Explanation Given Information: The volume of spherical container is equal to the volume of cylindrical container is equal to 26 inch 2 . Formula used: The Method of Lagrange multipliers, To find a minimum and maximum value for a function of the form f ( x , y ) for the given constrain g ( x , y ) = 0 follow following steps: Step1: First form the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ g ( x , y ) . Step2: Find the first partial derivative with respect to x , y , λ ( F x F y and F λ ). Step3: Now put F x F y and F λ to 0 and find the value of x and y from these equations. Step4: Determine the values of x and y yields a maxima or minima. Calculation: Let the volume of the cylindrical container is π r 2 h where r is radius and h is the height. Now given volume is 26 inch 3 , π r 2 h = 26 The surface area of the cylindrical container is S ( h , r ) = 2 π r ( r + h ) Now to find the minimum surface area constrain to π r 2 h − 26 = 0 From Lagrange function, S ( h , r ) = 2 π r h + 2 π r 2 − λ ( π r 2 h − 26 ) = 2 π r h + 2 π r 2 − λ π r 2 h + 26 λ Now the partially differentiate the above equation with respect to r , h and λ to obtain the values of S r , S h and S λ , S r = 2 π h + 4 π r − 2 λ π r h S h = 2 π r − λ π r 2 S λ = − π r 2 h + 26 Now put S r , S h and S λ equal to 0, S h = 0 2 π r − λ ( π r 2 ) = 0 2 π r = λ ( π r 2 ) λ = 2 r Now put S r = 0 and use λ = 2 r , 2 π h + 4 π r − 2 λ π r h = 0 2 π h + 4 π r − 2 ( 2 r ) π r h = 0 4 π r = 2 π h 2 r = h Now use this value in S λ = 0 − π r 2 h + 26 = 0 π r 2 h − 26 = 0 π r 2 ( 2 r ) = 26 π r 3 = 13 On further calculation, r 3 = 13 π r 3 = 4.138028 r = 1.6054 Now 2 r = h h = 2 r = 2 ( 1.6054 ) = 3

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ISBN: 9780136423287

Author: BITTINGER

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Chapter 6, Problem 37RE

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To calculate: The minimum surface area for given volume 26 inch2 for rectangular container having square top and bottom and cylindrical container also find which container will have smaller surface area.

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Chapter 6, Problem 36RE

Chapter 6, Problem 38RE

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The minimum surface area for given volume 26 inch 2 for rectangular container having square top and bottom and cylindrical container also find which container will have smaller surface area.

Solution Summary: The author calculates the minimum surface area for a given volume of 26 inch2 for rectangular container having square top and bottom.