Volume is a three-dimensional scalar quantity. The maximum volume of the box is 0.8m³.
A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.
Surface area of the box = 2x² + 3xy
Surface area of lid = 2xy
Total area = 2x² + 5xy = 6
y = (6-2x²)/5x
Volume of the box, V = x²y
V = x²[(6-2x²)/5x] = (6x/5)-(2x³/5)
Now, for the volume to be the maximum,
dV/dx = 0
(6/5)-(6x²/5) = 0
x² = 1
x = ±1
y = 4/5
Thus, the maximum volume of the box = x²y = (1)²×(4/5) = 0.8 m³
Hence, the maximum volume of the box is 0.8m³.
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