For the flow of heat `Q` through a heat conductor the following equation is valid:
`(dQ)/dt = k*S*(dT)/dx`
This means that by knowing the material of the heat conductor, and the rate of heat flow, one can determine the temperature difference along the heat conductor as long as we know its length and surface area.
The question in the problem is much simpler asking to determine only the heat flow rate (`(dQ)/dt` ). Since the system is in equilibrium the heat flow rate through the rod is equal to the heat rate that melts the ice.
`(dQ)/dt = m/t*lambda_(ice) =(6.8*10^-3)/60 *3.34*10^5 =37.85 J/s`
Now we can compute the temperature difference per length unit for the rod:
`S=pi*R^2 =pi*0.05^2 =7.85*10^-3 m^2`
`(Delta(T))/l =1/(k*S) *(dQ)/dt =37.85/(400*7.85*10^-3) =12 K/m`
The rate of heat flow through the rod is 37.85 J/s and the temperature difference per unit length along the rod is 12 K/m.