Problem 15
Suppose it is a very cold day and we go to your school football field. Let us say that the dimensions of your field are 100 meters by 60 meters. All of a sudden clouds appear in the sky and something peculiar happens. We see that snow starts falling according to a mathematical rule. At first 1 snow flake falls, then exactly 1 second later 2 snow flakes fall, then a second after that 4 snow flakes fall, then a second after that 8 flakes fall i.e. the snow flakes fall in 1 second intervals and after each second the amount of flakes doubles.
For simplicity let us assume that every flake that falls keeps its original volume i.e. there is no phase change to water or water vapour. Let us also assume that the flakes are little cubes of side length 1mm (0.001metres). Making things even simpler we also assume that the cubes start covering the field with perfect efficiency i.e. there will be no air pockets in the snow. Also let<92>s assume the clouds in the earth<92>s atmosphere can be very high. Much higher than possible!
If this strange snow pattern occurs for just 1 minute, how high will the snow be?
