Alright, It's time I wrapped my brain stem around this. Can someone please explain how best to go about rendering an image to fill a page that is 24x36?

It would seem that one would need to render an image that is 10800x7200 to acheive 300 DPI when using the Print Size Wizard for a sheet 36" wide x 24" tall. Something tells me this is overkill and that I could get away with a smaller image size. What would affect the output on a printer that I could use to cheat and have a smaller image? Would saved image format affect output on the printer, say tiff vs jpg? Could I get away with an image size of half the 10800x7200 and still have decent printed quality without jagged lines? I suppose the type of printer being used could affect output quality, but how would I make that determination?