back and look at the process in more detail.
posted by raedyn at 11:20 AM on October 18, 2005

Not a snark - but must you have your thumb out, one eye shut, pencil behind your ear and your tongue sticking out the corner of your mouth while viewing the site?

Some interesting stuff
posted by Smedleyman at 11:24 AM on October 18, 2005

Yeah, terrific site if you know what I mean
posted by donfactor at 11:36 AM on October 18, 2005

I love this stuff. I took a "we're going to explain the universe and how old everything is" course for non-scientists in college and this was the first thing our prof taught us. It's incredibly useful when you think about how much multiplication we (well, me at least) go through in our daily lives.

OK, justkevin, how does this sound- your car weighs a ton (2000 lbs). A penny weighs .01 lbs. Your car holds 15 gallons of gas, and averages 20 mph, so a full tank will push you, let's see... 205 X 10 is 200, add another half that, and your car goes 300 miles on a tank of gas. At \$3/gallon, that's about \$50 to fill your car. \$50/300mi is... well, 100/300 is .3, half that is \$.15 per mile. Call it \$.20, since we rounded down just before. A penny will take you 1/20 of a mile (.05mi).

The penny is 1:(2X10^5) the weight of the car. If it takes \$.20 to push the entire car 1 mile, it takes 20/(2X10^5), or 1/(10^4), or \$.0001 to push the penny a mile.

Ah, crap. Now my logic is failing me. How do I reconcile this all?

C'mon, what were you expecting?
posted by mkultra at 11:39 AM on October 18, 2005

On the other hand, if you don't have that penny in the car, you are destined to get a pile of change from that anal cashier when the bill is \$40.01 and you give her three twenties and she doesn't have any tens or fives, sorry.
posted by weapons-grade pandemonium at 12:08 PM on October 18, 2005

"Yeah, terrific site if you know what I mean
posted by donfactor at 11:36 AM PST on October 18 [!]"

Anal cashier, if you know what I mean.
posted by Smedleyman at 12:23 PM on October 18, 2005

mkultra and justkevin - so after 100 miles, the penny has cost a penny's worth of gas.

Damn. Those fast food wrappers on the floor are costing me a fortune.
posted by selfmedicating at 12:26 PM on October 18, 2005

mkultra -

I think you made it more coplicated than neccesary, but your logic is generally good.

If the car averages 20 mpg*, and a gallon costs \$3, then it costs \$3 to push the whole car 20 miles. \$3.00/20= \$0.15 per mile to push the whole car. (I'll round up to \$0.20 since the math is easier, and I'm in Canada where gas is actually more than that). Same answer, much simpler way of arriving at it.

The problem is with your assumption re: the weight of a penny. It's not .01 lbs, but 0.1 oz, which is about 0.001 lbs. So you were off by an order of magnitude.

The penny is 1/(2*10^6). It takes \$0.20 for the whole car to go a mile, so 20 cents times the weight of the penny = \$0.0000001 to push the penny a mile.

The question was:
how many miles will I drive before its extra weight has consumed its value in gasoline? Is it more or less than the expected lifespan of the car?

So we are solving for D in the (made up) equation D*cost to push the penny a mile = \$0.01

\$0.01/\$0.0000001 = 100,000 miles
which is less than the expected lifespan of the car

*I asssume you meant mpg, since mph has nothing to do with it.

On preview: I don't know where selfmedicating got his/her answer from, but it's just wrong. It doesn't make sense.
posted by raedyn at 12:40 PM on October 18, 2005

When I said "The penny is 1/(2*10^6)." I meant it is 1/(2*10^6) the weight of the car, of course.

(Damn I previewed that several times and never caught that. Sorry.)
posted by raedyn at 12:42 PM on October 18, 2005

smedleyman ... awesome comment

Those of us in our senior design class had an imitation of our professor that went exactly that way. He always was suggesting back of the envelope calculations as a starting point and when a number was calculated that way he would strike that pose. Our entire class found ourselves doing that outside of the classroom all the time.
posted by Phantomx at 12:46 PM on October 18, 2005

selfmedicating got his wrong answer from the last line of mkultra's calculation. he had 20/(2X10^5) when it should have been .20 which made the answer .0001 instead of .000001

which makes the answer 10,000 miles. which would also solve the order of magnitude difference between the answers
posted by Phantomx at 12:55 PM on October 18, 2005

eek, sorry, that last one was a post of half thoughts, I should really put more thought into it. I hope the point gets across though.
posted by Phantomx at 12:56 PM on October 18, 2005

On the penny, the problem is that weight affects gas mileage mostly when accelerating and climbing hills. When traveling at a steady speed weight only adds to rolling resistance (the tires squish out a little more, the axle bearings have a little more friction). Most of the energy is used to overcome wind resistance and parasitic drag on the drivetrain (pumping oil, turning the engine, spinning gears under oil in the trans...). For example, even though a motorcycle may weigh less than 1/4 of a car and have 1/2 - 1/10 the motor, it only gets 2-3 times the mileage (45-80 mpg depending on size, horsepower...). I would decrease your estimates by a factor of 5-10.
posted by 445supermag at 1:04 PM on October 18, 2005

As an after thought, if weight were proportional to mileage, then 3 250 lb. passengers would decrease the milage of the theoretical car by 1/4 (or is it 1/5?).
posted by 445supermag at 1:12 PM on October 18, 2005

The only way this clock could be more depressing would be if it asked a few questions, calculated your life expectancy and then ticked away at it.
posted by Skwirl at 1:19 PM on October 18, 2005

posted by alumshubby at 1:29 PM on October 18, 2005

Skwirl - that's a fantastic idea. There's the Death Clock which is similar but only takes into account sex, outlook, BMI, and smoking/non-smoking.

590,000,000 odd seconds left for me... (Sat Aug 11th, 2024 - ... WHAT?! I'll be dead by 46?!)

Yes, it *does* count down...
posted by PurplePorpoise at 1:39 PM on October 18, 2005

A great, useful post, indeed.

In recent weeks I have seen the "I am sorry for the length of my letter, but I had not the time to write a short one." attributed to Mark Twain and am happy to now know that it belongs to Blaise Pascal.
posted by bz at 2:23 PM on October 18, 2005

Fabulous. I use this sort of stuff all the time, especially the use-your-body-to-measure-stuff ones, and getting quick dirty estimates of, for instance, how many people a theater holds, how many books a bookshelf has, etc.
posted by signal at 6:19 PM on October 18, 2005

Gyan is MeFi's best poster.
posted by Kwantsar at 7:04 PM on October 18, 2005

I forgot, you take Paypal, right?
posted by Gyan at 8:06 PM on October 18, 2005

Now why would you go and cheapen my display of affection like that?
posted by Kwantsar at 6:17 PM on October 20, 2005