Here's how to solve this problem, which involves heat transfer and phase changes:

1. Define Variables:

* m_ice: Mass of ice = 30 g

* m_water: Mass of water = 500 g

* m_iron: Mass of iron container = 150 g

* T_ice: Initial temperature of ice = 0°C

* T_water: Initial temperature of water = 45°C

* T_final: Final temperature of the mixture (what we want to find)

* L_fusion: Latent heat of fusion of ice = 334 J/g

* c_water: Specific heat capacity of water = 4.18 J/(g°C)

* c_iron: Specific heat capacity of iron = 0.45 J/(g°C)

2. Set Up the Heat Balance Equation:

The total heat gained by the ice (melting and warming) must equal the total heat lost by the water and iron container as they cool down:

* Heat gained by ice = Heat lost by water + Heat lost by iron

3. Break Down the Heat Terms:

* Heat gained by ice:

* Melting: m_ice * L_fusion

* Warming: m_ice * c_water * (T_final - T_ice)

* Heat lost by water: m_water * c_water * (T_water - T_final)

* Heat lost by iron: m_iron * c_iron * (T_water - T_final)

4. Plug in the Values and Solve:

(30 g * 334 J/g) + (30 g * 4.18 J/(g°C) * (T_final - 0°C)) = (500 g * 4.18 J/(g°C) * (45°C - T_final)) + (150 g * 0.45 J/(g°C) * (45°C - T_final))

5. Simplify and Solve for T_final:

* 10020 J + 125.4 J/°C * T_final = 9390 J - 2090 J/°C * T_final + 3037.5 J - 67.5 J/°C * T_final

* 2283 J/°C * T_final = 1578.5 J

* T_final ≈ 0.69°C

Therefore, the final temperature of the mixture is approximately 0.69°C.