document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934652>Question 934652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An engineer constructs side-by-side parabolic arches to support a bridge over a road and a river. The arch over the road has a maximum height of 6 m and a width of 16 m. The river arch has a maximum height of 8 m, but its width is reduced by 4 m because it intersects the arch over the road. Without this intersection, the river arch would have a width of 24 m. A support footing is used at the intersection point of the arches. The engineer sketched the arches on a coordinate system. She placed the origin at the left most point of the road.<BR>\n" );
document.write( "(24, 8)<BR>\n" );
document.write( "(8, 6)<BR>\n" );
document.write( "a) Determine the system of equations that models the two arches. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #567926 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=567926\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the system of equations will be:\r<BR>\n" );
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document.write( "y1 = -6/64 * x^2 + 96/64 * x \r<BR>\n" );
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document.write( "y2 = -8/144 * x^2 + 384 / 144 * x - 3456 / 144\r<BR>\n" );
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document.write( "if you graph those equations, you will get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/122101.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the first equation is modeled by knowing that the zeroes of that eqution are at x = 0 and x = 16.\r<BR>\n" );
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document.write( "this means the factors of that equation are (x-0) * (x-16) = 0 which becomes x^2 - 16x = 0\r<BR>\n" );
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document.write( "the general equation becomes y = a * (x^2 - 16x)\r<BR>\n" );
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document.write( "you know that y = 6 when x = 8, so replace x in that equation to get a * (8^2 - 16(8)) = 6 which becomes a * (-64) = 6 which becomes -64*a = 6 which becomes a = -6/64.\r<BR>\n" );
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document.write( "your equation of y = a * (x^2 - 16x) becomes y = -6/64 * (x^2 - 16x) which becomes y = -6/64 * x^2 + 96/64 * x\r<BR>\n" );
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document.write( "that's the first equation that was graphed.\r<BR>\n" );
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document.write( "the second equation was solved for in a similar manner.\r<BR>\n" );
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document.write( "the roots were 12 and 36<BR>\n" );
document.write( "the factors were (x-12) * (x-36) = 0<BR>\n" );
document.write( "multiplying those factors out got x^2 - 48x + 432 = 0<BR>\n" );
document.write( "the general equation became y = a * (x^2 - 48x + 432)<BR>\n" );
document.write( "when x was 24, y was 8, so we got 8 = a * (24^2 - 48*24 + 432) which became 8 = 576 * a - 1152 * a + 432 * a which became 8 = -144 * a<BR>\n" );
document.write( "divide both sides of that equation by -144 and you get a = -8/144.<BR>\n" );
document.write( "your equation becomes y = -8/144 * x^2 + 384/144 * x - 3456/144<BR>\n" );
document.write( "that's the equation you see in the graph.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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