document.write( "Question 1208800: Let x and y be nonnegative real numbers. If x^2 + 5y^2 = 30 , then find the maximum value of x + y . \n" ); document.write( "

Algebra.Com's Answer #847263 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Let x and y be nonnegative real numbers. If x^2 + 5y^2 = 30, then find the maximum value of x + y.
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document.write( "Equation\r\n" );
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document.write( "    x^2 + 5y^2 = 30    (1)\r\n" );
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document.write( "represents an ellipse, centered at the origin of the coordinate system.\r\n" );
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document.write( "Equation \r\n" );
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document.write( "    x + y = c          (2)\r\n" );
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document.write( "represents straight line with the slope -1.  \r\n" );
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document.write( "They want you find a point on the ellipse with the maximum value x + y = c.\r\n" );
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document.write( "The value of \"c\" defines the position of the line in the plane: different values of \"c\"\r\n" );
document.write( "produce parallel lines, and changing of the value of \"c\" moves/translates the lines \r\n" );
document.write( "vertically up or down, leaving them parallel.\r\n" );
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document.write( "So, they actually want you find the tangent line to the given ellipse with maximum value of \"c\",\r\n" );
document.write( "which corresponds to the most high possible position of the tangent line.\r\n" );
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document.write( "Next, since the slope of the line is -1, from geometry intuition, it is clear that \r\n" );
document.write( "the tangency point on the ellipse lies in the first quadrant.\r\n" );
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document.write( "There is another parallel tangent line, but for this second line the tangency point \r\n" );
document.write( "is in the third quadrant.\r\n" );
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document.write( "        OK. This preface reveals the geometric essence of the problem.\r\n" );
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document.write( "                   Now I move on to the solution.\r\n" );
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document.write( "All lines x + y = c  have the slope -1.  \r\n" );
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document.write( "So, we are looking and searching for the points on the ellipse in QI and QIII, where \r\n" );
document.write( "the tangent line to the ellipse has the slope -1.\r\n" );
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document.write( "For it, I differentiate equation (1)\r\n" );
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document.write( "    2x*dx + 10y*dy = 0.\r\n" );
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document.write( "which is the same as\r\n" );
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document.write( "    2x*dx = - 10y*dy.\r\n" );
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document.write( "From this equation in differentials, I find the derivative  \r\n" );
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document.write( "     =  = \r\n" );
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document.write( "This derivative represents the slope of the tangent line to the ellipse in point (x,y).\r\n" );
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document.write( "I want the slope of the tangent line be -1, the same as for the family of lines (2).\r\n" );
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document.write( "So, I write this equation\r\n" );
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document.write( "     = -1.    (3)\r\n" );
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document.write( "Square both sides.  You will get\r\n" );
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document.write( "     = 1,  or  x^2 = 25y^2.\r\n" );
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document.write( "I transform the last equation this way, using equation (1)\r\n" );
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document.write( "    x^2 =  5*(5y^2) = 5*(30-x^2) = 150 - 5x^2,\r\n" );
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document.write( "    x^2 + 5x^2 = 150,\r\n" );
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document.write( "    6x^2 = 150,\r\n" );
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document.write( "    x^2 = 150/6 = 25,\r\n" );
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document.write( "    x = +/-  = +/- 5.\r\n" );
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document.write( "Thus for x= 5, the value of y in the first quadrant is (from equation (1))\r\n" );
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document.write( "     5^2 + 5y^2 = 30,  --->  25 + 5y^2 = 30, --->  5y^2 = 30-25 = 5,  y^2 = 5/5 = 1,  y =  = 1.\r\n" );
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document.write( "For x= -5, the value of y in the third quadrant is (from equation (1))\r\n" );
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document.write( "     5^2 + 5y^2 = 30,  --->  25 + 5y^2 = 30, --->  5y^2 = 30-25 = 5,  y^2 = 5/5 = 1,  y = - = -1.\r\n" );
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document.write( "Thus, two tangency points are  (5,1)  in the first quadrant  and  (-5,-1) in the third quadrant.\r\n" );
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document.write( "For the first  point  c = x + y = 5 + 1 = 6.\r\n" );
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document.write( "For the second point  c = x + y = -5 - 1 = -6.\r\n" );
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document.write( "We want the maximum \"c\", so we choose the first point with the value of c = 6.\r\n" );
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document.write( "ANSWER.  The maximum value of \"c\" is  6.\r\n" );
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