The volume of a cone is given by (1/3)*pi*r^2*h, where r is the radius and h is the height.
The volume is 408*pi and the radius is 12
408*pi = (1/3)*pi*12^2*h
=> 408*pi = 48*pi*h
=> h = 408*pi/(48*pi)
=> h = 8.5
The height of the cone is 8.5 cm.
All we have to do is to recall the identity that gives the volume of a cone:
V = pi*R^2*h/3
R is the radius of the base circle and h is the height of the cone.
Since we know the values of the volume and the radius, we'll replace them into the formula:
408*pi = pi*144*h/3
We'll divide by pi:
408*3 = 144*h
We'll divide by 144 both sides:
h = 408*3/144
h = 1224/144
h = 8.5 cm
The requested height of the cone is of 8.5 cm.