Melting of Ice with Inclusions and Liquid Level Change

Problem Statement

A cylindrical tank of cross-sectional area $S_0$S0​ contains a sufficient amount of water with density $\rho_w$ρw​.

A single large block of ice floats in this water. The ice block consists primarily of pure ice (mass $M$M, density $\rho_i$ρi​) and completely contains the following three foreign objects:

1. A metal lump of mass $m_1$m1​ and density $\rho_1$ρ1​ ($\rho_1 > \rho_w$ρ1​>ρw​)
2. Oil of mass $m_2$m2​ and density $\rho_2$ρ2​ ($\rho_2 < \rho_w$ρ2​<ρw​)
3. Air of volume $V_a$Va​

The ice block floats at rest on the water surface without touching the walls or bottom of the tank. Let the water level at this point be $h_{init}$hinit​.

After sufficient time passes and the ice fully melts, the metal lump sinks to the bottom, the oil separates and forms a uniform layer on top of the water, and all the air escapes into the atmosphere. Let the height of the topmost liquid surface (top of the oil layer) when the system is again at rest be $h_{final}$hfinal​.

Find the change in liquid level $\Delta h = h_{init} - h_{final}$Δh=hinit​−hfinal​.

Evaporation of water, changes in density due to temperature, and dissolution of oil in water can all be neglected. The mass of air is negligible.

Constraints

• $S_0 = 1.2 \times 10^{-2} \ \mathrm{m^2}$S0​=1.2×10−2 m2
• $M = 3.0 \ \mathrm{kg}$M=3.0 kg
• $\rho_i = 9.2 \times 10^2 \ \mathrm{kg/m^3}$ρi​=9.2×102 kg/m3
• $\rho_w = 1.0 \times 10^3 \ \mathrm{kg/m^3}$ρw​=1.0×103 kg/m3
• $m_1 = 2.7 \times 10^{-1} \ \mathrm{kg}$m1​=2.7×10−1 kg
• $\rho_1 = 7.5 \times 10^3 \ \mathrm{kg/m^3}$ρ1​=7.5×103 kg/m3
• $m_2 = 1.7 \times 10^{-1} \ \mathrm{kg}$m2​=1.7×10−1 kg
• $\rho_2 = 8.5 \times 10^2 \ \mathrm{kg/m^3}$ρ2​=8.5×102 kg/m3
• $V_a = 8.0 \times 10^{-4} \ \mathrm{m^3}$Va​=8.0×10−4 m3

Input Format

Find the numerical value of $\Delta h$Δh in $\mathrm{m}$m and give the answer as a positive integer equal to the value multiplied by $10^5$105.