Quick math . . . long post.

Back of the envelope calculations are wonderful things.

Anyone who's ever taken physics has worked their way through ugly, messy calculations and has probably stressed out (way more than necessary) about how many decimal places to report on an answer. But sometimes, you don't need specific answers, you just want to have an idea of what an answer would be like. What would it cost to fill your swimming pool with jello? How many french fries have I eaten in my life time? You know, life's important questions. And so, we have back-of-the-envelope calculations. You don't need to do a study or make measurements, you just want the fun answer. Well, start making guesses and see what comes out! And, the neat thing is, if you're good, you can actually get pretty darn close to the 'right' answer that would take 10 times as long.

Do you want to fill up a room with balloons (probably for some girl's birthday or something)? How many balloons do you need? Well, let's say your inflated balloon is round (always assume spherical!) and 10" across. But I sense an upcoming difficulty in dealing with feet and inches, so let's call it a 12" balloon. That's 0.5' in diameter. So it's volume is 4/3*Pi*r^3 = oh, wait that's hard math. So, we'll call Pi=3, so it is now 4*r^3, or 4*.5*.5*.5 = 2*.5*.5 = 1 * .5 = 0.5 ft^3 per balloon. Ok, now how big is your room? Well, I don't know who's birthday you are planning for, but let's say you decide the room is 15' x 16'. Again, hard math, so let's rearrange the room in our heads to 12' x 20'. That's probably close to the same size. Now how deep do you want to fill your room? To fully immerse a person, you'll need about 6' of balloons. So 12' x 20' x 6' = 240 x 6 which is about the same thing as 250 x 6 (which is easier to do) = 1500 ft^3. If each balloon is half a square foot, you'll need 3000 balloons. That's a lot of balloons!

But we've got to think about our problem just a bit more. Two things jump out in my mind. First, the balloons don't all pack together nicely. They're roundish after all. So what percentage of the room is really going to be full of balloons, and what is going to be gaps between the balloons. Well, I'm just going to pull a number out of the air and say . . . 70% balloons. So we don't need to fill 1500 ft^3, only 70% of that. Secondly, what about all the stuff in the room already? That's a lot of balloons worth of space! Of the bottom 6 feet of the room, how much is full of stuff? Not tons. There is a lot of air. But still, the bed and desk and electric guitar might count for 20%. So we have to fill 80% of our 1500 ft^3 with balloons, but then only 70% of that space is actually balloon space (as opposed to air). So . . 1500 x 0.8 x 0.7 = . . . well, now you might just need a calculator, but you've got one on your phone . . . 1500 x 0.8 x 0.7 = 840 ft^3. Double that to get your 1680 balloons. There you go. Start blowing them up!

Now, obviously, we guessed at a lot of stuff. But, we're actually probably not that far off. We made a good half dozen guesses or simplifications, and it is likely that some of them made us over estimate the number of balloons, while others causes us to under estimate the number. Happily, the errors that we didn't even know we were making help to cancel each other out much of the time. (Though sometimes we might accidentally choose estimates that are all either too high or too low.) Even if we're too high by 30%, you'd still need 1000 balloons to pull off what you're trying to do. So you might want to rethink your plans. If you only go for 3 feet deep, you can do it for more like 500 balloons. Yeah, that sounds like a good plan.

Now, let me just warn you from experience that getting all the balloons to stay in the room is a big problem, and it turns out that if you're in a dry place your balloons will start to build up static charges as you pile them in the room which can get high enough that they actually start popping. And if you've been working on blowing up hundreds of balloons, the last thing you want to hear is them popping.

Now, this has gone on for quite a while. But we can see that in a few minutes, we can take our balloon-bedroom-surprise through a quick calculation and see just how feasible it might be. We could continue by doing a quick cost analysis for 1000+ balloons, as well as time estimates for how many man hours it will take to blow them all up.

These calculations take a bit of a talent for making reasonable estimates of things that you've probably never measured (your friend's room, the size of a balloon, the time to blow it up) but I maintain that this is an excellent skill to have.

Finally, a little assignment for you to do. (Even though I know you probably won't.) (Except maybe Brett.) Assuming there is 33x10^6 km^3 of ice between the antarctic and greenland sheets, how much would the ocean levels rise if they all melted? See what sort of answer you can get without looking up any other values. I did it guestimate style as well as with looking up real values and my two answers were less than 4% different. (And they're not much further off the "official" values that someone told me at work today.)