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In general for most vegetables you want at least 12 inches deep, for tomatoes and large squash 18 inches deep, for herbs and small flowers 8 inches deep. Longer and wider is always better because the soil stays more evenly moist (up to 8ft long x 4 ft wide for raised beds and 8ft long x 3 ft wide for boxes).
The reason you divide by 1,728 is because ,1,728 cubic inches (12 X 12 X12 = 1,728) is 1 cubic foot. 46,080 divided by 1,728 equals 26.67cubic feet. In other words, you'll need around 27 cubic feet of soil to fill a 4-by-8-foot planter 10 inches deep.