Silly as it sounds, the length of a timeout depends upon how many TV commercials have been shown in the half. Besides the end of the quarter and the 2-minute warning, there are 8 additional "TV timeouts" which will extend the length of any given team timeout to 1 minute 50 seconds.
Once all of the TV timeouts have been used up, any remaining team timeouts taken will only last 30 seconds.
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It's not a math concept. "Thirty second timeout" is a pro football TV marketing concept. See the paragraph on "Commerical Breaks" in the article http://en.wikipedia.org/wiki/NFL_on_television#Commercial_breaks
You only get three timeouts per half ther is no thirty second time out in football you are thinking of Basketball where there are full timeouts and twenty second timeouts not nfl but rather nba
A 30 s time out is the minimum time you can take before restarting the game, so this is used to inform the other team to be ready for play although there is no difference between that and a full length time out . These are normally used as clock management timeouts and not play call related,