Problem $3$

A metallic sphere $($initial radius $r_0,$ density ${}_s)$ is suspended above a cylindrical tank already containing

the same metal in liquid state $($density ${}_1).$ The tank$\text{'}$s diameter is $D$ and the initial level of the liquid

metal in the tank is $h_0.$ The sphere is melting slowly, and its diameter is decreasing at a constant rate

$(\frac{m}{s}).$

Determine:

$($a$)$ the expression for the variation of the sphere's radius with time,

$($b$)$ the expression for the increase with time of the level of the liquid metal in the tank resulting from

the melting sphere,

$($c$)$ the final height of the liquid metal in the tank once the sphere is completely melted. Check your

answer by a global mass balance.