Hydrostatics and work done by a gas in a situation similar to a siphon

[chingel]

It isn't actually homework, it was just a problem proposed and then the answer was also given, but I don't understand it and I would like some help in trying to understand.

Homework Statement

In a cylindrical container the water level is at 30 cm. If you float a glass bowl in it, the water level will rise by 5,4 cm. What will be the water level if the glass bowl is drowned in the container? Density for glass = 2700 kg/m3, for water = 1000 kg/m3

Homework Equations

Here is the given answer which I don't understand:

$$\begin{split} &h_0=0,3m\\ &h_1=0,054m\\ &h ?\\ \\ &h_2ρ_{glass}=h_1ρ_{glass}-h_1ρ_{water}\\ &h_2=h_1(ρ_{glass}-ρ_{water})/ρ_{glass}\\ &h_2=3,4 cm\\ &h=h_0+h_2\\ &h=33.4cm\\ \end{split}$$

I don't have the slightest idea why are the densities and heights multiplied together and how does it give the answer. If anyone can shed some light on this I would be grateful.

The Attempt at a Solution

What I tried to do is that first I assumed the bottom of the container to have an area of $S (cm^{2})$. Then in order for the water level to rise 5,4 cm, the volume of water displaced must be $S*5,4 (cm^{3})$. For the bowl to displace that much water, it must weight as much as the water displaced, ie $m=S*5,4*1(g/cm^{3})=5,4S (g)$. For a glass object weighing that much, it's volume must be $5,4S/2,7=2S (cm^{3})$. Since when something is submerged underwater, it displaces water equal to the volume of the object, so the water level rises by 2 cm. What did I do wrong?Edit: sorry for the wrong title, I originally wanted to post two problems but I then decided to only post the hydrostatic buoyancy/water displacement problem.

 

[RTW69]

I don't understand their solution. I agree with your solution. You do a free body of a floating bowl: F_B=h₁*area*ρ_water*g=m_glass*g solve for m_glass

With submerged bowl:m_glass=ρ_glass*vol_bowl where vol_bowl=h₂*area. Area is the area of bottom of cylinder

Set the m_glass equal to each other and you get h₂=2 cm

If you find the answer let me know

 

[chingel]

It turned out that the answer calculated by you and me was correct and that they had revised their answer.