Explanation Given: The length of the wire is 12 feet. Radius of circle is x 2 π , Area of circle is π ( x 2 π ) 2 , Side of square is 12 − x 4 and Area of square is ( 12 − x 4 ) 2 . Calculation: The piece of length x is formed into a circle and let the range of x is 0 to 12. Assume that the end point be 0 and 12. The wire will be made as circle and square that is the length of the wire is equal to sum of circle and square. Let A ( x ) be the total area then A ( x ) = Area of circle + Area of square . A ( x ) = π ( x 2 π ) 2 + ( 12 − x 4 ) 2 = ( x 2 π ) 2 + ( 12 − x 4 ) 2 = x 2 4 π + ( 12 − x ) 2 16 That is, A ( x ) = x 2 4 π + ( 12 − x ) 2 16 . Find the critical number in the interval [ 0 , 12 ] to compute the absolute extrema. Differentiate the function A ( x ) = x 2 4 π + ( 12 − x ) 2 16 as follows. A ′ ( x ) = d d x ( x 2 4 π + ( 12 − x ) 2 16 ) = ( 2 ) x 2 − 1 4 π + ( 2 ) ( 12 − x ) 2 − 1 ( − 1 ) 16 = x 2 π − ( 12 − x ) 8 = 8 x − 2 π ( 12 − x ) 16 π That is, A ′ ( x ) = 4 x − π ( 12 − x ) 8 π . Set f ′ ( x ) = 0 and solve for x . 4 x − π ( 12 − x ) 8 π = 0 4 x − 12 π + x π = 0 x ( 4 + π ) − 12 π = 0 x ( 4 + π ) = 12 π On simplify further as follows, x = 12 π ( 4 + π ) = 37

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 6.1, Problem 59E

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To find: The length of wire used to form the circle which minimizethe sum of the areas enclosed by both figures.

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Chapter 6.1, Problem 58E

Chapter 6.1, Problem 60E

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Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 6.1 - YOUR TURN 1 Find the absolute extrema of the...Ch. 6.1 - Prob. 2YT Ch. 6.1 - Prob. 1WE Ch. 6.1 - Prob. 2WE Ch. 6.1 - Prob. 1E Ch. 6.1 - Find the locations of any absolute extrema for the...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

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