XKCD: Dimension Analysis

Just a verification that the units work and also evaluating the value:

Energy: This is measured in Joules, which are:

\rm kg \cdot m^2 \cdot s^{-2}

Pressure at the Earth’s core: This is a pressure, and so measured in Pascals. Pascals are in fact Newtons per Metre squared, or:

\rm kg \cdot m^{-1} \cdot s^{-2}

Gas Mileage: This is the amount of distance it goes per amount of petrol in the car. Therefore it is metres per cubic metres, or:

\rm m^{-2}

Distance: Distance is just metres.

Therefore the units of the equation are:

\frac{\rm kg \cdot m^{2} \cdot s^{-2}} {\rm kg \cdot m^{-1} \cdot s^{-2}} \times \frac{\rm m^{-2}}{\rm m} = \frac{\rm m^{2}} {\rm m^{-1}} \times \frac{\rm m^{-2}}{\rm m}  = \rm m^3 \times \rm m^{-3}

Which is of course dimensionless. Now what about the actual values?

Planck Energy: Wikipedia puts it at 1.956×109 J

Earth Core Pressure: Wikipedia puts it at between 330×109 Pa and 360×109 Pa

Prius Mileage: The American Fuel Economy website puts it at about 50 miles per gallon. Google can covert this to 21×106 m-2

Minimum Width of The English Channel: Answers.com says, via it’s Columbia Encyclopedia source, that 34×103 m is the minimum width.

Putting all this into the equation gets a value of around 3.6. Which is certainly near enough!

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