38' Gutter
I usually try to shoot for at least 1/8" per foot, for any application that involves channeling water, but, with eaves trough, aesthetics sometimes preclude that much. Nevertheless, for a 38' run, I would definitely go for two downspouts. You could have one at each end, but, again, for aesthetic purposes, I would look at placing them 9-10' in from the end, so that, using 1/8" per foot as an example, you would have a maximim drop of 1-1/4" in a ten foot length of eaves trough, versus 2-3/8" in a 19' run of it.
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