Ray,

You called this problem a related rates problem and you are correct. The fundamental question as I see it is how to you relate what is going on inside the filter to what is going on inside the cylinder. The water is flowing from the filter into the cylinder so the volume of the water in the filter is decreasing at the same rate as the volume of the water in the cylinder is increasing. To say this mathematically I need some variables.

Let t be the time measured in minutes, V_f be the volume of the filter measured in cubic centimeters and V_c be the volume of the cylinder measured in cubic centimeters. The fact I know then is

dV_f/dt = - dV_c/dt.            (*)

Suppose at some time t the height of the water in the filter is h_f, the radius of the surface of the water in the filter is r_f and the height of the water in the cylinder id h_c, all measured in centimeters. Thus you know that

V_c = π 6² h_c

and

V_f = 1/3 π r_f² h_f

Use similar triangles to express V_f in terms of h_f alone. Differentiate V_f and V_c with respect to t. Relate the expressions using equation (*). Substitute the values you know when h_f = 12 cm, nd solve for dh_c/dt.

Harley