If 10,800 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume (in cm3) of the box.

the largest possible volume (in cm³) of the box is 108,000 cm³

Step-by-step explanation:

first lets rep each side of the base by x cm; and height by y cm

now

Surface Area(SA) = base area + four sides area [open at top]

SA = x² + 2y(x + x)

= x² + 4xy = 10800 cm²

y = {10800 - x²} / 4x

y = 2700/x - x/4

now

Volume (V) = (x²) × (2700/x - x/4)

= 2700x - (x³)/4

by Differentiation

V' = 2700 - (3/4) × (x²)

Setting V' = 0,

2700 - (3/4) × (x²)  = 0;

Solving x = +/- 60 cm

As dimension cannot be negative, so - 60 is rejected.

As well V" = (-3/2) × (x)

At x = -60,

V" = 90, which is > 0; so volume is minimum;

while at x = 60,

V" = -90, which is < 0; so volume is maximum.

Therefore for maximum volume x = 60 cm

⇒ height y = 30 cm

Therefore the largest possible volume (in cm³) of the box

= 60 x 60 x 30 = 108,000 cm³

3+x

step-by-step explanation: