let

b = the average height of mountain B

c = the average height of mountain C

the average height of mountain A is 14,150 ft, and its average height is 8870 ft less than the sum of the average heights of mountains B and C

14,150 = (b + c) - 8870

(b + c) - 8870 = 14,150

the average height of mountain C is 272 ft less than four-fifths of the average height of mountain B

c = (4/5)b - 272

by solving the system of equations

(b + c) - 8870 = 14,150

c = (4/5)b - 272

we find

c = 10,080 ft

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the average height of mountain C is 10,080 ft.