Ooo, I'm a bit rusty at that, so let's see.

Volume of a cone: 1/3 pi*r^2*h

Your cone fills up at a speed of 12ft^3/min, so in one minute, the height goes from h to H as follows:

1/3*pi*R^2*H-1/3*pi*r^2*h = 12
We know that at max height, r_max = 8 and h_max = 16, and since the radius is proportional to the height (it's a cone), we have r = h/2

Therefore 1/3*pi*(H/2)^2*H-1/3*pi*(h/2)^2*h = 12
H^3 = (12*3*4/pi)+h^3
H = (144/pi + h^3)^(1/3)

With h=10, your new height after one minute is H ~ 10.150512 ft, so the speed in ft/min is the difference in height:
S = H-h ~ 0.150512ft/min

If you want to generalize for any given height h, what is the speed in ft/min:

S = H-h = (144/pi + h^3)^(1/3)-h

Generalized more for any cone of radius alpha*h:

S = {[(36/pi)+((alpha)^2*(h^3))]/(alpha^2)}^(1/3) - h

And furthermore, with a speed v in ft^3/min:

S = {[(3*v/pi)+((alpha)^2*(h^3))]/(alpha^2)}^(1/3) - h