At room temperature $(27^{\circ} \mathrm{C})$, the resistance of a heating element is $50 \Omega$. The temperature coefficient of the material is $2.4 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$. The temperature of the element, when its resistance is $62 \Omega$, is __________${ }^{\circ} \mathrm{C}$.

<p>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:</p> <p>$ R = R_0(1 + \alpha(T - T_0)) $</p> <p>Where:</p> <ul> <li>$ R $ is the resistance at temperature $ T $ (in this case, $ 62 \Omega $).</li> <li>$ R_0 $ is the resistance at reference temperature $ T_0 $ (in this case, $ 50 \Omega $).</li> <li>$ \alpha $ is the temperature coefficient of resistance (in this case, $ 2.4 \times 10^{-4} \, ^\circ\mathrm{C}^{-1} $).</li> <li>$ T $ is the unknown temperature we need to find.</li> <li>$ T_0 $ is the reference temperature, given as $ 27^\circ \mathrm{C} $.</li> </ul> <p>By substituting the given values into the formula, we get:</p> <p>$ 62 = 50(1 + 2.4 \times 10^{-4}(T - 27)) $</p> <p>First, divide both sides of the equation by 50:</p> <p>$ \frac{62}{50} = 1 + 2.4 \times 10^{-4}(T - 27) $</p> <p>Then solve for $ T $:</p> <p>$ 1.24 = 1 + 2.4 \times 10^{-4}(T - 27) $</p> <p>$ 0.24 = 2.4 \times 10^{-4}(T - 27) $</p> <p>$ \frac{0.24}{2.4 \times 10^{-4}} = T - 27 $</p> <p>$ 1000 = T - 27 $</p> <p>$ T = 1027 ^\circ\mathrm{C} $</p> <p>Thus, the temperature of the element when its resistance is $ 62 \Omega $ is $ 1027^\circ\mathrm{C} $.</p>