document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=18584>Question 18584</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If you have a string of length 50 cm, what are the dimensions of the rectangle of maximum area that you can enclose with your string?  Explain your reasoning.  What about a string of length k cm?\r<BR>\n" );
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document.write( "I've gotten up to the quadratic equation for the area, which is X^2 - 25X = Area.  Then you have to find the third component of the formula which I don't know how to do.  Then you have to state what x and y are, and I don't really know that, either.  This is finding a formula, and without information on x and y, I don't see how you can actually find the numbers.\r<BR>\n" );
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document.write( "The last part of question asks for a general formula, and the book says it's <BR>\n" );
document.write( "k/4 by k/4.  So, I have the answer, but no explanation, which is the most important part.  This doesn't make sense unless the rectangle is a square, because with a rectangle the two k's would be different.  Can you help me with this?\r<BR>\n" );
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document.write( "Thank you in advance.<BR>\n" );
document.write( "-Jessie </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #8899 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=8899\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ok, you can let x = the length of one side of the rectangle and y = the length of the other side of the rectangle.\r<BR>\n" );
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document.write( "The area is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+xy ALIGN=MIDDLE  ALT=\"A+=+xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the perimeter is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+2%28x%2By%29 ALIGN=MIDDLE  ALT=\"50+=+2%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve this in terms of y. Divide both sides by 2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25+=+x%2By ALIGN=MIDDLE  ALT=\"25+=+x%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract x from both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25-x+=+y ALIGN=MIDDLE  ALT=\"25-x+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now substitute this into the equation for the area.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+x%2825-x%29 ALIGN=MIDDLE  ALT=\"A+=+x%2825-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-x%5E2+%2B+25 ALIGN=MIDDLE  ALT=\"A+=+-x%5E2+%2B+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > What you have here is a quadratic equation whose graph is a parabola that opens downward. If you were to graph this, you would be graphing a function like: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-x%5E2+%2B+25 ALIGN=MIDDLE  ALT=\"y+=+-x%5E2+%2B+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you can see that y in this function plays the role of A, the area.<BR>\n" );
document.write( "You want to find the value of x that would make A (or y) a maximum.<BR>\n" );
document.write( "You could do this using differential calculus but there is a way to do it with algebra.<BR>\n" );
document.write( "Since the graph of the quadratic is a parabola opening downward, the vertex of the parabola would be at the maximum point on the curve, right? The x-coordinate, which represents the length of the side of your rectangle, is given by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F2a ALIGN=MIDDLE  ALT=\"x+=+-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is taken from the general form of quadratic equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In your quadratic equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2+%2B+25x ALIGN=MIDDLE  ALT=\"-x%5E2+%2B+25x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = -1, b = 25, and c = 0<BR>\n" );
document.write( "so we can find the x-coordinate of the vertex by:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-25%2F2%28-1%29+=+12.5 ALIGN=MIDDLE  ALT=\"-25%2F2%28-1%29+=+12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the length (x) of the rectangle must be 12.5 cm to get the maximum area.<BR>\n" );
document.write( "But what about the width (y)? <BR>\n" );
document.write( "Since the perimeter is 50 cm and this is twice the (length + width), the (length + width) is 25 cm, so the width is 25 cm - the length.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y+=+25 ALIGN=MIDDLE  ALT=\"x+%2B+y+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12.5+%2B+y+=+25 ALIGN=MIDDLE  ALT=\"12.5+%2B+y+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+25+-+12.5 ALIGN=MIDDLE  ALT=\"y+=+25+-+12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+12.5 ALIGN=MIDDLE  ALT=\"y+=+12.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, we end up with x (the length) = 12.5 cm and y (the width) = 12.5 cm.<BR>\n" );
document.write( "And this, of course, is a square whose sides are 12.5 cm\r<BR>\n" );
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document.write( "If the perimeter were k cm, then you would use the same technique and your quadratic equation would look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-x%5E2+%2B+wk%2F2 ALIGN=MIDDLE  ALT=\"A+=+-x%5E2+%2B+wk%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the x-coordinate of the vertex of the parabola would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-k%2F2%29%2F2%28-1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-k%2F2%29%2F2%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+k%2F4 ALIGN=MIDDLE  ALT=\"x+=+k%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The sides of the rectangle then would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+k%2F4 ALIGN=MIDDLE  ALT=\"x+=+k%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k%2F4 ALIGN=MIDDLE  ALT=\"y+=+k%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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