Yes, treating the sphere as a "solid of revolution" and using the disk method to find the volume is probably the best way to do this.
First start by setting up a coordinate system. You could do this by setting the origin at the center of the sphere but I suspect it will be a little simpler to think about if we set the origin at the bottom of the sphere so that the "height of the liquid" is its z coordinate.. Then the center will be at (0, 0, 15). Looked at from the side, the sphere appears to be a circle, in the xz- plane, with center at (0, 15). Its equation is . Each "disk" at a given z, seen from the side as here, appears to be a line segment from -x to + x where from the equation of the circle. Rotated around the z-axis, the disk has area . The volume from z= 0 to z= h is given by .
Find the volume of water poured in at for 4.5 seconds, set that integral equal to it and solve for h.