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1) Two spheres of different materials, one with double the radius and 1/4th wall thickness of the other are filled with ice. If the time taken for complete melting of ice in the large sphere is 25 minutes and that in smaller sphere is 16 minutes, what is the ratio of the thermal conductivities of larger one to smaller one?

I solved it in the following way:

Let R1 and R2 be the radius of the larger sphere and smaller sphere respectively. Let d1 and d2 be the thickness of the larger sphere and smaller sphere respectively. Let Q1 and Q2 be the amount of heat conducted by the larger sphere and smaller sphere respectively. Let t1 and t2 be the time taken for melting of ice by the larger sphere and smaller sphere respectively. Let K1 and K2 be the thermal conductivities of the larger sphere and smaller sphere respectively.

Given that R1 = 2R2, d1 = d2/4, t1 = 25 minutes, t2 = 16 minutes

Q1 = K1(4(pi)(R1^2)t1/d1

Q2 = K2(4(pi)(R2^2)t2/d2

Assuming that same quantity of ice is melted in both the spheres we have Q1 =Q2

K1(R1^2)t1/d1 = K2(R2^2)t2/d2

By solving I get,

K1/K2 = 1/25

But the book answer is 8/25. Which is correct?

I have another doubt.

2) Two solid spheres of radii R1 and R2 are made of same material and have similar surfaces. The spheres are raised to the same temperature and the allowed to cool under identical conditions. Assuming the spheres to be perfect conductors of heat, what is the ratio of their rate of loss of heat? Also what is the ratio of their rate of cooling?

I know the answer to first question which is R1^2/R2^2, since Rate of loss of heat is proportional to surface area. I have a doubt in the 2nd question. Doesn’t rate of cooling and rate of loss of heat mean the same? In that case the answer to the 2nd question is also R1^2/R2^2. But the book answer is R2/R1.