document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1077703>Question 1077703</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm working on an algebra project and I have a couple of questions that I need help with. The first question is: Write an expression for the width of any rectangle in terms of the length. The second question is: Write a function for the area of any rectangle in terms of the length. Also on this last question I need help on how to graph it and to find the vertex. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #692235 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=692235\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x equal the length of the rectangle\r<BR>\n" );
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document.write( "let w equal the width of the rectangle.\r<BR>\n" );
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document.write( "let y = the area of the rectangle.\r<BR>\n" );
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document.write( "area of a rectangle = length * width.\r<BR>\n" );
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document.write( "this translates to:\r<BR>\n" );
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document.write( "y = x * w\r<BR>\n" );
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document.write( "let k = the length of the rectangle minus the width of the rectangle.\r<BR>\n" );
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document.write( "you get k = x - w\r<BR>\n" );
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document.write( "solve this equation for w to get w = x - k\r<BR>\n" );
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document.write( "your equation for the area of the rectangle was:\r<BR>\n" );
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document.write( "y = x * w\r<BR>\n" );
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document.write( "replace w with x - k and the equation becomes:\r<BR>\n" );
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document.write( "y = x * (x - k)\r<BR>\n" );
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document.write( "simplify this equation to get:\r<BR>\n" );
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document.write( "y = x^2 - kx\r<BR>\n" );
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document.write( "this is a quadratic equation.\r<BR>\n" );
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document.write( "if we set y = 0, then it becomes:\r<BR>\n" );
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document.write( "x^2 - kx = 0\r<BR>\n" );
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document.write( "now it is in standard form of ax^2 + bx + c = 0, where a = 1 and b = -k and c = 0\r<BR>\n" );
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document.write( "the x-coordinate of the vertex of this equation is at x = -b/2a.\r<BR>\n" );
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document.write( "this becomes x = -(-k)/2) which becomes x = k/2.\r<BR>\n" );
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document.write( "the y-coordinate of the vertex of this equation is at y = f(-b/2a) which becomes y = f(k/2).\r<BR>\n" );
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document.write( "you solve for f(-b/2a) = f(k/2) by replacing x in the equation of y = x^2 - kx with (k/2)\r<BR>\n" );
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document.write( "the equation becomes y = (k/2)^2 - k*(k/2)\r<BR>\n" );
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document.write( "simplify this to get y = k^2/4 - k^2/2\r<BR>\n" );
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document.write( "place all fractions under a common denominator to get y = k^2/4 - 2k^2/4.\r<BR>\n" );
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document.write( "combine like terms to get y = -k^2/4.\r<BR>\n" );
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document.write( "your vertex is therefore at (k/2, -k^2/4)\r<BR>\n" );
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document.write( "what does this mean in real terms?\r<BR>\n" );
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document.write( "let's see:\r<BR>\n" );
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document.write( "assume the length of your rectangle is 7 and the width of your rectangle is 3.\r<BR>\n" );
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document.write( "since we set the equation up as y = x * w, where y is the area, x is the length, and w is the width, then we get:\r<BR>\n" );
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document.write( "y = x * w becomes y = 7 * 3 which becomes y = 21 which is the area of the rectangle.\r<BR>\n" );
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document.write( "all well and good, except that we want the equation in terms of x.\r<BR>\n" );
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document.write( "we did that earlier and we got:\r<BR>\n" );
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document.write( "y = x^2 - kx, where y is the area, x is the length and k is equal to x - w which is the length minus the width.\r<BR>\n" );
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document.write( "when x = 7 and w = 3, we get k = 7 - 3 = 4\r<BR>\n" );
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document.write( "the equation of y = x^2 - kx becomes y = x^2 - 4x\r<BR>\n" );
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document.write( "if we set y = 0, the equation becomes:\r<BR>\n" );
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document.write( "x^2 - 4x = 0\r<BR>\n" );
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document.write( "since this is a quadratic equation in standard form, we get:\r<BR>\n" );
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document.write( "a = 1<BR>\n" );
document.write( "b = -4<BR>\n" );
document.write( "c = 0\r<BR>\n" );
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document.write( "the x-coordinate of the vertex is at x = -b/2a which becomes x = 4/2 which becomes x = 2\r<BR>\n" );
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document.write( "the y-coordinate of the vertex is at f(-b/2a) which is at y = 2^2 - 4*2 which is at y = 4 - 8 = -4.\r<BR>\n" );
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document.write( "the vertex is therefore at (2,-4)\r<BR>\n" );
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document.write( "you can graph the equation of y = x^2 - 4x and it will look like this:\r<BR>\n" );
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document.write( "in this graph, y is the area and x is the length.\r<BR>\n" );
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document.write( "the width is assumed to be equal to the length minus 4.\r<BR>\n" );
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document.write( "what this means is:\r<BR>\n" );
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document.write( "when the length is 4, the width is 0<BR>\n" );
document.write( "when the length is 5, the width is 1<BR>\n" );
document.write( "when the length is 6, the width is 2<BR>\n" );
document.write( "when the length is 7, the width is 3\r<BR>\n" );
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document.write( "your area is equal to y.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "y is equal to 0 when x is equal to 4<BR>\n" );
document.write( "y is equal to 5 when x is equal to 5<BR>\n" );
document.write( "y is equal to 12 when x is equal to 6<BR>\n" );
document.write( "y is equal to 21 when x is equal to 7\r<BR>\n" );
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document.write( "vertical lines are placed on the graph to show you the value of y when x is the value indicated by the vertical line.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "what this says is that, as long as you know the value of k, you can find the area of the rectangle in terms of the length.\r<BR>\n" );
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document.write( "since we know the value of k, we can test this out.\r<BR>\n" );
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document.write( "the following graph shows the vertex as well as the value of y when x = 7.\r<BR>\n" );
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document.write( "we know that the value of k is equal to 7 - 3 which is equal to 4.\r<BR>\n" );
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document.write( "so we have k = 4.\r<BR>\n" );
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document.write( "and we have y = x^2 - kx which becomes y = x^2 - 4x when k = 4.\r<BR>\n" );
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document.write( "we also determined that the x-coordinate of the vertex of the equation would be at k/2, and we also determined that the y-coordinate of the vertex of the equation would be at -k^2/4\r<BR>\n" );
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document.write( "the k = 4, we get:\r<BR>\n" );
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document.write( "x-coordinate of the vertex of the equation is at 4/2 = 2\r<BR>\n" );
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document.write( "y-coordinate of the vertex of the equation is at -4^2/4 = -16/4 = -4.\r<BR>\n" );
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document.write( "since we already showed that the vertex of the equation is at (2,-4), then these values based on k = 4 are correct.\r<BR>\n" );
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document.write( "to summarize.\r<BR>\n" );
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document.write( "y is the area.<BR>\n" );
document.write( "x is the length<BR>\n" );
document.write( "k is the value of the length minus the width.\r<BR>\n" );
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document.write( "the equation becomes y = x^2 - kx.\r<BR>\n" );
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document.write( "the x-coordinate of the vertex of the equation is x = k/2\r<BR>\n" );
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document.write( "the y-coordinate of the vertex of the equation is y = -k^2/4\r<BR>\n" );
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document.write( "this should work with any value of length and width.\r<BR>\n" );
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document.write( "let's assume, arbitrarily, that the length is 50 and the width is 75.\r<BR>\n" );
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document.write( "this means that k = 50 - 75 = -25.\r<BR>\n" );
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document.write( "y = x^2 - kx becomes y = x^2 + 25x\r<BR>\n" );
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document.write( "x-coordinate of the vertex becomes y = k/2 = -25/2 = -12.5\r<BR>\n" );
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document.write( "y-coordinate of the vertex becomes y = -k^2/4 = -156.25\r<BR>\n" );
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document.write( "when the length is 50, the width is 75 and the area is equal to 50 * 75 = 3750.\r<BR>\n" );
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document.write( "all of this can be seen on the following graph of the equation of y = x^2 + 25x.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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