Explanation Given information: A rancher will use 600 m of fencing to build a corral in the shape of a semicircle on top of a rectangle. Formula used: Area of semicircle = π 2 r 2 Perimeter of semicircle = π 2 d + d where d is the diameter Area of rectangle = length×Breadth Perimeter of rectangle= 2 ( length + Breadth ) Calculation: Let x be the width of the rectangle and y be the height of the rectangle. Also, the diameter of the semicircle is equal to the width of the rectangle, so the radius of the semicircle is x 2 . The perimeter of the window is, P = π 2 x + 2 x + 2 y = 600 1200 = π x + 4 x + 4 y 4 y = 1200 − π x − 4 x y = 1 4 ( 1200 − π x − 4 x ) Area of the window is A = π 2 ( x 2 ) 2 + x y A = π 2 ( x 2 4 ) + x [ 1 4 ( 1200 − π x − 4 x ) ] = π 8 x 2 + ( 1 4 ) ( 1200 x − π x 2 − 4 x 2 ) We need to maximize this area, for this differentiate the above equation, we get, d A d x = π 8 ( 2 x ) + ( 1 4 ) ( 600 − π ( 2 x ) − 4 ( 2 x ) ) For maximum area we substitute d A d x = 0 for the above equation and solve for x

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 4.6, Problem 15E

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Chapter 4.6, Problem 14E

Chapter 4.6, Problem 16E

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Chapter 4 Solutions

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Find the dimensions that maximize the area of the corral.

Solution Summary: The author calculates the dimensions that maximize the area of the corral. A rancher will use 600m of fencing to build the semicircle on top of a rectangle.

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