Explanation Given information: By U.S. postal rule, the length plus the girth of a package cannot exceed 84 inches. Formula used: Volume of the package is V = l × w × h , where l , w , h are length, width and height of the package. Second derivative test: Suppose that f ( x , y ) is a function and ( a , b ) is a point at which ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 , and let D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 . It has four different cases to determine a relative maximum and minimum. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) . If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Calculation: The sum of the length and the girth of a package cannot exceed 84 inches. 84 = l + ( 2 x + 2 y ) By solving this equation for l , l = 84 − 2 x − 2 y . Therefore, the volume of the package is V = ( 84 − 2 x − 2 y ) × x × y . ⇒ V = 84 x y − 2 x 2 y − 2 x y 2 . If V ( x , y ) has maximum at ( x , y ) = ( a , b ) , then ∂ V ∂ x ( a , b ) = 0 and ∂ V ∂ y ( a , b ) = 0 . So find partial derivatives ∂ V ∂ x and ∂ V ∂ y of the function V = 84 x y − 2 x 2 y − 2 x y 2 . Differentiate V = 84 x y − 2 x 2 y − 2 x y 2 with respect to x by treating y as constant. ∂ V ∂ x = ∂ ∂ x ( 84 x y − 2 x 2 y − 2 x y 2 ) ⇒ ∂ V ∂ x = 84 y − 4 x y − 2 y 2 . Differentiate V = 84 x y − 2 x 2 y − 2 x y 2 with respect to y by treating x as constant. ∂ V ∂ y = ∂ ∂ y ( 84 x y − 2 x 2 y − 2 x y 2 ) ⇒ ∂ V ∂ y = 84 x − 2 x 2 − 4 x y . By the first derivative test, set the first partial derivatives equal to zero and solve the system of equations. ∂ V ∂ x = 0 84 y − 4 x y − 2 y 2 = 0 ⇒ 2 y ( 42 − 2 x − y ) = 0 By using the zero product property, y = 0 or 42 − 2 x − y = 0 . As y = 0 , the height of the package is zero, which is not possible. So, 42 − 2 x − y = 0 . Now, solve the equation 42 − 2 x − y = 0 for y . ⇒ y = 42 − 2 x . Similarly, ∂ V ∂ y = 0 84 x − 2 x 2 − 4 x y = 0 ⇒ 2 x ( 42 − x − 2 y ) = 0 By using the zero product property, x = 0 or 42 − x − 2 y = 0 . As x = 0 , the width of the package is zero, which is not possible

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 7.3, Problem 49E

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To calculate: The dimensions of the rectangular package of greatest volume that can be mailed, where U.S. postal rule is the length plus the girth of a package cannot exceed 84 inches. That is 84=(length)+(girth)=l+(2x+2y).

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Chapter 7.3, Problem 48E

Chapter 7.3, Problem 50E

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Chapter 7 Solutions

Calculus & Its Applications (14th Edition)

Ch. 7.1 - Let f(x,y,z)=x2+y(xz)4. Compute f(3,5,2).Ch. 7.1 - Prob. 2CYU Ch. 7.1 - Let f(x,y)=x23xyy2. Compute f(5,0),f(5,2) and...Ch. 7.1 - Prob. 2E Ch. 7.1 - Let g(x,y,z)=x/(yz). Compute g(1,1),g(0,1) and...Ch. 7.1 - Let f(x,y,z)=x2ey2+z2. Compute f(1,1,1) and...Ch. 7.1 - Let f(x,y)=xy. Show that f(2+h,3)f(2,3)=3h.Ch. 7.1 - Let f(x,y)=xy. Show that f(2,3+k)f(2,3)=2k.Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...

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The dimensions of the rectangular package of greatest volume that can be mailed, where U.S. postal rule is the length plus the girth of a package cannot exceed 84 inches. That is 84 = ( length ) + ( girth ) = l + ( 2 x + 2 y ) .

Solution Summary: The author calculates the dimensions of the rectangular package of greatest volume that can be mailed, where the length plus the girth of a package cannot exceed 84 inches.

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To calculate: The dimensions of the rectangular package of greatest volume that can be mailed, where U.S. postal rule is the length plus the girth of a package cannot exceed 84 inches. That is 84=(length)+(girth)=l+(2x+2y).

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Chapter 7 Solutions