document.write( "Question 1159790: A semi-ellipse and a parabola rests on the same base 60 meters wide and 20 meters high. Using the common base as x-axis, compute the difference of ordinates at points 25 meters from the center of the base. \n" ); document.write( "
Algebra.Com's Answer #782884 by Edwin McCravy(20060)\"\" \"About 
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document.write( "We want to know the differences in heights at the green line:\r\n" );
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document.write( "The ellipse's equation is of the form:\r\n" );
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document.write( "\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "where a=center to vertex = (0,0) to (30,0) = 30\r\n" );
document.write( "and b = center to covertex = (0,0) to (0,20) = 20\r\n" );
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document.write( "Equation of ellipse:\r\n" );
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document.write( "\"x%5E2%2F30%5E2%2By%5E2%2F20%5E2=1\"\r\n" );
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document.write( "\"x%5E2%2F900%2By%5E2%2F400=1\"\r\n" );
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document.write( "The parabola's equation is of the form:\r\n" );
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document.write( "\"y=a%28x-h%29%5E2%2Bk\"\r\n" );
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document.write( "where the vertex (h,k) = (0,20)\r\n" );
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document.write( "\"y=a%28x-0%29%5E2%2B20\"\r\n" );
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document.write( "\"y=ax%5E2%2B20\"\r\n" );
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document.write( "It goes through (30,0)\r\n" );
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document.write( "\"0=a%2830%29%5E2%2B20\"\r\n" );
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document.write( "\"0=900a%2B20\"\r\n" );
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document.write( "\"-20=900a\"\r\n" );
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document.write( "\"%28-20%29%2F%28900%29=a\"\r\n" );
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document.write( "\"-1%2F45=a\"\r\n" );
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document.write( "\"y=%28-1%2F45%29x%5E2%2B20\" <-- equation of parabola\r\n" );
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document.write( "We find the ordinate of the ellipse at x=25\r\n" );
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document.write( "\"x%5E2%2F900%2By%5E2%2F400=1\"\r\n" );
document.write( "\"4x%5E2%2B9y%5E2=3600\"\r\n" );
document.write( "Substitute x=25\r\n" );
document.write( "\"4%2825%29%5E2%2B9y%5E2=3600\"\r\n" );
document.write( "\"4%28625%29%2B9y%5E2=3600\"\r\n" );
document.write( "\"2500%2B9y%5E2=3600\"\r\n" );
document.write( "\"9y%5E2=1100\"\r\n" );
document.write( "\"y%5E2=1100%2F9\"\r\n" );
document.write( "\"y=sqrt%281100%2F9%29\"\r\n" );
document.write( "\"y=sqrt%281100%29%2Fsqrt%289%29\"\r\n" );
document.write( "\"y=sqrt%28100%2A11%29%2F3\"\r\n" );
document.write( "\"y=10sqrt%2811%29%2F3\"\r\n" );
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document.write( "We find the ordinate of the parabola at x=25\r\n" );
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document.write( "\"y=%28-1%2F45%29x%5E2%2B20\"\r\n" );
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document.write( "\"y=%28-1%2F45%29%2825%29%5E2%2B20\"\r\n" );
document.write( "\"y=%28-1%2F45%29%28625%29%2B20\"\r\n" );
document.write( "\"y=-625%2F45%2B20\"\r\n" );
document.write( "\"y=-125%2F9%2B20\"\r\n" );
document.write( "\"y=-125%2F9%2B180%2F9\"\r\n" );
document.write( "\"y=55%2F9\"\r\n" );
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document.write( "We subtract the two y-coordinates:\r\n" );
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document.write( "\"difference=10sqrt%2811%29%2F3-55%2F9\"\r\n" );
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document.write( "\"difference=30sqrt%2811%29%2F9-55%2F9\"\r\n" );
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document.write( "\"difference=%2830sqrt%2811%29-55%29%2F9\" <--exact difference\r\n" );
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document.write( "\"4.944304857\" <--approximate difference\r\n" );
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document.write( "Edwin
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