document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177813>Question 1177813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A parabolic arch spans a stream 200 feet wide. How high above the stream must the arch be to give a minimum clearance of 40 feet over a channel in the center that is 120 feet wide?<BR>\n" );
document.write( "I have tried setting up various equations, but none have worked so far. Thank you for any help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #806929 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=806929\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the origin of the coordinate system be the center of the stream at the surface; then the vertex of the parabola is at (0,k).  The equation is then of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where, because the parabola opens downward, we know a is some negative constant.<br><BR>\n" );
document.write( "We can determine a and k using the information in the problem:<br><BR>\n" );
document.write( "(1) The arch is 200 feet wide.  This means 100 feet either side of the center of the stream the height of the arch is zero:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+a%28100%5E2%29%2Bk ALIGN=MIDDLE  ALT=\"0+=+a%28100%5E2%29%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > [1]<br><BR>\n" );
document.write( "(2) We need a minimum vertical clearance of 40 feet everywhere in a channel 120 feet wide in the center of the stream.  That means at 60 feet either side of the center of the stream the height must be 40:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40+=+a%2860%5E2%29%2Bk ALIGN=MIDDLE  ALT=\"40+=+a%2860%5E2%29%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > [2]<br><BR>\n" );
document.write( "Solve the pair of equations [1] and [2].<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+10000a%2Bk ALIGN=MIDDLE  ALT=\"0+=+10000a%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40+=+3600a%2Bk ALIGN=MIDDLE  ALT=\"40+=+3600a%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Subtract the second equation from the first to eliminate k.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-40+=+6400a ALIGN=MIDDLE  ALT=\"-40+=+6400a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-40%2F6400+=+-1%2F160 ALIGN=MIDDLE  ALT=\"a+=+-40%2F6400+=+-1%2F160\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Substitute that value in [1] to find k, which is the height of the arch we are to find.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28-1%2F160%29%2810000%29%2Bk+=+-62.5%2Bk ALIGN=MIDDLE  ALT=\"0+=+%28-1%2F160%29%2810000%29%2Bk+=+-62.5%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+62.5 ALIGN=MIDDLE  ALT=\"k+=+62.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: The height of the arch above the stream must be 62.5 feet.<br><BR>\n" );
document.write( "CHECK:  The equation of the parabola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=f%28x%29+=+%28-1%2F160%29x%5E2%2B62.5 ALIGN=MIDDLE  ALT=\"y=f%28x%29+=+%28-1%2F160%29x%5E2%2B62.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%2860%29+=+%28-1%2F160%293600%2B62.5+=+-22.5%2B62.5+=+40 ALIGN=MIDDLE  ALT=\"f%2860%29+=+%28-1%2F160%293600%2B62.5+=+-22.5%2B62.5+=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28100%29+=+%28-1%2F160%2910000%2B62.5+=+-62.5%2B62.5+=+0 ALIGN=MIDDLE  ALT=\"f%28100%29+=+%28-1%2F160%2910000%2B62.5+=+-62.5%2B62.5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, showing a width of 200 feet and a height of 40 feet 60 feet either side of the center of the arch.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C200%2C-120%2C120%2C-20%2C80%2C%28-1%2F160%29x%5E2%2B62.5%2C40%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C200%2C-120%2C120%2C-20%2C80%2C%28-1%2F160%29x%5E2%2B62.5%2C40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );