document.write( "Question 1153288: A goat is tied outside a triangular fenced garden at point A. The sides of the fence are AB = 8 m, BC = 9 m, and CA = 12 m. If the rope with which the goat is tied is 14 m long, find the area over which the goat can graze outside the fence.\r
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Algebra.Com's Answer #775537 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "The figure below is drawn to scale.  The red line is the rope, with \r\n" );
document.write( "the goat pulling it tight.  The circle has center A and radius 14 m.\r\n" );
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document.write( "The goat can graze anywhere in the circle except in the triangle and\r\n" );
document.write( "in the area marked with an X in the figure below:\r\n" );
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document.write( "The desired area consists of three sectors.\r\n" );
document.write( "1. The big sector whose center is A and has radius 14 m. Its\r\n" );
document.write( "central angle is 360°-A \r\n" );
document.write( "2. The smallest sector whose center is C and has radius 2 m. Its\r\n" );
document.write( "central angle is 180°-C.  \r\n" );
document.write( "3. The next to smallest sector whose center is B and has radius 6 m.  \r\n" );
document.write( "We will need the exterior angle at B to calculate its area. That will\r\n" );
document.write( "be the sum of the two remote interior angles, A+C\r\n" );
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document.write( "We find all three angles of the triangle using the law of cosines.\r\n" );
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document.write( "The formula for the area of a sector is \r\n" );
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document.write( "The area of the big sector: Its radius is r = 14 m. \r\n" );
document.write( "Its central angle is θ = 360°-A = 360°-48.58881136° = 311.4111886°.\r\n" );
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document.write( "The area of the smallest sector: Its radius is r = 2 m. \r\n" );
document.write( "Its central angle is θ = 180°-C = 180°-41.80907919° = 138.1909208°.\r\n" );
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document.write( "The area of the next to smallest sector: Its radius is r = 6 m. \r\n" );
document.write( "Its central angle is θ = A+C = 48.58881136° + 41.80907919° = 90.39789055°\r\n" );
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document.write( "Finally we add the three sectors together to get the entire grazing area:\r\n" );
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document.write( "532.6447557 + 4.823773129 + 28.39933489 = 565.8678637 m².\r\n" );
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document.write( "The correct answer is B.\r\n" );
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document.write( "Edwin

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