document.write( "Question 616393: zoe the goat is tied to a rope to one corner of a 15 meter by 25 meter rectangular barn. over what area of the field can zoe graze is the rope is
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\n" ); document.write( "4. zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used? \n" ); document.write( "
Algebra.Com's Answer #387683 by Edwin McCravy(20056)\"\" \"About 
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zoe the goat is tied to a rope to one corner of a 15 meter by 25 meter rectangular barn. over what area of the field can zoe graze is the rope is
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\n" ); document.write( "1. 10 meters long?
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document.write( "That's 3/4ths of a circle with a radius of 10.\r\n" );
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document.write( "The area of a full circle with radius 10 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A10%5E2\" = \"100pi\" \r\n" );
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document.write( "Multiply that by 3/4ths gives \"3%2F4\"·\"100pi\" = \"75pi\" = 235.6 mē \r\n" );
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\n" ); document.write( "2. 20 meters long?
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document.write( "That's 3/4ths of a circle with a radius of 20. plus 1/4th of a circle\r\n" );
document.write( "with a radius of 5\r\n" );
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document.write( "The area of a full circle with radius 20 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A20%5E2\" = \"400pi\" \r\n" );
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document.write( "Multiply that by 3/4ths gives \"3%2F4\"·\"400pi\" = \"300pi\" = 942.48 mē \r\n" );
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document.write( "The area of a full circle with radius 5 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A5%5E2\" = \"25pi\" \r\n" );
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document.write( "Multiply that by 1/4th gives \"1%2F4\"·\"25pi\" = \"6.25pi\" = 19.63 mē\r\n" );
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document.write( "Total area = 942.48+19.63 = 962.11 mē.\r\n" );
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\n" ); document.write( "3. 30 meters long?
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document.write( "That's 3/4ths of a circle with a radius of 30. plus 1/4th of a circle\r\n" );
document.write( "with a radius of 15 at the top, plus 1/4 of a circle with radius 5 on\r\n" );
document.write( "the right.\r\n" );
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document.write( "The area of a full circle with radius 30 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A30%5E2\" = \"900pi\" \r\n" );
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document.write( "Multiply that by 3/4ths gives \"3%2F4\"·\"900pi\" = \"900pi\" = 2827.43 mē \r\n" );
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document.write( "The area of a full circle with radius 10 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A10%5E2\" = \"100pi\" \r\n" );
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document.write( "Multiply that by 1/4th gives \"1%2F4\"·\"100pi\" = \"25pi\" = 78.54 mē\r\n" );
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document.write( "The area of a full circle with radius 5 is\r\n" );
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document.write( "A = \"pi%2Ar%5E2\" = \"pi%2A5%5E2\" = \"25pi\" \r\n" );
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document.write( "Multiply that by 1/4th gives \"1%2F4\"·\"25pi\" = \"6.25pi\" = 19.63 mē\r\n" );
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document.write( "Total area = 2827.43 mē + 78.54 mē + 19.63 mē = 2925.60 mē.\r\n" );
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\n" ); document.write( "zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used?
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document.write( "To have exactly 400 mē of grazing area, it would be like the first case,\r\n" );
document.write( "where we just have three-quarters of just one circle.  Let the radius\r\n" );
document.write( "be R.  \r\n" );
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document.write( "The area of a full circle with radius R is\r\n" );
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document.write( "A = \"pi%2AR%5E2\"  \r\n" );
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document.write( "Multiply that by 3/4th gives \"3%2F4\"·\"pi%2AR%5E2\" \r\n" );
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document.write( "Setting that = 400\r\n" );
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document.write( "\"3%2F4\"·\"pi%2AR%5E2\" = 400\r\n" );
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document.write( "Multiply through by 4\r\n" );
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document.write( "\"3pi%2AR%5E2\" = 1600\r\n" );
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document.write( "       Rē = \"1600%2F%283pi%29\"\r\n" );
document.write( "       Rē = 169.7652726\r\n" );
document.write( "        R = 13.03 m\r\n" );
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document.write( "So the rope would have to be 13.03 m long or longer.\r\n" );
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document.write( "Edwin
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