A metal rod AB of length 10 x has its one end A in ice at $0^{\circ} \mathrm{C}$ and the other end B in water at $100^{\circ} \mathrm{C}$. If a point P on the rod is maintained at $400^{\circ} \mathrm{C}$, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is $540 \mathrm{cal} / \mathrm{g}$ and latent heat of melting of ice is $80 \mathrm{cal} / \mathrm{g}$. If the point P is at a distance of $\lambda \mathrm{x}$ from the ice end A , find the value of $\lambda$. [Neglect any heat loss to the surrounding.]