Apparent depth with multiple indices of refraction

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To determine the apparent depth of a penny located at the bottom of a barrel of water with a layer of oil on top, Snell's law must be applied at both the water-oil and oil-air boundaries. The apparent depth through just the water is calculated to be 0.75 meters. To generalize this for the oil layer, the observer must account for the refractive indices of both water (1.33) and oil (1.5). The solution involves using Snell's law twice to find the total apparent depth as seen from the air. Understanding the refraction at each interface is crucial for accurate calculations.
KBriggs
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Homework Statement


A penny is located at the bottom of a barrel of water 1m deep. There is a 20cm thick layer of oil on top of the water. To an observer at normal incidence, what is the apparent depth of the penny. n for water is 1.33, for oil it is 1.5


Homework Equations


Snell's law, lots of ray diagrams


The Attempt at a Solution


I have already shown that to an observer looking through just the water, the apparent depth is 0.75m. I am just not sure how to generalize this for the other layer. Can someone point me in the right direction?
 
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The method is the same, only you have to use Snell's law twice, at the water oil, and oil/air boundaries.
 
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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