document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1053197>Question 1053197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the set of all rectangles whose perimeter is 50 cm. For any one rectangle, once its width W is measured, it is possible to do some computations that yield the area of the rectangle. Verify this assertion by producing an explicit expression for area A as a function of width W. Find the domain and range of your function.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #668420 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=668420\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p = 2 * l + 2 * w<BR>\n" );
document.write( "a = l * w\r<BR>\n" );
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document.write( "l = length<BR>\n" );
document.write( "w = width<BR>\n" );
document.write( "a = area<BR>\n" );
document.write( "p = perimeter\r<BR>\n" );
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document.write( "if the perimeter is always 50, then p = 2 * l + 2 * w becomes:\r<BR>\n" );
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document.write( "50 = 2 * l + 2 * w\r<BR>\n" );
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document.write( "if you divide both sides of this equation by 2, you get\r<BR>\n" );
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document.write( "25 = l + w\r<BR>\n" );
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document.write( "if you know the width, then you can solve for l to get:\r<BR>\n" );
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document.write( "l = 25 - w\r<BR>\n" );
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document.write( "since a = l * w, you can replace l with 25 - w to get:\r<BR>\n" );
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document.write( "a = (25 - w) * w\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "a = 25w - w^2\r<BR>\n" );
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document.write( "an example of how this works is as follows:\r<BR>\n" );
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document.write( "let's say that the perimeter is 50 and the width is 5.\r<BR>\n" );
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document.write( "you can solve for area by using the formula of a = 25w - w^2\r<BR>\n" );
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document.write( "since w = 5, this means that a = 25*5 - 25.\r<BR>\n" );
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document.write( "this results in a being equal to 100.\r<BR>\n" );
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document.write( "let's see if this makes sense.\r<BR>\n" );
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document.write( "p = 50 and w = 5\r<BR>\n" );
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document.write( "p = 2 * l + 2 * w\r<BR>\n" );
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document.write( "you get 50 = 2 * l + 10\r<BR>\n" );
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document.write( "solve for l to get l = 40/2 = 20\r<BR>\n" );
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document.write( "you get l = 20 and w = 5\r<BR>\n" );
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document.write( "p = 2*l + 2*w = 40 + 10 = 50 ***** perimeter checks out ok.\r<BR>\n" );
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document.write( "you get a = l*w = 20*5 = 100 ***** area check out ok.\r<BR>\n" );
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document.write( "looks like it works.\r<BR>\n" );
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document.write( "as long as the perimeter is always 50 and as long as you know the width, you can solve for the area using the formula of a = 25w - w^2.\r<BR>\n" );
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