given in problem,

the radius of the hemispherical bowl, R = 9 cm

the internal base radius of the cylindrical vessel, r = 6 cm

the height of the water in the cylindrical vessel be h.

for that,

volume of the hemispherical bowl = volume of the cylindrical vessel

[2/3] Π r³ = Πr²h

[2/3]  r³ = r²h

[2/3] *9*9*9 = 6*6*h

2*3*81=36*h

6*81=36*h

486=36*h

h=486/36

h=13.5 cm

∴ The height of the water in the cylindrical vessel is 13.5 cm.