A copper slug whose mass `m_(c )` is 75 g is heated in a laboratory oven to a temperature T of `312^(@)C` . The slug is then dropped into a glass beaker containing a mass `m_(w ) = 220g` of water. The heat capacity `C_(b)` of the beaker is 45 cal `//`K. The initial temperature `T_(i)` of the water and the beaker is `12^(@)C`. Assuming that the slug, beaker, and water are an isolated system and the water does not vaporize, find the final temperature `T_(f )` of the system at thermal equilibrium.

Explanation

To find the final temperature \( T_f \) of the system at thermal equilibrium, we can use the principle of conservation of energy, which states that the heat lost by the copper slug will be equal to the heat gained by the water and the beaker.

### Step 1: Identify the heat transfer equations
The heat lost by the copper slug can be expressed as:
\[ Q_{c} = m_{c} C_{c} (T_{f} - T_{c}) \]
where:
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