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.]

Options

A. 3
B. 6
C. 9
D. 12

Answer

C

Explanation

Heat current =` (T_1 -T_2)/(l//KA)`
Let m be the amount of ice that melts per unit time and same amount of water evaporates per unit time, then we can write the following:
For ice
` m xx 80 = (400)/(lamda x // kA) = (k xx 400 xx A)/(lamda x)` ....(i)
for water
` m xx 540 = (300)/((10x - lamdax) // kA) =(k xx 300 xx A)/((10x - lamda x )) ` ......(ii)
equation (i)/equation (ii)
` 80/540 = 400/300 xx ((10 - lamda))/(lamda)`
` rarr 1/54 = 1/6 xx ((10 - lamda))/(lamda)`
` rArr 3 lamda = 270 - 27 lamda`
` rarr 30 lamda = 270`
` rArr lamda = 9`