Explanation Given: A storage room with volume 1000 cubic feet loses heat from all the four walls and the ceiling. The ratio of the heat loss per unit area through the walls to the heat loss per unit area through the ceiling is 3:5 and the floor is insulated. Formula used: The function has a relative minima at critical point ( a , b ) if f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 > 0 and f x x ( a , b ) > 0 . The function has a relative maxima at critical point ( a , b ) f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 > 0 and f x x ( a , b ) < 0 . A point ( a , b , f ( a , b ) ) is said to be a saddle point if f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 < 0 . The test fails when f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 = 0 Calculation: Let the room have a length of x feet, breadth of y feet and a height of z feet. Now surface area of the room would of the surface area of the walls, the roof and the floor. A = ( 2 x y + 2 y z ) + x y + x y Let the heat loss per unit area be k units. Use this to obtain the heat loss function asL: H ( x , y , z ) = 3 a ( 2 x y + 2 y z ) + 5 a x y = a ( 6 z ( x + y ) + 5 x y ) It has been provided that the volume of the room is 10 cubic feet. V = x y z 1000 = x y z z = 1000 x y Substitute this in the heat function and obtain: H ( x , y ) = a ( ( 6000 x y ) ( x + y ) + 5 x y ) Now equate both the partial derivatives of the heat function to 0 to obtain the critical points. H x ( x , y ) = 5 a ( y x 2 − 1200 x 2 ) 5 a ( y x 2 − 1200 x 2 ) = 0 y = 1200 x 2 And, H x ( x , y ) = 5 a ( x y 2 − 1200 y 2 ) 5 a ( x y 2 − 1200 y 2 ) = 0 x = 1200 y 2 As x and y can’t be negative, it can be concluded that x and y are equal

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13, Problem 7PS

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To calculate: The dimensions of a storage room where the total heat loss in minimum. And given is a storage room with volume 1000 cubic feet loses heat from all the four walls and the ceiling. The ratio of the heat loss per unit area through the walls to the heat loss per unit area through the ceiling is 3:5 and the floor is insulated.

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Chapter 13, Problem 6PS

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Calculus: Early Transcendental Functions (MindTap Course List)

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Solution Summary: The author explains that a storage room with volume 1000 cubic feet loses heat from all the four walls and the ceiling.

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