Prove, that if a given rectangle can be covered by 100 circles each of r=2, then it can be covered by 400 circles each of r=1.

First off the idea seems plausible since the areas are the same.

Second the converse is not true: a √2 by 400√2 rectangle can be covered by 400 circles of r=1 but would require 164 of r=1
[(400√2)/(2√3)]

The smallest square that a single r=2 circle can cover is 2√2 by 2√2 and it's easy to see that four r=1 circles can cover this.

The trick is to show that this square can be widened or made taller and 4 times as many r=1 as r=2 will always suffice.

I can picture this but I can't put it into words.

Posted by Jer on 2012-01-30 12:33:57