document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208681>Question 1208681</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two positive numbers whose sum is 8 and the product is minimum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847084 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847084\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "As tutor greenestamps points out, there is no minimum since the results get closer to 0. <BR>\n" );
document.write( "I'll assume your teacher meant to say \"maximum\" instead of \"minimum\".\r<BR>\n" );
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document.write( "Let x be one of the numbers.<BR>\n" );
document.write( "8-x must be the other number if we want them both to add to 8.<BR>\n" );
document.write( "x+(8-x) = 8.\r<BR>\n" );
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document.write( "Their product is x(8-x) = -x^2+8x\r<BR>\n" );
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document.write( "This is a parabola that opens downward.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28500%2C500%2C-3%2C10%2C-5%2C20%2C%0D%0Agraph%28500%2C500%2C-3%2C10%2C-5%2C20%2C-100%2C-x%5E2%2B8x%29%2C%0D%0Acircle%284%2C16%2C0.05%29%2Ccircle%284%2C16%2C0.07%29%2Ccircle%284%2C16%2C0.09%29%2Ccircle%284%2C16%2C0.11%29%2C%0D%0Alocate%284-0.5%2C16%2B1.1%2C%22%284%2C16%29%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can use graphing tools like <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> or <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> (among many other tools) to confirm this is the correct graph.<BR>\n" );
document.write( "If you are familiar with a TI83, then it's probably best to use that instead.\r<BR>\n" );
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document.write( "The key thing to note on the graph is the vertex (4,16)<BR>\n" );
document.write( "Here's how we find the coordinates of that location.\r<BR>\n" );
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document.write( "Consider the parabola y = ax^2+bx+c.<BR>\n" );
document.write( "Its vertex is located at (h,k) where h = -b/(2a)<BR>\n" );
document.write( "In the case of y = -x^2+8x we have a = -1, b = 8, c = 0 (side note: since a < 0, the parabola opens downward to produce a highest point).\r<BR>\n" );
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document.write( "Let's plug in those values:<BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -8/(2*(-1))<BR>\n" );
document.write( "h = 4<BR>\n" );
document.write( "This x coordinate of the vertex.<BR>\n" );
document.write( "Another way to find this value is to apply the midpoint formula to the roots x = 0 and x = 8. This works due to the parabola's symmetry.\r<BR>\n" );
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document.write( "That x value is then plugged into the equation to find its paired y coordinate.<BR>\n" );
document.write( "y = -x^2+8x<BR>\n" );
document.write( "y = -4^2+8*4<BR>\n" );
document.write( "y = -16+32<BR>\n" );
document.write( "y = 16<BR>\n" );
document.write( "Or you could say<BR>\n" );
document.write( "y = x*(8-x)<BR>\n" );
document.write( "y = 4*(8-4)<BR>\n" );
document.write( "y = 4*4<BR>\n" );
document.write( "y = 16<BR>\n" );
document.write( "This would tell us that the largest possible product of x and 8-x is 16, and it happens when both values are 4.\r<BR>\n" );
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document.write( "------------------------------<BR>\n" );
document.write( "A real world application: <BR>\n" );
document.write( "A famer has 16 meters of fencing and wishes to form the largest rectangle in terms of area. <BR>\n" );
document.write( "The farmer should make a 4 by 4 square to max out the area. <BR>\n" );
document.write( "perimeter = 16 meters ---> side length = 16/4 = 4 meters ----> area = 4^2 = 16 square meters<BR>\n" );
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