#### 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.