document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012199>Question 1012199</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A parkway 20 meters wide is spanned by a parabolic arc 30 meters long along the horizontal. If the parkway is centered, how high must the vertex of the arch be in order to give a minimum clearance of 5 meters over the parkway? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628165 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=628165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  A parkway 20 meters wide is spanned by a parabolic arc 30 meters long along the horizontal.<BR>\n" );
document.write( " If the parkway is centered, how high must the vertex of the arch be in order to give a minimum clearance of 5 meters over the parkway?<BR>\n" );
document.write( ":<BR>\n" );
document.write( "that means the 30m arch starts and ends 5m on either side of the 20m highway<BR>\n" );
document.write( "From the information given we can obtain the following x,y pairs<BR>\n" );
document.write( "x=0; y=0<BR>\n" );
document.write( "x=5; y=5<BR>\n" );
document.write( "x=25; y=5<BR>\n" );
document.write( "Using the form ax^ + bx + c = y; this parabola passes thru the origin, therefore c = 0<BR>\n" );
document.write( "Two equations<BR>\n" );
document.write( "x=5; y=5<BR>\n" );
document.write( "25a + 5b = 5<BR>\n" );
document.write( ":<BR>\n" );
document.write( "x=25; y=5<BR>\n" );
document.write( "625a + 25b = 5<BR>\n" );
document.write( "simplify divide by 5<BR>\n" );
document.write( "125a + 5b = 1<BR>\n" );
document.write( ":<BR>\n" );
document.write( "using elimination<BR>\n" );
document.write( "125a + 5b = 1<BR>\n" );
document.write( " 25a + 5b = 5<BR>\n" );
document.write( "---------------------subtraction eliminates b, find a<BR>\n" );
document.write( "100a + 0 = -4<BR>\n" );
document.write( "a = -4/100<BR>\n" );
document.write( "a = -.04<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Find b using eq 25a + b = 5<BR>\n" );
document.write( "-.04(25) + 5b = 5<BR>\n" );
document.write( "-1 + 5b = 5<BR>\n" );
document.write( "5b = 5 + 1<BR>\n" );
document.write( "b = 6/5<BR>\n" );
document.write( "b = 1.2<BR>\n" );
document.write( ":<BR>\n" );
document.write( "the equation: y = -.04x^2 + 1.05x<BR>\n" );
document.write( "Looks like this, green line is 5m height<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-5%2C+32%2C+-5%2C+18%2C+-.04x%5E2%2B1.2x%2C+5%2C+9%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-5%2C+32%2C+-5%2C+18%2C+-.04x%5E2%2B1.2x%2C+5%2C+9%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ":<BR>\n" );
document.write( "\"how high must the vertex of the arch be\" find the axis of symmetry x = -b/(2a)<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1.2%29%2F%282%2A-.04%29 ALIGN=MIDDLE  ALT=\"%28-1.2%29%2F%282%2A-.04%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 15<BR>\n" );
document.write( "Find y when x=15<BR>\n" );
document.write( "y = -.04(15^2) + 1.2(15)<BR>\n" );
document.write( "y = -9 + 18<BR>\n" );
document.write( "Vertex pair: 15, 9<BR>\n" );
document.write( "y = 9 meters above the center of the high-way, blue line<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );