A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

Answer 1

x is side of base and y is height. Tnen A = 4xy + x². Now V = x²y = 4000; y = 4000/x²;

A = 16000/x + x²; A' = - 16000/x² + 2x = 0; x³ = 8000; x = 20.

Now A'' = 32000/x³ + 2, at x = 20 A'' 32000/20³ + 2 = 6 > 0 and we have min.

y = 4000/20² = 10. So, box must 20cm×20cm×10cm to have minimum amount of material: min A = 16000/20 + 20² = 1200 cm²