Explanation Given: The box is rectangular in shape and opened from top. The volume of the box is 36 ft 3 . The length of the box is 2 x , where x is the width of the base of the box. Formula used: Volume function of the resulting open box will be: V ( x ) = l × b × h Here, l , b , h refers to the length, breadth, depth of the open box, respectively. Calculation: The box is rectangular. Consider the width of the base of the box as x . Then, the length of the box is 2 x . Also, consider the depth of the box as y . Calculate the volume of the box: V = l b h Substitute 2 x for l , x for b and y for h : V = 2 x ⋅ x ⋅ y = 2 x 2 y Further, substitute 36 ft 3 for V ; 36 = 2 x 2 y y = 36 2 x 2 = 18 x 2 …...…... (1) Now, the closed box has six sides but, in this case, the box is opened from one side and thus has five sides only. Calculate the surface area of the bottom of the box as the top portion is open. A bottom = 2 x ⋅ x = 2 x 2 The sides length of the box has a depth of y and the width as x . Also, all the rest four sides have equal area. Thus, calculate the surface area of the rest four sides of the box. A rest sides = 2 x y + x y + 2 x y + x y = 6 x y When the surface area of the box is less, then the material needed will also be less. Thus, calculate the amount of the material needed for the box or the surface area of the box. M ( x ) = A bottom + A rest sides Substitute 2 x 2 for A bottom , and 6 x y for A rest sides . M ( x ) = 2 x 2 + 6 x y Further, substitute 18 x 2 for y from equation (1). M ( x ) = 2 x 2 + 6 x ( 18 x 2 ) = 2 x 2 + 108 x Now, to get the minimum amount of materials, take the first derivative of the function M ( x ) and equate to zero

Mathematics with Applications, Books a la Carte, and MyLab Math with Pearson eText - Title-Specific Access Card Package (12th Edition)

12th Edition

ISBN: 9780134862651

Author: Lial

Publisher: PEARSON

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Chapter 12.3, Problem 42E

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To calculate: The dimensions of the box built with the minimum amount of materials.

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Chapter 12.3, Problem 41E

Chapter 12.3, Problem 43E

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Mathematics with Applications, Books a la Carte, and MyLab Math with Pearson eText - Title-Specific Access Card Package (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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