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Hairs on a Head

Date: 12/10/2002 at 19:50:39
From: Landon
Subject: How Many?

How many hairs are on your head?

I don't know how to solve this or where to start.


Date: 12/10/2002 at 21:06:32
From: Doctor Ian
Subject: Re: How Many?

Hi Landon,

One way to start would be to draw a very small square on your head
(maybe 5 mm on a side), and count the number of hairs that lie inside
it. Actually, you'd have to have someone else do it for you; or you
could assume that one head is like another, and do it to someone else. 

Anyway, then you'd need to estimate the surface area of your head in
square millimeters. Since a rough answer would probably be good 
enough, you could assume that your head is a sphere, and estimate what
percent of the sphere is covered by hair. 

For example, suppose you decide that your head is a sphere with a
radius of 10 cm, and you count 50 hairs in a 5 mm by 5 mm area. Also, 
you decide that your hair covers about 50% of your head. 

The surface area of a sphere is 

  A = 4 pi r^2

    = 4 pi (100 mm)^2

    = 120,000 mm^2

And 50% of that would be 60,000 mm^2. 

We said that you have 50 hairs per 25 mm^2, so the total number of
hairs would be

                 50 hairs
   60,000 mm^2 * -------- = 120,000 hairs 
                 25 mm^2

Here are a few things to consider:

1) I just made these numbers up, so the actual answer may differ from
this considerably - although I just did a search in Google and found
this, 

   The Nuclear Age: Its Physics and History - 
   Profs. Goldstein and Sherwin
   http://www.tufts.edu/as/physics/courses/physics5/estim_97.html 

which arrives at an answer that's in the same ballpark.  (They assume a 
diameter of 10 in, where I assumed a radius of 10 cm, and make other 
assumptions that differ from mine.) 

2) You'd probably want to count hairs in a few places, and compute an
average number of hairs per square. Also, you have to decide what to
do about hairs that fall exactly on the line you draw. (One way to
handle them would be to consider half of them inside the square, and
the other half outside.) 

3) If you're 12 years old, your head is probably smaller than average, 
but the percentage of your head covered with hair is probably higher 
than average. So these might just cancel out.

I hope this helps. Write back if you'd like to talk more about this, 
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Statistics

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