Two spheres of different materials one with double the radius and one-fourth wall thickness of the other are filled with ice. If the time taken for complete melting of ice in the larger sphere is `25` minutes and for smaller one is `16` minutes, the ratio of thermal conductivities of the materials of larger sphere to that of smaller sphere is:

Options

A. `4:5`
B. `5:4`
C. `8:25`
D. `25:8`

Answer

C

Explanation

`(DeltaQ)/(Deltat)=(KADeltaT)/(triangle)impliesDeltaQ=KA((DeltaT)/(Deltax))Deltat`
Assuming the thickness of the spheres to be small, we have for smaller sphere:
`("rate of heat flow")("time")=("volume of ice melted")(rho L)`
i.e., `K_1(4pir^2)(Delta theta)/(d)16=(4)/(3)pir^3rhoL` ...(i)
For larger sphere:
`K_2[4pi(2r)^2](Deltatheta)/((d)/(4))25=(4pi)/(3)(2r)^3rhoL` ...(ii)
dividing Eq. (ii) by Eq. (i)
`K_2//K_1=8//25`