Explanation Given: The volume of the box is 16,000 c m 3 The cost of the material for top and bottom of the box is $3 per square centimeter. The cost of the material for sides of the box is $1.50 per square centimeter. Calculation: Let x be the length of the side of the top and bottom for the box. Let y be the depth of the box. Since the top and bottom are in square shape, the area of top and bottom is ( x ) ( x ) = x 2 Since the cost of the material for top and bottom of the box is $3 per square centimeter, the cost of material for top and bottom is ( 3 ) ( 2 x 2 ) . The sides are in rectangle shape, the area of one side is x y . Thus, the area of 4 sides is 4 x y . Since the cost of the material for sides of the box is $1.50 per square centimeter, the cost of material for 4 sides is ( 1.50 ) ( 4 x y ) The total cost is as follows, C ( x ) = ( 3 ) ( 2 x 2 ) + ( 1.50 ) ( 4 x y ) = 6 x 2 + 6 x y The volume of the box is given by, V = ( x ) ( x ) ( y ) = x 2 y From the given volume of the box is 16,000. x 2 y = 16 , 000 y = 16 , 000 x 2 Substitute y = 16 , 000 x 2 in C ( x ) , C ( x ) = 6 x 2 + 6 x y = 6 x 2 + 6 x ( 16 , 000 x 2 ) = 6 x 2 + 96 , 000 x The first derivative of C ( x ) is as follows, C ( x ) = 6 x 2 + 96 , 000 x C

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 6.2, Problem 23E

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To find: The dimensions of the box which gives minimum total cost and also find minimum total cost.

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Chapter 6.2, Problem 22E

Chapter 6.2, Problem 24E

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Chapter 6 Solutions

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 6.1 - YOUR TURN 1 Find the absolute extrema of the...Ch. 6.1 - Prob. 2YT Ch. 6.1 - Prob. 1WE Ch. 6.1 - Prob. 2WE Ch. 6.1 - Prob. 1E Ch. 6.1 - Find the locations of any absolute extrema for the...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

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