Respuesta :
Step-by-step explanation:
√19200cm²
=138.56cm
then the highest possible volume
=(138.56)³
=2660195.926cm³
The largest possible volume of the box is; V = 25600 cm³
Let us denote the following of the square box;
Length = x
Width = y
height = h
Formula for volume of a box is;
V = length * width * height
Thus; V = xyh
but we are dealing with a square box and as such, the base sides are all equal and so; x = y. Thus;
V = x²h
The box has an open top and as such, the surface are of the box is;
S = x² + 4xh
We are given S = 19200 cm². Thus;
19200 = x² + 4xh
h = (19200 - x²)/4x
Put (19200 - x²)/4x for h in volume equation to get;
V = x²(19200 - x²)/4x
V = 4800x - 0.25x³
To get largest possible volume, it will be dimensions when dV/dx = 0. Thus;
dV/dx = 4800 - 0.75x²
At dV/dx = 0, we have;
4800 - 0.75x² = 0
0.75x² = 4800
x² = 4800/0.75
x² = 6400
x = √6400
x = 80 cm
From h = (19200 - x²)/4x;
h = (19200 - 80²)/(4 × 80)
h = (19200 - 6400)/3200
h = 4 cm
Largest possible volume = 80² × 4
Largest possible volume = 25600 cm³
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