document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1154243>Question 1154243</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The table lists a readings for three years. <BR>\n" );
document.write( "​(a) If the quadratic relationship between the carbon dioxide concentration C and the year t is expressed as Upper C equals at squared plus bt plus c​, where t equals 0 corresponds to​ 1962, use a system of linear equations to determine the constants​ a, b, and​ c, and give the equation.<BR>\n" );
document.write( "​(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its 1962 level.<BR>\n" );
document.write( "Year<BR>\n" );
document.write( "1962<BR>\n" );
document.write( "1992<BR>\n" );
document.write( "2022\r<BR>\n" );
document.write( "\n" );
document.write( "CO2<BR>\n" );
document.write( "318<BR>\n" );
document.write( "341<BR>\n" );
document.write( "371<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #776656 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=776656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "The table lists a readings for three years.\r<BR>\n" );
document.write( "\n" );
document.write( "(a) If the quadratic relationship between the carbon dioxide concentration <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+ ALIGN=MIDDLE  ALT=\"t+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=at%5E2%2Bbt%2Bc ALIGN=MIDDLE  ALT=\"C=at%5E2%2Bbt%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+ ALIGN=MIDDLE  ALT=\"t+\"   DISABLED_style= \"max-width:90%; height:auto;\" >equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" >, use a system of linear equations to determine the constants<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and give the equation.\r<BR>\n" );
document.write( "\n" );
document.write( "table:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Year ALIGN=MIDDLE  ALT=\"Year\"   DISABLED_style= \"max-width:90%; height:auto;\" >|<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CO2 ALIGN=MIDDLE  ALT=\"CO2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=level ALIGN=MIDDLE  ALT=\"level\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" >|<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=318 ALIGN=MIDDLE  ALT=\"318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1992 ALIGN=MIDDLE  ALT=\"1992\"   DISABLED_style= \"max-width:90%; height:auto;\" >|<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=341 ALIGN=MIDDLE  ALT=\"341\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2022 ALIGN=MIDDLE  ALT=\"2022\"   DISABLED_style= \"max-width:90%; height:auto;\" >|<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=371 ALIGN=MIDDLE  ALT=\"371\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=at%5E2%2Bbt%2Bc ALIGN=MIDDLE  ALT=\"C=at%5E2%2Bbt%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" >  is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=318 ALIGN=MIDDLE  ALT=\"318\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=318=a%2A0%5E2%2Bb%2A0%2Bc ALIGN=MIDDLE  ALT=\"318=a%2A0%5E2%2Bb%2A0%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=318%29 ALIGN=MIDDLE  ALT=\"highlight%28c=318%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=at%5E2%2Bbt%2Bc ALIGN=MIDDLE  ALT=\"C=at%5E2%2Bbt%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >................year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1992 ALIGN=MIDDLE  ALT=\"1992\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=341 ALIGN=MIDDLE  ALT=\"C=341\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=30 ALIGN=MIDDLE  ALT=\"t=30\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=318 ALIGN=MIDDLE  ALT=\"c=318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=341=a%2A30%5E2%2Bb%2A30%2B318 ALIGN=MIDDLE  ALT=\"341=a%2A30%5E2%2Bb%2A30%2B318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900a%2B30b=341-318 ALIGN=MIDDLE  ALT=\"900a%2B30b=341-318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900a%2B30b=23 ALIGN=MIDDLE  ALT=\"900a%2B30b=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=23%2F30-900a%2F30 ALIGN=MIDDLE  ALT=\"b=23%2F30-900a%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=23%2F30-30a ALIGN=MIDDLE  ALT=\"b=23%2F30-30a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq.1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=at%5E2%2Bbt%2Bc ALIGN=MIDDLE  ALT=\"C=at%5E2%2Bbt%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >................year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2022 ALIGN=MIDDLE  ALT=\"2022\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=371 ALIGN=MIDDLE  ALT=\"C=371\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=60 ALIGN=MIDDLE  ALT=\"t=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=318 ALIGN=MIDDLE  ALT=\"c=318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=371=a%2A60%5E2%2Bb%2A60%2B318 ALIGN=MIDDLE  ALT=\"371=a%2A60%5E2%2Bb%2A60%2B318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600a%2B60b=371-318 ALIGN=MIDDLE  ALT=\"3600a%2B60b=371-318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600a%2B60b=53 ALIGN=MIDDLE  ALT=\"3600a%2B60b=53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60b=53-3600a ALIGN=MIDDLE  ALT=\"60b=53-3600a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=+53%2F60-60a ALIGN=MIDDLE  ALT=\"b=+53%2F60-60a\"   DISABLED_style= \"max-width:90%; height:auto;\" >........eq.2\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "from eq.1 and eq.2 we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2F30-30a=53%2F60-60a ALIGN=MIDDLE  ALT=\"23%2F30-30a=53%2F60-60a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30a%2B60a=53%2F60-23%2F30 ALIGN=MIDDLE  ALT=\"30a%2B60a=53%2F60-23%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90a=7%2F60 ALIGN=MIDDLE  ALT=\"90a=7%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=7%2F5400%29 ALIGN=MIDDLE  ALT=\"highlight%28a=7%2F5400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "go to\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=23%2F30-30a ALIGN=MIDDLE  ALT=\"b=23%2F30-30a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq.1, plug in<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a ALIGN=MIDDLE  ALT=\"+a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=23%2F30-30%287%2F5400%29 ALIGN=MIDDLE  ALT=\"b=23%2F30-30%287%2F5400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=131%2F180%29 ALIGN=MIDDLE  ALT=\"highlight%28b=131%2F180%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "your equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28C=%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318%29 ALIGN=MIDDLE  ALT=\"highlight%28C=%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
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document.write( "(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" > level.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" > level is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=318 ALIGN=MIDDLE  ALT=\"318\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+ ALIGN=MIDDLE  ALT=\"t+\"   DISABLED_style= \"max-width:90%; height:auto;\" >years will  double which is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2%2A318=636 ALIGN=MIDDLE  ALT=\"C=2%2A318=636\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=636=%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318 ALIGN=MIDDLE  ALT=\"636=%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........sole for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318-636=0 ALIGN=MIDDLE  ALT=\"%287%2F5400%29t%5E2%2B%28131%2F180%29t%2B318-636=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2F5400%29t%5E2%2B%28131%2F180%29t-319=0 ALIGN=MIDDLE  ALT=\"%287%2F5400%29t%5E2%2B%28131%2F180%29t-319=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "using quadratic formula, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-1965%2F7+-+%2815+sqrt%2870753%29%29%2F7 ALIGN=MIDDLE  ALT=\"t+=+-1965%2F7+-+%2815+sqrt%2870753%29%29%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> we need only positive root, we will disregard negative solution \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%2815+sqrt%2870753%29%29%2F7+-+1965%2F7 ALIGN=MIDDLE  ALT=\"t+=+%2815+sqrt%2870753%29%29%2F7+-+1965%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> exact solution\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+288.6 ALIGN=MIDDLE  ALT=\"t+=+288.6\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=288.6 ALIGN=MIDDLE  ALT=\"288.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > years from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" > is:  the year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2250 ALIGN=MIDDLE  ALT=\"2250\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the amount of carbon dioxide in the atmosphere will double from its <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1962 ALIGN=MIDDLE  ALT=\"1962\"   DISABLED_style= \"max-width:90%; height:auto;\" > level.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );