Explanation Given data: y The sum of length and girth of the package is greater than 84 inches and less than 108 inches. The package is rectangular with rectangular ends and 40 inches in length as shown in figure. Method/Formula used: A function y = f ( x ) has maxima or minima at x = a when f ′ ( a ) = 0 . The point x = a is maxima when f ″ ( a ) < 0 , and minima when f ″ ( a ) > 0 . Calculation: The shape of Linda’s package is shown in above figure. The package is rectangular with rectangular ends. Let the dimensions of the ends of the package are x and y and the length that is given 40 inches. Therefore, the girth of the package is 2 ( x + y ) inches. Now, the sum of the length and the girth is 40 + 2 ( x + y ) = L (say) = constant. The volume (in inch 3 ) of the package is V = 40 ⋅ x ⋅ y = 40 x y ... (1) The surface area (in inch 2 ) is S = 2 ( 40 ⋅ x + 40 ⋅ y + x ⋅ y ) S = 80 x + 80 y + 2 x y ... (2) Largest volume: The volume the package is V = 40 x y We have 40 + 2 ( x + y ) = L ⇒ y = L − 40 − 2 x 2 Therefore, volume of the box V = 40 x y = 40 x ( L − 40 − 2 x 2 ) = 20 x ( L − 40 − 2 x ) V = ( 20 x L − 800 x − 40 x 2 ) ... (3) Now, differentiating result with respect to ‘ x’ d V d x = 20 L − 800 − 80 x , and d 2 V d x 2 = − 80 . For maxima and minima of function V ( x ) ; d V d x = 0 ⇒ 20 L − 800 − 80 x = 0 ⇒ 20 L − 800 − 80 x = 0 ⇒ x = 20 L − 800 80 = L 4 − 10 Now, ( d 2 V d x 2 ) x = L 4 − 10 = − 80 < 0 , therefore, for x = L 4 − 10 the volume of package is maximum. We have y = L 2 − x − 20 = L 2 − ( L 4 − 10 ) − 20 = L 4 − 10 , ∵ x = L 4 − 10 Thus, for largest volume we have x = y = L 4 − 10 Now, according to given condition the sum of length and girth of the package is greater than 84 inches and less than 108 inches , i

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ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3.4, Problem 46A

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To find: The largest volume and surface area of the package.

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Chapter 3.4, Problem 45A

Chapter 3.4, Problem 47A

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Chapter 3 Solutions

Calculus

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The largest volume and surface area of the package.

Solution Summary: The author calculates the largest volume and surface area of Linda's package. The package is rectangular with rectangular ends and 40 inches in length.

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To find: The largest volume and surface area of the package.

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