Answer
The sphere is made of iron.
Work Step by Step
The heat energy lost by the metal sphere will be equal to the heat energy gained by the mercury. We can find the specific heat $c_s$ of the sphere:
$m_s~c_s~\Delta T_s = m_m~c_m~\Delta T_m$
$c_s = \frac{m_m~c_m~\Delta T_m}{m_s~\Delta T_s}$
$c_s = \frac{(4.08~kg)(140~J/kg~C^{\circ})(99.0^{\circ}C-20.0^{\circ}C)}{(0.500~kg)(300^{\circ}C-99.0^{\circ}C)}$
$c_s = 449~J/kg~C^{\circ}$
Since this specific heat is close to the specific heat of iron, we can conclude that the sphere is made of iron.
