SOLUTION: A beverage container is made of aluminum (SG = 2.7) with a thickness of 0.1 mm. It's cylindrical, with a top and bottom of Al. The diameter is 64 mm, the height is 128 mm. /////

    You should use the thickness of the container's wall 0.1 mm, when you calculate the mass 
    of the container, but you can neglect this thickness when you calculate the volume of the water 
    inside the container.

(a)  First consider the empty container (with no water inside).

     Then it is clear that the center of gravity is the center of this empty cylinder.

     So, the position of the center of gravity of the empty container is half 
     the height of the container.

     The mass of the container you can easily calculate as the mass of the empty cylinder
     with the given dimensions and the SG of aluminum.


     So, I suppose that you calculate this mass  of the container on your own.

     You just know that the mass   of the empty container is located at the height 
     H/2, where H is the height of the container.



(b)  Second, when you fill the water to the height h of the container, the center of the gravity 
     of water is at the height h/2 and the mass of water  you can compute easily

                   = .



(c)  So, you have the mass  concentrated at the height H/2,
         and you have the mass of water  concentrated at the height h/2.


     OBVIOUSLY, the mass of the combined system is   and the height 
     of center of mass of the combined system is  .    (1)


     Notice that this term,  is zero when h= 0 (then  is zero)
     and  is zero when h = H, so at these two values of h
     the gravity center is at half of the height H at h= 0 or h= H.



(d)  After that, your task is to combine everything together in formula (1).

     You will have the position of the center as a quadratic function of h.

     To get the minimum of the center of mass height, you find the minimum of this function on h.