document.write( "Question 1194325: The circumference and radius of a circle \"T\" are unknown. Two point A and B lie on circle whose arc length is 200 feet but the length of AB chord is 150 feet.
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Algebra.Com's Answer #826514 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The circumference and radius of a circle \"T\" are unknown.
\n" ); document.write( "Two point A and B lie on circle whose arc length is 200 feet
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\n" ); document.write( "(B) Can we find area of the circle \"T\" if we know the area of segment from AB chord to AB curve?
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document.write( "In a coordinate plane make a sketch: draw a circle T (with unknown radius r);\r\n" );
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document.write( "mark the appropriate points A and B  (A in Q1, B in the Q4 symmetrically to A about the x-axis).\r\n" );
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document.write( "Draw the chord AB.\r\n" );
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document.write( "Let the central angle for the arc AB be  radians.\r\n" );
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document.write( "So from the given part, you have these two equations\r\n" );
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document.write( "     = 200 feet,  or    = 100         (1)  (from the arc length)\r\n" );
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document.write( "and\r\n" );
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document.write( "     = 150,  or   = 75.      (2)  (from the chord length).\r\n" );
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document.write( "Dividing equation (2) by equation (1), you get\r\n" );
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document.write( "     = ,  or   = .    (3)\r\n" );
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document.write( "This equation is not algebraic equation; it is transcendent equation and can not be solved \r\n" );
document.write( "by the Algebra methods.  But it can be solved numerically with any prescribed precision.\r\n" );
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document.write( "There are many solvers for solving such equations in the Internet.\r\n" );
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document.write( "I used one such free of charge common use solver at this link\r\n" );
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document.write( "https://www.wolframalpha.com/widgets/view.jsp?id=a7d8ae4569120b5bec12e7b6e9648b86\r\n" );
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document.write( "It gave me the value of  = 1.2757  radians.\r\n" );
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document.write( "        To check it,  compare both sides of equation (3)\r\n" );
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document.write( "            sin(1.2757) = 0.9568;  0.75*1.2757 = 0.9568,  \r\n" );
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document.write( "        so these values are consistent.\r\n" );
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document.write( "Next, knowing , we can find \"r\" from equation (1)\r\n" );
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document.write( "    r =  =  = 78.3883  (rounded).\r\n" );
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document.write( "    +--------------------------------------------------------------+\r\n" );
document.write( "    |   Now we have everything to answer the problem's questions.  |\r\n" );
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document.write( "Question (A)\r\n" );
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document.write( "    The area of the segment is the area of the sector minus the area of the triangle\r\n" );
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document.write( "        area of the segment =  -  = \r\n" );
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document.write( "                            =  -  = \r\n" );
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document.write( "                            = 7838.8264 - 1709.7699 = 6129.0565 square units  (rounded)    ANSWER\r\n" );
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document.write( "Question (B)\r\n" );
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document.write( "    After solving the problem, I just know the radius of the circle; therefore, I can calculate\r\n" );
document.write( "    the area of the circle.  It is\r\n" );
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document.write( "        area of the circle =  =  = 19304.2084  (rounded).    ANSWER\r\n" );
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