How Is the Area of a Floating Slab Calculated Using Buoyancy Principles?

[walker]

Question is: A plastic slab has a thickness of h and a density p_s . when a swimmer of mass m is resting on it, the slab floats in fresh water with its top at the same level as the water surface. find the area of the slab. give your answer in terms of density of water p_w , p_s , h and m

so what I've done is used achimedes principle where all forces are in equilibrium. that is B = Mg

M being the mass of the swimmer and slab of plastic combined

so by setting this equation equal to the buoyancy force in terms of pressure I get: Mg = p_w*g*V where p_w is the density of fresh water and since V equals h*A I can sub for V and get Mg = p_w*g*h*A then i can just solve for A but the answer A = M/(p_w*h) doesn't use all the terms given in the question. am i just reading into the question too much or am i missing something here?

 

[Chi Meson]

I got an answer by setting up two equations:

find a statement for the mass of the slab using its density, h and the unknown area.

Find another statement for the mass of the slab using the equation you have above (but call the combined mass "M+m" with M being the mass of the slab).

With two terms for the mass of the slab, set them equal to each other so that M goes away. Then do some algebraic gymnastics and you get an answer with the proper variables.

 

[kyle8921]

Your approach using Archimedes' principle is correct. You have correctly identified that the buoyancy force (p_w*g*V) is equal to the weight of the swimmer and slab (Mg). However, your final answer of A = M/(p_w*h) is missing the density of the plastic slab (p_s). The correct answer should be A = M/(p_s - p_w)*h. This takes into account the fact that the buoyancy force is equal to the difference between the densities of the object and the surrounding fluid. So, the final answer should include all the terms given in the question.