3. (15 points). A thin-walled cylindrical glass, filled with maple syrup to $1 / 4$, floats in a container of water, submerging to the middle. The same glass, but filled with water to $1 / 2$, floats in a container of syrup, also submerging to the middle. What fraction of the glass can be filled with syrup so that it does not sink in water? And what fraction of the glass can be filled with syrup so that it does not sink in syrup?
1.3 Equal volumes of two immiscible liquids with different densities were poured into a long thin tube, filling it halfway. The tube was then bent into a ring, placing it in a vertical plane (see fig.). The angle that the segment passing through the boundary of the liquids and the center of the ring makes with the vertical is $\alpha$. Find the density of the lighter liquid if the density of the heavier liquid is known and equal to $\rho_{1}$.

In mechanical equilibrium, the hydrostatic pressures in the left and right parts of the ring will be equal.

In the right part:

$$
P_{R}=\rho_{1} g h_{1}=\rho_{1} g R\left(1-\cos \left(\frac{\pi}{2}-\alpha\right)\right)=\rho_{1} g R(1-\sin (\alpha)) ;
$$

In the left part:

$$
P_{R}=P_{L} \Longrightarrow \rho_{1}(1-\sin (\alpha))=\rho_{1}(1-\cos (\alpha))+\rho_{2}(\cos (\alpha)+\sin (\alpha))
$$

Finally:

$$
\rho_{2}=\rho_{1} \cdot \frac{\cos \alpha-\sin \alpha}{\cos \alpha+\sin \alpha}=\rho_{1} \cdot \frac{1-\tan \alpha}{1+\tan \alpha} .
$$