(a) For the sake of simplicity in presentation let us denote the quantities pertaining to the copper block by subscript 1 and those to the steel block by the subscript 2.
The volume change of the blocks will be very small. These expansions occur against atmospheric pressure, making the work negligible compared to the heat. Therefore, from the first law we note that the change in the internal energy of each block is accounted for by the heat flow only.
\begin{equation*}
\Delta U_{1} \approx Q_{\text{into }1}.
\end{equation*}
Since no change of phase takes place we can write the \(Q\) in terms of specific heat. Although specific heat is a function of temperature, we will assume that the specific heat does not change much so that
\begin{equation*}
Q = m c \Delta T.
\end{equation*}
Let \(T_f\) be the final temperature of the two blocks. Then, the change in the energy of the copper block will be:
\begin{equation*}
\Delta U_{1} = m_{1} c_{1} (T_f-T_1).
\end{equation*}
Similarly, the change in the energy of the steel block will be
\begin{equation*}
\Delta U_{2} = m_{2} c_{2} (T_f-T_2).
\end{equation*}
The total energy of the two blocks does not change since the two blocks do not interact with any other systems. Therefore,
\begin{equation*}
\Delta U_{1} + \Delta U_{2} = 0,
\end{equation*}
which gives
\begin{equation*}
m_{1} c_{1} (T_f-T_1) + m_{2} c_{2} (T_f-T_2) = 0.
\end{equation*}
Solving for \(T_f\) we get
\begin{equation*}
T_f = \frac{m_{1} c_{1} T_1 + m_{2} c_{2} T_2 }{m_{1} c_{1} + m_{2} c_{2} }.
\end{equation*}
Now, we put in the following numbers.
\begin{align*}
\amp m_1 = 50\ \text{kg},\ \ c_1 = 390\ \text{J/kg.K}, \ \ T_1 = 400\ \text{K}\\
\amp m_2 = 75\ \text{kg},\ \ c_2 = 500 \ \text{J/kg.K}, \ \ T_2 = 300\ \text{K}
\end{align*}
Therefore
\begin{align*}
T_f \amp = \frac{50\ \text{kg}\times 390\ \text{J/kg.K}\times 400\ \text{K} + 75\ \text{kg}\times 500 \ \text{J/kg.K}\times 300\ \text{K} }{50\ \text{kg}\times 390\ \text{J/kg.K} + 75\ \text{kg}\times 500 \ \text{J/kg.K} }\\
\amp = 334\ \text{K}.
\end{align*}
