A box with a square base and open top must have a volume of 32000 cm³. Find the dimensions of the base that minimize the amount of material used. A. 20 cm × 20 cm B. 30 cm × 30 cm C. 40 cm × 40 cm D. 50 cm × 50 cm Answer: C) 40 cm × 40 cm Explanation: V = x²h = 32000 → h = 32000/x². Surface area S = x² + 4xh = x² + 128000/x. S' = 2x - 128000/x² = 0 → x³ = 64000 → x = 40. Dimensions are 40 cm × 40 cm.

imo class 12 application of derivatives

A box with a square base and open top must have a volume of 32000 cm³. Find the dimensions of the base that minimize the amount of material used.

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A box with a square base and open top must have a volume of 32000 cm³. Find the dimensions of the base that minimize the amount of material used.

A. 20 cm × 20 cm
B. 30 cm × 30 cm
C. 40 cm × 40 cm
D. 50 cm × 50 cm

Answer: C) 40 cm × 40 cm

Explanation: V = x²h = 32000 → h = 32000/x². Surface area S = x² + 4xh = x² + 128000/x. S' = 2x - 128000/x² = 0 → x³ = 64000 → x = 40. Dimensions are 40 cm × 40 cm.

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