A metal rod AB of length 10x has its one end A in ice at 0°C and the other end B in water at 100°C. If a point P on the rod is maintained at 400°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 λx from the ice end A. Find the value of λ. (Neglect any heat loss to the surrounding).

Let dmice/dt be the rate of melting of ice and dmvapour/dt be the rate of evaporation of water.
Given that equal amounts of water and ice evaporate and melt per unit time, therefore dmice/dt = dmvapour/dt = m
Heat transferred from P to A (ice) is given by:
Q1 = KA(400 - 0)t/(10 - λ)x = m × 80
Heat transferred from P to B (water) is given by:
Q2 = KA(400 - 100)t/λx = m × 540
Dividing both equations:
[KA(400)t/(10 - λ)x]/[KA(300)t/λx] = (m × 80)/(m × 540)
400λ/300(10 - λ) = 80/540
4λ/3(10 - λ) = 4/27
27(4λ) = 12(10 - λ)
108λ = 120 - 12λ
120λ = 120
λ = 1