數(shù)學英語：32 How to Use Math to Make Estimates

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　　In the first “secret-agent math” article, “How to Use Math to Send Encrypted Messages,” we learned how to decrypt messages secured with absolutely unbreakable encryption using nothing more than simple integer arithmetic. It was all great fun, but by the end of the article you, the character in our story had been kidnapped! Today, the plot continues—read on to find out what happens.

　　But first, the podcast edition of this article was sponsored by Go to Meeting. With this meeting service, you can hold your meetings over the Internet and give presentations, product demos and training sessions right from your PC. For a free, 45 day trial, visit GoToMeeting.com/podcast.

　　How Long Can You Survive in a Sealed Room?

　　At the end of the last “secret-agent math” article, you’d been whisked away from a coffee shop by a group of kidnappers. Now, several hours later after a long and bumpy car ride you hear the engine shut off. You’re pulled from the trunk, led down a long hallway, and then tossed into a small dimly-lit room. Your captors tell you that the room has been tightly sealed, that it has no ventilation, and that you’d better hope your employers meet their ransom demands before you run out of breathable air. Needless to say, you’re a bit troubled by all this. But you’re also a secret agent—so instead of sulking you decide to start working on getting yourself out.

　　What to do first? Well, you realize you first need to figure out how much time you have to escape before you run out of air so that you know just how quickly you need to work. And for the first time in our story, you—the secret agent you, that is—are stumped. You realize that while you learned all sorts of things in school and in your spy training, nobody ever taught you how to calculate this sort of thing. Your breathing picks up as you get nervous. Realizing that your reaction is wasting air, you take a moment, steady your breath, and think. You resolve to start from the beginning, take things step-by-step, and figure it out.

　　How Can You Calculate the Volume of a Room?

　　You reason that if you figure out how big the room is and how big each of your breaths are, then you should be able to figure out how many breaths of air the room contains. So, how big is the room? Walking from one side to the other you find that it’s about three steps across. You know that one of your steps is a little more than three feet, so you figure the room is about ten feet wide (3 x 3 = 9, plus a little more because your steps are more than 3 feet is about 10). The room looks pretty square so you figure it must also be about ten feet long. And the height of the ceiling seems a little higher than the heights of your ceilings at home, which you can just touch if you jump and know are eight feet high. You jump in your cell and find that you can reach to within about one foot of the ceiling, so you figure the ceiling must be about nine feet high. So, you’re stuck in a room that’s 10 feet wide, 10 feet deep, and 9 feet high—and that means you have 10 feet x 10 feet x 9 feet = 900 cubic feet of air! In other words, if you had a box that’s one foot long on each side, you’d have 900 of those boxes full of air. Though that’s not terrible, it’s not the best news either.

　　What is the Volume of a Single Breath?

　　But what’s the volume of one of your breaths? When you’re faced with a question like this, the quick and dirty tip is to try and use everyday references to help you make educated guesses (just as you used your ceilings at home as a reference to figure out the height of the ceiling in your cell). At one point or another most of us have exhaled into and filled up a plastic sandwich bag just before popping it. The volume of air in one of your breaths must, therefore, be about the same as the volume of air in an inflated sandwich bag. But how much air is that? Well, an inflated bag has an odd shape, so it’s hard to say exactly; but a sandwich bag is very roughly the same size as an orange. And an orange is very roughly the same size as a box about three inches on a side. So, the volume of one of your breaths must be about 3 inches x 3 inches x 3 inches = 27 cubic inches. Or, since three inches is the same as one-quarter foot, one of your breaths has a volume of 1/4 foot x 1/4 foot x 1/4 foot = 1/64 cubic feet.

　　How Many Breaths of Air are in the Room?

　　At this point you mutter to yourself: “If the volume of air in a single breath would fit into a cube three inches on a side, then all I have to do is figure out how many of those cubes of air will fit inside this room—that’s the number of breaths I have left before….” Your mind races as you worry that you may have already used up most of these cubes of air. But, again, you calm down and go on to reason that since you know the volume of air in the room, and the volume of air in one of your breaths, the number of breaths in the room is just the volume of the room divided by the volume of a single breath. Or, since each breath has a volume of 1/64 cubic feet, that means that each cubic foot of air must contain 64 breaths. But you also know that you have 900 total cubic feet of air in the room, which means that the room contains 64 x 900 = 57,600 breaths. What a relief—that’s a lot of breaths! Isn’t it?

　　How Much Time Do You Have to Escape?

　　Well, just how long does that actually give you to escape? You realize that you need one last piece of information: How often do you breathe? So, you count the number of breaths you take in a minute. On average, most people breathe 15 to 20 times a minute. Since you’re under a fair bit of stress, it’s no surprise that you’re breathing about 20 times per minute. That comes out to 28,800—or approximately 30,000—breaths per day (that’s 20 breaths per minute, times 60 minutes per hour, times 24 hours per day). Dividing the 57,600—or roughly 60,000—total breaths in the room by the 30,000 breaths you take per day, you find that you have about a two-day supply of air. Now that really is a relief! Feeling calm and confident that your capable colleagues will rescue you within this two day time-frame, you decide to stop worrying and instead pull out your phone to post a funny picture of you sitting in your cell to Facebook. And then, you take a nap.

　　Don’t Be Afraid to Think!

　　So, what’s really the point of all this? Well, as our secret agent has learned, math isn’t about solving a bunch of problems within a fixed time-limit (unless you’re taking the SAT). It’s about facing situations that you’re not familiar with and drawing upon the knowledge you’ve accumulated to devise solutions. So here’s the quick and dirty tip: Think! Think your way through problems. If it seems too hard, try anyway. The more you try to make estimates and figure things out using math, the more frequently you’ll be successful.

　　Wrap Up

　　Okay, that’s all the math we have time for today. Thanks again to our sponsor this week, Go to Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.

　　Please email your math questions and comments to................You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.

　　Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!