document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=751408>Question 751408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A parabola shaped tunnel is 10m high at the centre and spans 18m wide at the base. A 7.5 m high vehicle needs to use the tunnel;, what is the maximum width of the vehicle that can fit through the tunnel? <BR>\n" );
document.write( "I need to draw a diagram to represent the tunnel on a coordinate number plane , and find the equation of the parabola. using algebra and coordinate geometry to determinate the maximum width of the truck<BR>\n" );
document.write( " I try to solve this problem many times but my results seems to be wrong, please help me I'm so stuck<BR>\n" );
document.write( " Ally </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #457180 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=457180\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The parabola is upside down and has the vertex at (0,10). Notice how this is the highest point.\r<BR>\n" );
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document.write( "The two roots are (-9,0) and (9,0). The horizontal distance between these two points is exactly 18 m. Basically how I found these two points is that I divided the distance of 18 m in half to get 9 m, then you walk 9 m in each direction to land on (-9,0) and (9,0)\r<BR>\n" );
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document.write( "So we have the three points: (0,10), (-9,0) and (9,0)\r<BR>\n" );
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document.write( "The first point is the vertex. The first point is also the y-intercept.\r<BR>\n" );
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document.write( "The next two points are the x-intercepts or roots.\r<BR>\n" );
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document.write( "Let's use the roots to find the equation\r<BR>\n" );
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document.write( "x = -9 or x = 9\r<BR>\n" );
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document.write( "x+9 = 0 or x-9 = 0\r<BR>\n" );
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document.write( "k(x+9)(x-9) = 0 ... for some fixed number k (we don't know it yet, but we'll find it soon)\r<BR>\n" );
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document.write( "k(x^2 - 81) = 0\r<BR>\n" );
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document.write( "The equation so far is y = k(x^2 - 81)\r<BR>\n" );
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document.write( "Since the y-intercept is (0,10), we can plug in x = 0 and y = 10 to find k\r<BR>\n" );
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document.write( "y = k(x^2 - 81)\r<BR>\n" );
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document.write( "10 = k(0^2 - 81)\r<BR>\n" );
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document.write( "10 = k(0 - 81)\r<BR>\n" );
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document.write( "10 = k(-81)\r<BR>\n" );
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document.write( "10 = -81k\r<BR>\n" );
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document.write( "-81k = 10 \r<BR>\n" );
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document.write( "k = 10/(-81)\r<BR>\n" );
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document.write( "k = -10/81\r<BR>\n" );
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document.write( "So the equation that models the tunnel is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-expr%2810%2F81%29%28x%5E2-81%29 ALIGN=MIDDLE  ALT=\"y+=+-expr%2810%2F81%29%28x%5E2-81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which distributes to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-expr%2810%2F81%29%28x%5E2%29-expr%2810%2F81%29%28-81%29 ALIGN=MIDDLE  ALT=\"y+=+-expr%2810%2F81%29%28x%5E2%29-expr%2810%2F81%29%28-81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-expr%2810%2F81%29%28x%5E2%29%2B10 ALIGN=MIDDLE  ALT=\"y+=+-expr%2810%2F81%29%28x%5E2%29%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation in standard form is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-expr%2810%2F81%29x%5E2%2B10 ALIGN=MIDDLE  ALT=\"y+=+-expr%2810%2F81%29x%5E2%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we're told that the truck is 7.5 m, so this means that the height y is 7.5, so we can plug in y = 7.5 and solve for x to find the clearance\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.5+=+-expr%2810%2F81%29x%5E2%2B10 ALIGN=MIDDLE  ALT=\"7.5+=+-expr%2810%2F81%29x%5E2%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4.5 ALIGN=MIDDLE  ALT=\"x+=+4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4.5 ALIGN=MIDDLE  ALT=\"x+=+-4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4.5 ALIGN=MIDDLE  ALT=\"x+=+4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4.5 ALIGN=MIDDLE  ALT=\"x+=+-4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the height is exactly 7.5 m, which means that the truck's width must be less than 2*4.5 = <font color=\"red\">9 feet</font> across so it can fit in the tunnel.  </SPAN>\n" );
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