Physics density?

Question

A Styrofoam slab has thickness h and density ρs. 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.

Answer 1

When an object is floating, the net force on it will be zero. This happens when the volume of the object submerged displaces an amount of liquid whose weight is equal to the weight of the object.

A = area of styrofoam slab
H = thickness of slab
p = density of slab
g = acceleration of gravity
w = density of water

Mass of slab + man = AHp + m
Weight = (AHp + m)g

Since the top of the slab is level with the water surface the whole slab is underwater so the amount of water displaced is AHw

Weight of water = AHwg

Set the two weights equal to each other:
(AHp + m)g = AHwg
m = AH(w - p)
A = m/[H(w - p)]