document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=843192>Question 843192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> making a rain gutter<BR>\n" );
document.write( "a rain gutter is formed by bending up the sides of a 30-inch-wide rectangular metal sheet<BR>\n" );
document.write( "(a) find a function that models the cross- sectional area of the gutter in terms of x<BR>\n" );
document.write( "(b) find the value of x that maximizes the cross-sectional area of the gutter<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 #507992 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=507992\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the perimeter of a rectangle is equal to 2L + 2W.<BR>\n" );
document.write( "the area of a rectangle is equal to LW.\r<BR>\n" );
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document.write( "the rectangle is the cross section of the gutter.\r<BR>\n" );
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document.write( "you know that the perimeter is equal to 30, so you set up an equation as shown below:\r<BR>\n" );
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document.write( "2L + 2W = 30\r<BR>\n" );
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document.write( "solve for L in terms of W.\r<BR>\n" );
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document.write( "you will get L = (30 - 2W) / 2 which simplifies to L = 15 - W\r<BR>\n" );
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document.write( "substitute for L in the equation of Area = LW to get:\r<BR>\n" );
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document.write( "Area = (15-W) * W\r<BR>\n" );
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document.write( "After you simplify, your equation becomes:\r<BR>\n" );
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document.write( "Area = 15W - W^2\r<BR>\n" );
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document.write( "rearrange the terms to get:\r<BR>\n" );
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document.write( "Area = -W^2 + 15W\r<BR>\n" );
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document.write( "this equation is now in the standard form of a quadratic equation which is equal to ax^2 + bx + c\r<BR>\n" );
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document.write( "a = -1<BR>\n" );
document.write( "b = 15<BR>\n" );
document.write( "c = 0\r<BR>\n" );
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document.write( "since the coefficient of the w^2 term in this quadratic is negative, the graph will point up and open down.\r<BR>\n" );
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document.write( "this means the max/min point of the quadratic equation is a max point.\r<BR>\n" );
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document.write( "the x-coordinate of the maximum point in the graph of this equation is equal to -b/2a which is equal to -15/-2 which is equal to 7.5.\r<BR>\n" );
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document.write( "the y-coordinate of the maximum point in the graph of this equation is equal f(7.5) which is equal to -(7.5)^2 + (15 * 7.5) which is equal to 56.25.\r<BR>\n" );
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document.write( "all you do is replace w in the original equation with 7.5 and solve for the area.\r<BR>\n" );
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document.write( "-w^2 + 15w becomes -(7.5)^2 + 15*7.5\r<BR>\n" );
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document.write( "the maximum area is at 56.25.\r<BR>\n" );
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document.write( "it occurs when W is equal to 7.5\r<BR>\n" );
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document.write( "When W is equal to 7.5, the perimeter of the gutter becomes 2L + 2(7.5) = 30  which simplifies to 2L + 15 = 30\r<BR>\n" );
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document.write( "solve for L to get L = 7.5.\r<BR>\n" );
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document.write( "the maximum cross sectional area is when both L and W are equal to 7.5.\r<BR>\n" );
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document.write( "the graph of the equation of the area of the cross section of the gutter is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C20%2C-10%2C60%2Cx%2A%2815-x%29%2C56.25%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C20%2C-10%2C60%2Cx%2A%2815-x%29%2C56.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the formula that was used to generate this graph was:\r<BR>\n" );
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document.write( "y = (15 - x) * x\r<BR>\n" );
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document.write( "length is represented by (15 - x) and width is represented by x.\r<BR>\n" );
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document.write( "the horizontal line at y = 56.25 is there to show you what the top of the curve maxes out at 56.25.\r<BR>\n" );
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