Answer: 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
