Explanation Given Information: The top is made with heavy-duty cardboard and the sides are made with lightweight cardboard. The volume of the box is 2 cubic feet. Formula used: To solve a maxima - minima problem using substitution, 1. Solve the constraint equation for one of the variables, and then substitute in the objective function. 2. Minimize/Maximize the resulting function of two variables. Calculation: Let l represent the length, w represent the width and h represent the height of the cubic box. Now, the volume of this cubic box is 2 cubic feet, then, l w h = 2 And, let L represent the cost of lightweight cardboard and H represent the cost of heavy-duty cardboard. So, the total cost of making as only the top is made with heavy duty carboard will be, C = ( H + L ) l w + 2 L l h + 2 L w h Subject to l w h = 2 Now, minimize the objective function C = ( H + L ) l w + 2 L l h + 2 L w h Now, solve the constraint equation for one of the variables, l = 2 w h …… (1) Thus, substitute 2 w h for l in the objective function as, C ( w , h ) = ( H + L ) ( 2 w h ) w + 2 L ( 2 w h ) h + 2 L w h = 2 ( H + L ) h + 4 L w + 2 L w h Now, minimize the resulting function of two variables. First obtain the partial derivative with respect to w as, C w = − 4 L w 2 + 2 L h Obtain critical point by setting C w = 0 as, − 4 L w 2 = − 2 L h h = 2 w 2 Now, obtain the derivative with respect to h as, C h = − 2 ( H + L ) h 2 + 2 L w Set C h = 0 , − 2 (

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ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Chapter 8.4, Problem 36E

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To calculate: The dimensions of the box in terms of the price per square foot of heavy-duty cardboard and lightweight cardboard if only the bottom of the box are to be made with heavy duty cardboard and the sides are to be made with lightweight cardboard and volume of the box is 2 cubic feet.

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Chapter 8.4, Problem 35E

Chapter 8.4, Problem 37E

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The dimensions of the box in terms of the price per square foot of heavy-duty cardboard and lightweight cardboard if only the bottom of the box are to be made with heavy duty cardboard and the sides are to be made with lightweight cardboard and volume of the box is 2 cubic feet.

Solution Summary: The author explains how to solve a maxima-minima problem using substitution: solve the constraint equation for one of the variables, and then substitute in the objective function.

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To calculate: The dimensions of the box in terms of the price per square foot of heavy-duty cardboard and lightweight cardboard if only the bottom of the box are to be made with heavy duty cardboard and the sides are to be made with lightweight cardboard and volume of the box is 2 cubic feet.

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