A square sheet of metal has side 12 cm. Four equal squares are cut from the corners and the sides are turned up to form an open box. The maximum volume of the box is:

A square sheet of metal has side 12 cm. Four equal squares are cut from the corners and the sides are turned up to form an open box. The maximum volume of the box is:

• A. 128 cm³
• B. 64 cm³
• C. 256 cm³
• D. 32 cm³

Answer: A) 128 cm³

Explanation: Let cut square side = x. Box dimensions: 12 − 2x by 12 − 2x by x. V = x(12 − 2x)² = 4x(6 − x)² = 4(x³ − 12x² + 36x). dV/dx = 4(3x² − 24x + 36) = 12(x² − 8x + 12) = 12(x − 2)(x − 6). x = 2 gives max. V = 2 × 8² = 128 cm³.