3000/220 = 13.6A.. I suspect you need to uprate somewhat. Call it 20A to leave some headroom.. I believe the minimum wire size for a 20A breaker is 12awg, so your 10awg is fine. The wire needs to be sized to handle the current, the breaker is sized to protect the wire.
Local electrical codes might be different, but that seems safe to me. I'm not an electrician and have never run power in Florida, so for whatever my opinion is worth.
12 ga would likely be fine if the distance is short. The NEC is very conservative on wire sizes in branch circuits. Look at this chart: http://www.usawire-cable.com/pdfs/nec ampacities.pdf . The ampacity of typical 90 deg building wire is actually 30 amps but is arbitrarily limited to 20 amps overcurrent protection. This is good from the standpoint of limiting voltage drop, which can easily be an issue in garage circuits depending on distances.
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