Heat And Thermodynamics

A metal rod AB of length 10x has its one end A in ice at 0\(^\circ\)C and the other end B in water at 100\(^\circ\)C. If a point P on the rod is maintained at 400\(^\circ\)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 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of \(\lambda x\) from the ice end A, find the value of \(\lambda\). (Neglect any heat loss to the surrounding.)