Suppose a closed rectangular box has length twice it's breadth and has constant volume V. Determine the dimensions of the requiring least surface area.[ lagrange's method of undetermined multipliers]
Answers
Given : a closed rectangular box has length twice it's breadth and has constant volume V
To Find : Determine the dimensions of the requiring least surface area
Solution:
Breadth = x
Length = 2x
Height = h
Volume = x * 2x * h = V
=> h = V/2x²
Surface area = 2 ( lb + bh + lh )
S = 2 ( 2x² + xV/2x² + 2xV/2x²)
=> S = 2 ( 2x² + V/2x + V/x)
=> S = 4x² + V/x + 2V/x
=> S = 4x² + 3V/x
dS/dx = 8x - 3V/x²
dS/dx = 0
=> 8x - 3V/x² = 0
=> 8x³ = 3V
=> x³ = 3V/8
=> x = ∛(3V) / 2
d²S/dx² = 8 + 6V/x³ > 0 hence least area
Dimensions are :
Breadth = x ∛(3V) / 2
Length = 2x = ∛(3V)
Height = h = V/2x² = 2∛(V/9)
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