let

h = the slant height of the cone

r = the radius of the cone

A = 12,560 cm^2

H = the height of the cylinder

slant height of cone is 9 times the radius of cone:

h = 9r

the area of the cone is A = r^2 pi + r pi h

r^2 pi + r pi h = 12560

plug h = 9r into the last equation:

r^2 pi + r pi * 9r = 12560

r^2 pi + 9 r^2 pi = 12560

10 r^2 pi = 12560

r^2 pi = 1256

r = (1256/pi)^0.5

pi = 3.14

r = 20

the volume of the cylinder is equal to the volume of the cone:

r^2 pi * H = (1/3) r^2 pi (h^2 - r^2)^0.5 divide both sides by r^2 pi

H = (1/3) (h^2 - r^2)^0.5

H = (1/3) ((9r)^2 - r^2)^0.5

H = (1/3) r*(81 - 1)^0.5

H = (1/3) r*(81 - 1)^0.5

H = (1/3)*20*8.94

H = 59.6 cm is the height of the cone