The coefficient of area expansion \(\beta\) of a rectangular sheet of a solid in terms of the coefficient of linear expansion \(\alpha\) is:
1. \(2\alpha\)
2. \(\alpha\)
3. \(3\alpha\)
4. \(\alpha^2\)

A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are \(5.012\) m and \(5.00\) m, respectively at \(27^{\circ}\text{C}.\) To what temperature should the ring be heated so as to fit the rim of the wheel?
\(\left(\text{Given:}~\alpha~\text{for iron}= 1.20 \times 10^{-5}~^\circ{\text{C}^{-1}}\right)\)
1. \(128^{\circ}\text{C}\)
2. \(118^{\circ}\text{C}\)
3. \(227^{\circ}\text{C}\)
4. \(218^{\circ}\text{C}\)