Explanation Given information: Construct a box using 12 ft 2 of material. Calculation: Let x be the length of the box, y be the width of the box, and z be the height of the box. The surface area of the box with the dimensions x , y , and z is expressed as, S = x y + 2 y z + 2 z x (1) The surface area of the material is 12 ft 2 . Substitute 12 ft 2 for S in Equation (1). x y + 2 y z + 2 z x = 12 The volume of the box is V = x y z . Let consider the given function as follows: f ( x , y , z ) = x y z Find the gradient ( ∇ f ) . ∇ f = y z i + x z j + x y k Show the constraint as follows: g ( x , y ) = x y + 2 y z + 2 z x − 12 = 0 (2) Find the gradient ( ∇ g ) using Equation (2). ∇ g = ( y + 2 z ) i + ( x + 2 z ) j + ( 2 y + 2 x ) k Find the values of x , y , and λ that satisfies the following equation: ∇ f = λ ∇ g Substitute y z i + x z j + x y k for ∇ f and ( y + 2 z ) i + ( x + 2 z ) j + ( 2 y + 2 x ) k for ∇ g

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Chapter 14.7, Problem 59E

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Construct a rectangular box with maximum volume of the dimensions.

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Chapter 14.7, Problem 58E

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