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. (Use any variable or symbol stated above along with the following as necessary: ρw for the density of water.)

Answer 1

 A = area of styrofoam 
M = mass of stryofoam = A*h*rho_s 
m = mass of swimmer 

Total mass = m + M = m + A*h*rho_s 
Downward force = g*(total mass) = g*[m + A*h*rho_s] 

The slab is completely submerged. 
Buoyant force = g*(mass of water displaced) = g*[A*h*rho_w] 

Equate these 
g*[m + A*h*rho_s] = g*[A*h*rho_w] 
m + A*h*rho_s = A*h*rho_w 
A*h*[rho_w - rho_s] = m 
A = m/[h*(rho_w - rho_s)]