document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209187>Question 1209187</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A homeowner has 80 feet of fence to enclose a rectangular garden. What dimensions for the garden give the maximum area? </SPAN>\n" );
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document.write( "Answer: <font color=red>20 feet by 20 feet</font>\r<BR>\n" );
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document.write( "Short explanation:<BR>\n" );
document.write( "Given some amount of fencing P, the dimensions of the max area rectangle (which turns out to be a square) is P/4 by P/4<BR>\n" );
document.write( "We have P = 80 feet of fencing lead to P/4 = 80/4 = 20 which is the dimensions of the square.\r<BR>\n" );
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document.write( "Longer explanation\r<BR>\n" );
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document.write( "x = length\r<BR>\n" );
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document.write( "perimeter of a rectangle = 2*width+2*length<BR>\n" );
document.write( "80 = 2*width+2*length<BR>\n" );
document.write( "40 = width + length  ........ divide both sides by 2<BR>\n" );
document.write( "width = 40-length<BR>\n" );
document.write( "width = 40-x\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "length = x<BR>\n" );
document.write( "width = 40-x\r<BR>\n" );
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document.write( "which leads to:<BR>\n" );
document.write( "area = length*width<BR>\n" );
document.write( "area = x*(40-x)<BR>\n" );
document.write( "area = 40x-x^2<BR>\n" );
document.write( "area = -x^2+40x\r<BR>\n" );
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document.write( "Compare the equation y = -x^2+40x with the template y = ax^2+bx+c<BR>\n" );
document.write( "a = -1, b = 40, c = 0<BR>\n" );
document.write( "The vertex (h,k) is the highest point of this parabola. <BR>\n" );
document.write( "This is because a = -1 is negative. The parabola opens downward.<BR>\n" );
document.write( "Therefore, finding the vertex will help us max out the area.\r<BR>\n" );
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document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -40/(2*(-1))<BR>\n" );
document.write( "h = 20<BR>\n" );
document.write( "This is the x coordinate of the vertex.<BR>\n" );
document.write( "You can use a graphing tool like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to verify.\r<BR>\n" );
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document.write( "The area maxes out when the length is x = 20 feet.<BR>\n" );
document.write( "The width is 40-x = 40-20 = 20 feet.\r<BR>\n" );
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document.write( "<font color=red>The dimensions are 20 feet by 20 feet.</font>\r<BR>\n" );
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document.write( "Extra info: The area is 20*20 = 400 square feet.\r<BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "Factor -x^2+40x = 0 to get -x(x-40) = 0<BR>\n" );
document.write( "From here we can quickly see that the roots are x = 0 and x = 40.<BR>\n" );
document.write( "It turns out that the x coordinate of the vertex is the midpoint of these roots. <BR>\n" );
document.write( "This is due to the parabola's mirror symmetry.<BR>\n" );
document.write( "Add the roots and divide in half: (0+40)/2 = 20<BR>\n" );
document.write( "The x coordinate of the vertex is x = 20.<BR>\n" );
document.write( "The length is x = 20 and the width is 40-x = 40-20 = 20.<BR>\n" );
document.write( "We have a <font color=red>20 by 20 square</font>.\r<BR>\n" );
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document.write( "Side note: you could use the quadratic formula to solve -x^2+40x=0, but it would be overkill in my opinion.<BR>\n" );
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