What is the ratio between the depths they will sink into the ice?

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When two copper vessels of different masses are placed on ice, their sinking depths will vary due to their differing weights. The larger vessel (50g) will exert more pressure on the ice, causing it to sink deeper compared to the lighter vessel (20g). The cooling rate to 0 degrees Celsius will also differ, with the heavier vessel taking longer to cool due to its greater thermal mass, which retains heat more effectively. This phenomenon can be explained by the conservation of energy, as the larger vessel has more energy to lose before reaching thermal equilibrium with the ice. Ultimately, the ratio of their sinking depths is influenced by both their weights and the area of contact with the ice.
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q.if two copper vessels of mass 20g and 50g are heated to 85 degrees celcius and are put onto similar ice blocks, what is the ratio between the depths they will sink into the ice?
 
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Some more information would be rather helpful here. For instance, I would think the area where ice and copper touch are probably different: different weight+same material -> different size.
 
Assuming that the bases of the copper vessels were the same size, consider: which of these vessels would take longer to cool to 0 degrees (the assumed temperature of the ice)?

THen answer the following question: "why"? (Using conservation of energy as your guide).
 
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