Answer :
Dimension of Box = 12 * 12 * 6 cm with maximum capacity whose top is open & surface area = 432 cm²
Step-by-step explanation:
Let say Dimension of the box = L , B & H
Surface Area = LB + 2(L + B)H = 432
Volume = LBH
Capacity is maximum when volume is maximum.
For Maximum Volume for given Surface Area
Length = Breadth
L = B = x
=> x² + 4xh = 432
=> h = (432 - x²)/4x
Volume = x²h
= x² (432 - x²)/4x
= x(432 - x²)/4
= 108x - x³/4
V = 108x - x³/4
dV/dx = 108 - 3x²/4
putting dV/dx = 0
=> 3x²/4 = 108
=> x² = 36 * 4
=> x = 12
d²V/dx² = -6x/4 is -ve hence Volume is maximum
Length = 12
Breadth = 12
h = (432 - x²)/4x = (432 - 12²)/4(12)
=> h = 6
Dimension of Box = 12 * 12 * 6 cm
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