We can start solving this problem by first understanding that the resistance of a material changes with temperature, and this change can be quantified using the temperature coefficient of resistance $ \alpha $. The relationship between the resistance of a material at any temperature $ T $ and its resistance at a reference temperature $ T_0 $ is given by the formula:

$ R = R_0(1 + \alpha(T - T_0)) $

Where:

• $ R $ is the resistance at temperature $ T $ (in this case, $ 62 \Omega $).
• $ R_0 $ is the resistance at reference temperature $ T_0 $ (in this case, $ 50 \Omega $).
• $ \alpha $ is the temperature coefficient of resistance (in this case, $ 2.4 \times 10^{-4} \, ^\circ\mathrm{C}^{-1} $).
• $ T $ is the unknown temperature we need to find.
• $ T_0 $ is the reference temperature, given as $ 27^\circ \mathrm{C} $.

By substituting the given values into the formula, we get:

$ 62 = 50(1 + 2.4 \times 10^{-4}(T - 27)) $

First, divide both sides of the equation by 50:

$ \frac{62}{50} = 1 + 2.4 \times 10^{-4}(T - 27) $

Then solve for $ T $:

$ 1.24 = 1 + 2.4 \times 10^{-4}(T - 27) $

$ 0.24 = 2.4 \times 10^{-4}(T - 27) $

$ \frac{0.24}{2.4 \times 10^{-4}} = T - 27 $

$ 1000 = T - 27 $

$ T = 1027 ^\circ\mathrm{C} $

Thus, the temperature of the element when its resistance is $ 62 \Omega $ is $ 1027^\circ\mathrm{C} $.