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Two spherical containers made of different materials are filled with ice. One sphere has twice the radius of the other, while its wall thickness is one-fourth that of the smaller sphere. If the ice in the larger sphere melts completely in 25 minutes and that in the smaller sphere in 16 minutes, what is the ratio of the thermal conductivity of the material of the larger sphere to that of the smaller sphere?

1. 4: 5
2. 5: 4
3. 25: 8
4. 8: 25

Correct answer: 8: 25

Solution

Melt time t ~ mass*thickness/(K*area) ~ r^3*d/(K*r^2) = r*d/K, so K ~ r*d/t. K_large/K_small = (2r*(d/4)/25)/(r*d/16) = (1/50)*16 = 8/25, i.e. 8:25.

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