journey through life: rant about handwash soap

all right now i feel like i want to make some little calculation after seeing a handwash soap advertisement. they say that washing your hands with that soap eradicates 99.99% of the bacterium that exist there, so let's calculate a bit.
warning: latex from an online latex equation editor dead ahead!

From this source, I obtained that average E. coli bacteria is ${\color{black} \inline 2 \: \mu m}$ long and ${\color{black} \inline 0.5 \: \mu m}$ wide. That means its average area is ${\color{black} \inline 1 \: \mu m^2}$ . Meanwhile from another source, the average length of an adult male's hand is ${\color{black} \inline 189 \: mm}$ , and the average breadth is ${\color{black} \inline 84 \: mm}$ , which means that the average area is

${\color{black} \inline 2 \times 15876 \: mm^2 = 31752000000 \: \mu m^2}$
It is multiplied by 2, because... average people have 2 hands. Yeah, common sense.

Now, the average hand can be fully occupied by ${\color{black} \inline \frac{31752 \: mm^2}{1 \: \mu m^2} = 31752000000}$ bacterium.

Everytime you wash your hand, there's always ${\color{black} \inline 0.01 \% = 0.0001}$ of the bacterium left, and as the amount of bacterium existing is always a non-negative integer (no way you can have a one and a quarter of a bacterium, and it's still living), we can set up an inequality.

${\color{black} \inline 31752000000r^n < 1}$
${\color{black} \inline 31752000000(0.0001)^n < 1}$
${\color{black} \inline (0.0001)^n < \frac{1}{31752000000}}$
${\color{black} \inline 10^{-4n} < 3.149 \times 10^{-11}}$
${\color{black} \inline -4n < \log 3.149 - 11}$
${\color{black} \inline 11 - \log 3.149 < 4n}$
${\color{black} \inline 11 - \log 3.149 < 4n}$
${\color{black} \inline 4n > 10.502}$
${\color{black} \inline n > 2.6255}$
where ${\color{black} \inline r}$ is the ratio, that is, ${\color{black} \inline r = 0.0001}$ and ${\color{black} \inline n}$ is the number you have to repeat washing your hands to ensure no bacteria exists anymore.

Conclusion

Since ${\color{black} \inline n}$ must be a non-negative integer also, to ensure the amount of bacterium existing on your hand is less than 1, (which means 0 as it is a non-negative integer) you must wash your hand with that handwash soap at least 3 times to ensure total eradication of the average E. coli bacteria.

Source (opens in new window) :
1. ^ Bacterial cell structure - Wikipedia
2. ^ Hand - Wikipedia

Sunday, July 4, 2010

rant about handwash soap

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