1790, "channel, trough," a dialectal combination of chute and shoot (n.1).
Entries linking to shute
chute n.1
1725, American English, "fall of water" (earlier shoot, 1610s), from French chute "fall," from Old French cheoite "a fall," fem. past participle of cheoir "to fall," from Latin cadere"to fall," from PIE root *kad- "to fall." Meaning "inclined tube, trough" is from 1804; that of "narrow passage for cattle, etc." first recorded 1871. In North America, absorbing some senses of similar-sounding shoot (n.1).
shoot n.1
"young branch of a tree or plant," mid-15c., from shoot (v.). Also "heavy, sudden rush of water; a river-fall or rapid," especially one through which a canoe or timber can "shoot" (1610s); "artificial channel for water running down" (1707); "conduit for coal, etc." (1844). In some senses influenced by or confused with chute (n.1).