If 1300 square centimeters of material is available to make a box with a square base and an open top find the largest possible volume of the box.?

Answer 1

Let the side of the base of the box be s

Let the height of the box be h

The surface area of the box is A = s² + 4sh → h = (A - s²)/4s

The the volume of the box is V = s²h = s²(A - s²)/4s = s(A - s²)/4

The volume is a max when dV/ds = 0

dV/ds = d/ds (sA/4 - s³/4) = A/4 - 3s²/4 = 0

→ 3s²/4 = A/4

→ s² = A/3

→ s = √(A/3)

→ h = (A - s²)/4s

= (A - A/3)/(4√(A/3))

= (2A/3)/(4√(A/3))

= ½√(A/3)

→ V = s²h

= (√(A/3))² ½√(A/3)

= ½(A/3)√(A/3)

= ½(A/3)^1.5

= ½(1300 cm²/3)^1.5

≈ 4510.28 cm³

Answer 2

It is 4510.28 cm3.