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 one 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 `540cal//g` and latent heat of melting of ice is `80cal//g`. If the point P is at a distance of `lambdax` from the ice end A, find the value `lambda`. [Neglect any heat loss to the surrounding.]

Explanation

For heat flow from P to 0
`L_f(dm_1)/(dt)=(KA400)/(lambdax)……(i)`
For heat flow from P to B
`L_(vap)(dm_2)/(dt)=(KA300)/(10x-lambdax)….(ii) [Given (dm_1)/(dt)=(dm_2)/(dt)]`
On solving (i) and (ii), we get `lambda=9`