document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=428079>Question 428079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with this word problem.\r<BR>\n" );
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document.write( "A decorative slate centerpiece for a kitchen island has the shape of a rectangle with a semicircular section attached at each end.The radii of the semicircles measure x/2 inches.The perimeter of the centerpiece is 38 inches.Write a completely factored polynomial whose values give the area of the centerpiece. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #297636 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=297636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the radius of each of the semi-circles is equal to x/2.\r<BR>\n" );
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document.write( "since the semi-circles are attached at each end of the rectangle, then the width of the rectangle is equal to the diameter of each semi-circle which is equal to 2 * the radius of each semi-circle which is equal to x.\r<BR>\n" );
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document.write( "if you take the 2 semi-circles and put them together then you have a complete circle.\r<BR>\n" );
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document.write( "the perimeter of the centerpiece is equal to 38 inches.\r<BR>\n" );
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document.write( "this encompasses the 2 lengths of the rectangle and the circumference of the 2 semi-circles put together.\r<BR>\n" );
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document.write( "the perimeter is therefore equivalent to 2 times the length of the rectangle plus the perimeter of the full circle.\r<BR>\n" );
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document.write( "we know that x is the width of the rectangle.\r<BR>\n" );
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document.write( "we set y equal to the length of the rectangle.\r<BR>\n" );
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document.write( "the perimeter of the centerpiece is therefore equal to 2*y + 2*pi*r.\r<BR>\n" );
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document.write( "r is the radius of the circle.\r<BR>\n" );
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document.write( "since r = x/2, then this formula becomes:\r<BR>\n" );
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document.write( "perimeter of centerpiece = 2*y + 2*pi*(x/2)\r<BR>\n" );
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document.write( "since the perimeter of centerpiece is equal to 38, then we get:\r<BR>\n" );
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document.write( "38 = 2*y + 2*pi*(x/2)\r<BR>\n" );
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document.write( "this reduces to 38 = 2*y + pi*x\r<BR>\n" );
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document.write( "if we solve for y, we get:\r<BR>\n" );
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document.write( "2y = 38 - pi*x which becomes:\r<BR>\n" );
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document.write( "y = (38 - pi*x) / 2\r<BR>\n" );
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document.write( "we now have a value for y in terms of x.\r<BR>\n" );
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document.write( "we have:\r<BR>\n" );
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document.write( "width of rectangle is x.<BR>\n" );
document.write( "length of rectangle is (38 - pi*x) / 2\r<BR>\n" );
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document.write( "the area of the centerpiece is going to be the area of the rectangle plus the area of the circle.\r<BR>\n" );
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document.write( "remember that, if you put 2 semi-circles together, you get 1 circle.\r<BR>\n" );
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document.write( "the area of the centerpiece is therefore equal to length * width + pi*r^2.\r<BR>\n" );
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document.write( "pi*r^2 is the area of the circle.\r<BR>\n" );
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document.write( "length is equal to (38-pi*x)/2\r<BR>\n" );
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document.write( "width is equal to x\r<BR>\n" );
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document.write( "r is equal to (x/2)\r<BR>\n" );
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document.write( "area of the centerpiece is therefore equal to (38-pi*x)/2 * x + pi*(x/2)^2\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "(38*x)/2 - (pi*x^2)/2 + (pi*x^2)/4\r<BR>\n" );
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document.write( "bringing everything up to a common denominator gets us:\r<BR>\n" );
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document.write( "(76*x)/4 - (2*pi*x^2)/4 + (pi*x^2)/4\r<BR>\n" );
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document.write( "now that we have a common denominator, we can add all these together to get:\r<BR>\n" );
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document.write( "((76*x) - (2*pi*x^2) + (pi*x^2)) / 4\r<BR>\n" );
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document.write( "we combine like terms to get:\r<BR>\n" );
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document.write( "area of the centerpiece is equal to ((76*x) - (pi*x^2))/4\r<BR>\n" );
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document.write( "that's our equation.\r<BR>\n" );
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document.write( "it looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2876%2Ax%29+-+%28pi%2Ax%5E2%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28%2876%2Ax%29+-+%28pi%2Ax%5E2%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "a picture of the centerpiece and it's equivalent structure is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/centerpiece_20110329_1.jpg\" alt = \"***** picture not found *****\" />\r<BR>\n" );
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document.write( "the perimeter is equal to the 2 lengths of the rectangle plus the cicumference of the circle.\r<BR>\n" );
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document.write( "the area is equal to the area of the rectangle plus the area of the circle.\r<BR>\n" );
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document.write( "in the picture, the lengths of the rectangle are the horizontal lines of the rectangle.\r<BR>\n" );
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document.write( "in the picture, the widths of the rectangle are the vertical lines of the rectangle.\r<BR>\n" );
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document.write( "the width of the rectangle is shown as x.\r<BR>\n" );
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document.write( "the length of the rectangle is not shown in the picture, but was previously shown as y which was then calculated to be (38-pi*x)/2.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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