Mathematics, atoms, hair and snots

I have the habit of using one of my first classes of every course in teaching my students to do calculations that, seemingly, can turn out to be impossible to carry out. This apparent difficulty to take them to good end is given by the absence of information, of excellent information.

The physicist of Italian origin Enrico Fermi (1901-1954), who was one of the most visible heads in the development of the famous project Manhattan, which would conclude with the construction of the first atomic bomb, was possessing an amazing facility to solve certain type of problems, like that I describe you in the first paragraph. Departing from a few meager information, it was capable of obtaining a few good estimations, approaches amazingly precise to the solutions of the raised problems. In his honor, there are called they to these problems or questions problems of Fermi. And to solve them, Fermi was trying always to decompose the original problem in simpler others, was crumbling it until to each of these micro-problems it could assign a simple answer to him.

To explain yourselves of what these problems consist, I will put to you three examples of those that I usually propose to my students. They are these:

1. How many atoms are there in a human body?

2. What is the length of the hair that exists in a feminine head?

3. How many people are there, right now in the world, the nose being poked?

You will not deny to me that they have substance: truth? Do you understand now why do I say what I say in the previous paragraphs? How devils can give itself a solution come closer fellow men questions? So, exactly that, this is what I prepare to tell you right now.

Let's begin for the first question. How many atoms are there in a human body? Let's see, the body is formed by a more or less diverse series of constituent chemical elements, but we do not know exactly how many are of every type. Nevertheless, yes we know that a big percentage of our body is a water. Let's take, then, as the first approach that all our body is a water. Being this percentage of 70 %, this still does not mean that we commit 30 % of error, since exactly this other 30 % is formed by other atoms, although they are not of water. Well, a basic chemistry knowledge says to us that every water molecule possesses three atoms: two of hydrogen and one of oxygen. The following modest step is a knowledge how much weighs a water molecule or, what is the same, every atom that constitutes it. This also we learned it in the school. In a water mol there is the number of Avogadro (approximately 600.000 trillions) of molecules and every mol weighs 18 grams. only it reduces to us to assume a middleweight for a human body. Let's put 70 kg. It turns out to be trivial to deduce that in a human body there are, then, approximately 3900 water thick chile sauces and, therefore, 1028 atoms. Decisive problem!

We go now with the second one of the raised questions. To try to estimate the entire length of the hairs that populate a feminine head (masculine also it costs) it is possible to decompose the problem in these simpler three: first, to find out the area of the scalp; then, the number of hairs for unit of area and, finally, the length of a typical hair. Let's see. The palm of a completely widespread hand usually includes approximately 20 cm. A human head has an approximate diameter of a span. If I suppose that the form of the head is spherical and that the scalp occupies half of this one, using the expression of the area of a sphere (4 times pi for the square of the radio of the same one), it is obtained that the scalp occupies an extension of approximately 600 square centimeters. The following step consists of using the imagination or, alternatively, a pair of hair of being started and of verifying that more or less they possess a width (position one next of other) 1 mm in a graduated rule. This does approximately 400 hairs for square centimeter in our scalp. Therefore, multiplying two numbers dear till now, it is had that in the head there are approximately 240.000 hairs. Supposing that a woman has, in average, his long hair at a height of the shoulders and taking for this distance approximately 10 cm, one concludes that the entire length of all his hair is 24 km. Impressive: no?

The third and last question is the one that more I like of them. Without alertness cameras I can still know how many you present yourself approximately there is in this moment doing filths, looking for oil in his nasal orifices. For it, I will set off of a quite obvious mathematical beginning and that says to me that the fraction of time that someone uses in a certain activity is equal to the fraction of the people who realizes precisely this activity in the same moment. Said more simply, if I use 10 % of my time in flying in plane, then more or less 10 % of the world population will be flying in a certain moment.

Well, then the question is: how long do we use in poking our nose after the day? Ten seconds? It seems small: do not you believe?. Let's see: how are 1000 seconds? On the contrary, it seems too much: is not it true? Let's take, then, the intermediate magnitude order, that is to say, approximately 100 seconds a day (slightly less than 2 minutes). If we eliminate of our calculation the people with a pair of nostril to compensate with those who eat the snots often (the filthy children), and who support it of course mistaken, it designs of that to feed on small balls seems to help to the infantile immune system to recognize certain types of virus and harmful bacteria (to see Sophie's comment, further down), simply we will be able to establish a very simple proportion that will provide the solution to us to our problem raised originally. The quotient between the number of hurgadores and the world population (rounding, approximately 6000 millions) has to be equal to the quotient between the time used in be poking and the duration of one day. The result, amazing, undoubtedly: 10 million persons are gathering right now, scratching or annoying hairs being started.

What amazing things can be done by the mathematics! Up to the next edition of the Carnival of Mathematics!

P.D. If you like this type of puzzles, problems and questions and you want to promote your ingenuity, you will find much more material in the book of Lawrence Weinstein and titled John A. Adam Guesstimation: Solving the world's problems on the back of to cocktail napkin.