Q.3A rectangular box, which is open at the top, has a capacity of 256 cubic feet. Determine the dimensions of the box such that the least material is required for the construction of the box. Use Lagrange, s method of multipliers to obtain the solution.
Answers
Given:
Capacity of rectangular box = 256 cubic feet.
To find:
The dimensions of the box
Explanation:
Let x, y, z be the length, breadth , height of the box
volume = xyz =256
xyz - 256 = 0 (1)
∅(x, y, z) = x y z - 256
Let S be the material surface of the box.
S = x y + 2yz + 2zx
∂S/∂x = y + 2x and ∂∅/∂x = yz
∂S/∂y = x + 2z and ∂∅/∂y = xz
∂S/∂z= 2y + 2x and ∂∅/∂z = xy
By Langrage's method multiplier we have:
[(∂S/∂x) + λ(∂∅/∂x)]= 0 y + 2z +λyz = 0 (2)
[(∂S/∂y) + λ(∂∅/∂y)]= 0 x + 2z +λyz = 0 (3)
[(∂S/∂z) + λ(∂∅/∂z)]= 0 2y + 2x +λyz = 0 (4)
Multiplying by x we get,
xy + 2 xz + λ xyz = 0
xy + 2 xz + 256λ = 0
xy + 2xz = -256λ (5)
Multiplying(3) by y we get,
xy + 2 yz +λ xyz = 0
xy + 2 yz +256λ = 0
xy + 2yz = -256λ (6)
Multiplying (4) by z , we get
2yz +2xz +λ xyz = 0
2yz +2xz =-256λ = 0 (7)
From 5 and 6 we have,
xy + 2xz = xy +2yz
From 6 and 7
xy +2yz = 2yz = 2yz + 2xz
xyz = 256
(y)(y) (y/2) = 256= y³ = 512 y = 8
x = 8 , y= 8 , z = 4
Answer = 8 , 8 , 4.