Equilibrium and find static friction

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The discussion centers around a homework problem involving equilibrium and static friction. A participant expresses confusion over a previously provided solution, noting discrepancies in the process and units used. They emphasize that the solution fails to incorporate the given dimensions, which are crucial to the problem. The forum rules prompt them to share their own attempt at a solution, including relevant equations. Clarity on the correct application of dimensions and equations is essential for solving the problem accurately.
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Homework Statement


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Homework Equations

The Attempt at a Solution


I had someone help me with this problem (not on here) and their process and units don't seem to match to me. Or is this correct?
 
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That solution is clearly not right. It makes no use of the given dimensions, and these are easily seem to be relevant.
As required by the forum rules, please now post your attempt at a solution. Start with listing the standard equations you know which are likely to be relevant.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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