Why Does the Styrofoam Slab Float at Water Level When a Swimmer Rests on It?

[djeitnstine]

This (textbook) question seems so simple yet I have been having the hardest time solving it :S I know there's something completely obvious I'm missing.

Homework Statement

A styrofoam slab has a thickness h and density $$\rho_{s}$$. When a swimmer of mass m is resting on it, the slab floats in fresh water with its tip at the same level as the water surface. Find the area of the slab.

Homework Equations

$$\Sigma F=F_{buoyant}-Mg=0$$
$$F_{buoyant}=Mg$$
$$F_{buoyant} = \rho_{f}ghA$$

(Archimedes Principal: Any object completely or partially submerged in a fluid experiences an upward buoyant force whose magnitude is equal to the weight of the fluid displaced by the object)

When totally submerged $$\Sigma F= (\rho_{f}-\rho_{o})V_{o}g$$

Where $$\rho_{o}$$ is the density of the object

The Attempt at a Solution

I don't even know. I wrote out a bunch of stuff. I know the density of water is $$1.00 (10^{3} \frac{kg}{m^{3}})$$. Of course I tried substituting and that gets me no where. I tried making a free body diagram and all that says is that the Buoyant force is equal to weight of the swimmer and the board (duh).

$$\Sigma F= F_{buoy} - F_{board} - F_{swimmer} = 0$$

Also a simple manipulation showed that $$(\rho_{f}-\rho_{s}) \Delta h A = m_{s}$$. I think is right?

Honestly I think some more numbers are missing :S

 

[djeitnstine]

Ok I read the solution and it was only a formula that they wanted... So I was correct.