In a container with mercury, water is added to a height of $9\mathrm{.}0$ cm and on top of this layer, another layer of oil $12$ cm thick is added. Under these conditions, a cylinder is placed that is $32$ cm high and is made up of half iron and half aluminum. Indicate the submerged height of the cylinder in mercury in cm$,$ if the density of iron is $7800$ kg$/$m$^3,$ The density of aluminum is $2700$ Kg$/$m$^3$ and the density of oil is $900$ kg$/$m$^3.$

The correct answer is $10\mathrm{.}81$