Question

A metal rod $AB$ of length $10x$ has its one end $A$ in ice at $0^{\circ }$ and the other end $B$ in water at ${100}^{\circ }C$. If a point $P$ on the rod is maintained at ${400}^{\circ }$, 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 $540cal/g$ and latent heat of melting of ice is $80cal/g$. 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).

Answer 1

$\frac{d{m}_{ice}}{dt}=\frac{d{m}_{vapour}}{dt}$
$Q_1=\frac{KA400}{t}=m\times 80$
$Q_1=\frac{KA(400-100)t}{(10-\lambda )x}t=m\times 540$
Dividing both, $\lambda =9$.