$p_L=p_P$

$\cancel{p_a}+p_o=\cancel{p_a}+p_w$

$p_o=p_w$

$p_o={\rho }_{o}gH$

$p_w={\rho }_{w}gl$

$l=2x$

$H=h_0+x$

$x=H-h_0$

${\rho }_o\cancel{g}H={\rho }_w\cancel{g}l$

${\rho }_{o}H={\rho }_w\cdot 2x$

${\rho }_{o}H=2{\rho }_w\left(H-h_0\right)$

${\rho }_{o}H=2{\rho }_{w}H-2{\rho }_w{h}_0|-2{\rho }_{w}H$

${\rho }_{o}H-2{\rho }_{w}H=-2{\rho }_w{h}_0|\cdot (-1)$

$2{\rho }_{w}H-{\rho }_{o}H=2{\rho }_w{h}_0$

$H\left(2{\rho }_w-{\rho }_o\right)=2{\rho }_w{h}_0|:\left(2{\rho }_w-{\rho }_o\right)$

$H=\frac{2{\rho }_w{h}_0}{2{\rho }_w-{\rho }_o}$

$H=\frac{2\cdot 1\frac{\mathrm{g}}{{\mathrm{cm}}^3}\cdot 8\mathrm{cm}}{2\cdot 1\frac{\mathrm{g}}{{\mathrm{cm}}^3}-0,82\frac{\mathrm{g}}{{\mathrm{cm}}^3}}=\frac{16\cancel{\frac{\mathrm{g}}{{\mathrm{cm}}^3}}\cdot \mathrm{cm}}{1,18\cancel{\frac{\mathrm{g}}{{\mathrm{cm}}^3}}}\approx 13,6\mathrm{cm}$