Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 77 / 4 / b / Solution

The horizontal area is $A=10^6{\mathrm{m}}^2$ and the total mass from part (a) is $m=2.55\times 10^{11}\mathrm{kg}$. The iceberg melt heat budget includes warming to $0^{\circ }\mathrm{C}$ followed by fusion. With temperature $T(z)=-0.6z$ in the upper $50$ m and $-30^{\circ }\mathrm{C}$ below,The two integrals are $525000$ and $6750000\mathrm{kg}\mathrm{K}{\mathrm{m}}^{-2}$, so using the stated ice specific heat capacity,The latent heat of fusion contribution isThereforeThis produces liquid water at $0^{\circ }\mathrm{C}$; warming the meltwater further would require additional heat.