A copper wire 10.0 mm in diameter is drawn down to 8.0 mm. What is the percent cold work (reduction in area)?

Answer: A. 36%

Percent cold work is the reduction in cross-sectional area, %CW = (A₀ − A_d)/A₀ × 100. Because area is proportional to the square of the diameter, %CW = (10.0² − 8.0²)/10.0² × 100 = (100 − 64)/100 × 100 = 36%. 20% is the reduction in diameter, (10.0 − 8.0)/10.0, and 56% is the mistake of dividing by the area after drawing ((100 − 64)/64 ≈ 0.56). The greater the cold work, the higher the dislocation density and the more the metal work-hardens: strength rises but ductility falls. The amount of cold work also affects the recrystallization temperature and the grain size after recrystallization.