BS206 Answers for Topic 7 2019.docx - Problems for Tutorial associated with Topic 7(for week 9 Problem One Consider the following constrained

BS206 Answers for Topic 7 2019.docx - Problems for Tutorial...

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Problems for Tutorial associated with Topic 7 (for week 9). Problem One. Consider the following constrained optimisation problem: Max: z = xy +2x s.t. 4x + 2y =60 a) solve this problem by using the substitution method. b) solve this problem by using the total differential or equal slope method. c) solve this problem by using the Lagrangian method. (a) Sub Method z = xy + 2 x 1 4 x + 2 y = 60 2 Rearrange (2) as y = 60 4 x 2 = 30 2 x and then substitute in (1): z = x ( 30 2 x ) + 2 x z = 30 x 2 x 2 + 2 x z = 32 x 2 x 2 Maximise: d z dx = 32 4 x = 0 x* ¿ 32 4 =8 Plug back into (2) to get y: y*=14 Plug back into (1) to get z: z*= (8)(14) + 2(8)=96 Thus max = (8,14,96) (b) Equal Slope Method 1
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z ( x , y ) = xy + 2 x g ( x, y ) = 60 4 x 2 y Find slope of z(x,y) by setting total differential to 0: dz = ∂z ∂ x .dx + ∂ z ∂ y dy = 0 dz = [ y + 2 ] .dx + [ x ] .dy = 0 [ y + 2 ] .dx =− [ x ] .dy y + 2 x = dy dx = slope Find slope of g(x,y) by setting total differential to 0: dg = ∂g ∂x .dx + ∂g ∂ y dy = 0 dg = [ 4 ] .dx + [ 2 ] .dy = 0 [ 4 ] .dx =− [ 2 ] .dy 4 2 = - 2 = dy dx = slope
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  • One '19
  • Greg Moore

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