my question is about physics.?

Question

an industrial process has a cooling operation.
a large amount of hot solution (4000kg)must be cooled from a temperature of 48 c degree to 40c degree by the addition of ice.
treating the solution as warm water (shc=42oooj/kg c).
how much ice should be added?
a) if the ice teperature = 0 c degree
b) if the ice came from a freezer at -20 c degree? thank you all.

Answer 1

If the solution is to be treated as warm water, its heat capacity is 4,184 J/kg°C (not 42,000)
a)
m((80*4,186) J/kg + (40 °C)(4,186 J/kg°C)) = (4,000 kg)(4,184 J/kg°C)(8 °C)

m(80 + 40) = (4,000 kg)(8)
m = 4,000/15 kg ≈ 266.7 kg

b)
m(80 + 40 + 20/2) = (4,000 kg)(8)
m = (32,000 kg) / (130) ≈ 246.2 kg