Question

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 $25$ minute and for smaller one is $16$ minute, the ratio of thermal conductivities of the materials of larger spheres to that of smaller sphere is

• A. 4:05
• B. 5:04
• C. 25:08:00
• D. 8:25

Answer 1

The Correct Option is D

Radius of small sphere $=r$
Thickness of small sphere $=t$
Radius of bigger sphere $=2 r$
Thickness of bigger sphere $=t / 4$
Mass of ice melted $=($ volume of sphere $) \times$ (density of ice)
Let $K_{1}$ and $K_{2}$ be the thermal conductivities of larger and smaller sphere.
For bigger sphere,
$\frac{K_{1} 4 \pi(2 r)^{2} \times 100}{t /4}=\frac{\frac{4}{3} \pi(2 r)^{3} \rho L}{25 \times 60}$
For smaller sphere,
$\frac{K_{2} \times 4 \pi r^{2} \times 100}{t}=\frac{\frac{4}{3} \pi r^{3} \rho L}{16 \times 60}$
$\therefore \frac{K_{1}}{K_{2}}=\frac{8}{25}$