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Wow, that's a cool question. I want to have a crack at it!
The addition of the ice cube would raise the water in the glass because it will displace the water currently in the glass. As the ice cube melts, it will not change the water level because the water released from the melting ice was already "pushing" on the water around it. You are just transforming the frozen water to a liquid state.
In fact, after the initial "plop" of the ice cube into the cup (which made the water level spike in the first place) once the ice cube melts you may see a decrease in water level because the frozen water expanded while in ice cube form. It took up more volume in the cup while frozen than it will when melted. On the other hand, depending on the shape and size of the cube, some of the cube might have been "sticking out of the water" in the way that ice bergs do. That piece would not be displacing water initially, but would add to the water supply as it melted. In this case, the water might only rise a fraction with the melting of the ice cube.
The trick is, you have to know how big the cube (and its shape) to figure out how much of it would be sticking out of the water after the inital "plop."
Having the metal inside of the ice cube doesn't change the scenario a lot, depending on how big the chunk is. The metal is taking the place of water that would have taken up volume in the ice cube. The ice cube, in essence, is the same size whether filled with water or metal chunks. The metal would eventually be released and sink to the bottom of the glass where it would take up space, but had the metal not been in the ice in the first place, water would have been there, and so the level would have risen the same amount anyway after melting. The variable here, I think, is weight. If the metal were heavy enough it might sink the ice cube initially. That would eliminate the "ice-berg" effect so that wouldn't be a variable.
Overall, my conclusion is that the initial drop of the ice cube would cause the water level to rise (because it takes up space the water now can't use,) but as it melts you should see a slight reduction in water level to account for the fact that the frozen ice took up more space than its liquid form did. This assumes that the entire ice cube is under water...if a little bit was sticking above the water, you would not see the drop from the frozen ice "contracting" as it became liquid because the melting water from the part of the cube above the water, initially not displacing any water in the cup, would now do so.
Man, does that make any sense at all?
When an ice cube with a metal piece inside floating on water melts, the level of water will decrease.
To understand why this is so let us consider the action of the ice and the metal piece separately.
The ice floats on water because it has lower density than water. When ice is put in water it displace a volume of water which weighs same as the weight of ice. After melting, the volume of ice becomes same as that of the water that it was displacing. Therefore, there is no change in level of water due to melting of ice.
Now let us examine effect of metal piece. The combined density of ice and metal ice is still less than that of water. That is why the ice cube with metal piece floats. In this case, additional volume of water displaced due to weight of metal piece, weighs same as the weight of metal piece. This is more than the volume of metal which has higher density than water. When the ice melts, the metal piece no longer is able to float ion water and sinks. At this time the the volume of water displaced by the metal piece that has sunk is equal to just the volume of metal piece. Thus the volume of water displace by metal piece reduces on melting of ice. Therefore. the level of water will reduce after melting of ice.
Please note that, if at any time, before complete melting of ice, the metal piece embedded in ice sinks in the water, temporarily the level of water will rise. The exact amount of rise will depend on the remaining quantity of ice, If only the metal piece sinks, but the ice cubes remains floating, then also the water level will reduce to the same level as after complete melting of ice.
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