– The mass of the block $m_b = 0.5text{ kg}$ and of the water $m_w = 0.5text{ kg}$

– The starting temperature of the block $T_b = 54.5text{textdegree}text{C}$ and the water $T_w = 20.0text{textdegree}text{C}$

– The specific heat capacities of the block $c_b = 390 frac{text{J}}{text{kg}text{textdegreetext{C}}}$ and the water $c_w = 4186 frac{text{J}}{text{kg}text{textdegreetext{C}}}$

Thermal energy is transferred from the block to the water through heat. This leads to a change in temperature:

$$
begin{gather*}
Q_b + Q_w = 0 \
m_b , c_b (T – T_b) + m_w , c_w(T – T_w) = 0
end{gather*}
$$

Rewriting we have:

$$
begin{align*}
& (m_b , c_b , T_b + m_w , c_w)T = m_b , c_b , T_b + m_w , c_w , T_w \
& T = frac{m_b , c_b , T_b + m_w , c_w , T_w}{m_b , c_b + m_w , c_w}
end{align*}
$$

Now using $m_b = m_w$

$$
begin{align*}
T &= frac{c_b , T_b + c_w , T_w}{c_b + c_w} = frac{390 frac{text{J}}{text{kg}text{textdegree}text{C}} cdot 54.5text{textdegree}text{C} + 4186frac{J}{text{kg}text{textdegree}text{C}} cdot 20.0text{textdegree}text{C}}{ left(390 + 4186 right)frac{text{J}}{text{kg}text{textdegree}text{C}}} \
T &= 22.95 text{textdegree}text{C}
end{align*}
$$

– The specific heat capacities of lead and water are:

$$
begin{align*}
& c_l = 128 frac{text{J}}{text{kg}text{textdegree}text{C}} \
& c_w = 4186 frac{text{J}}{text{kg}text{textdegree}text{C}}
end{align*}
$$

– The masses of lead and water are:

$$
begin{align*}
& m_l = 225text{ g} = 0.235 text{ kg} \
& m_w = 177text{ g} = 0.177 text{ kg}
end{align*}
$$

– The initial temperatures of lead and water are:

$$
begin{align*}
& T_l = 84.2text{textdegree}text{C} \
& T_w = 21.5text{textdegree}text{C}
end{align*}
$$

The changes in thermal energy of the lead and water are related to their changes in temperature as follows:

$$
begin{align*}
Q_l &= m_l , c_l (T – T_l) \
Q_w &= m_w , c_w (T – T_w)
end{align*}
$$

Since the heat is transferred from one body to the other the following holds:

$$
begin{align*}
& Q_l + Q_w = 0 \
& m_l , c_l (T – T_l) + m_w , c_w (T – T_w) = 0
end{align*}
$$

Rearranging for temperature:

$$
begin{align*}
T &= frac{m_l , c_l , T_l + m_w , c_w , T_w}{m_l , C_l + m_w , T_w} = \
&= frac{ left(0.235text{ kg} right) left(128 frac{text{J}}{text{kg}text{textdegree}text{C}} right) left(84.2text{textdegree}text{C} right) + left(0.177text{ kg} right) left(4186 frac{text{J}}{text{kg}text{textdegree}text{C}} right) left(21.5text{textdegree}text{C} right)} { left(0.235text{ kg} right) left(128 frac{text{J}}{text{kg}text{textdegree}text{C}} right) + left(0.177text{ kg} right) left(4186 frac{text{J}}{text{kg}text{textdegree}text{C}} right)} = 23.9 text{textdegree}text{C}
end{align*}
$$