document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153986>Question 1153986</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.\r<BR>\n" );
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document.write( "Year.      <BR>\n" );
document.write( "1962<BR>\n" );
document.write( "1982<BR>\n" );
document.write( "2002\r<BR>\n" );
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document.write( "CO2<BR>\n" );
document.write( "319<BR>\n" );
document.write( "340<BR>\n" );
document.write( "368<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 #776331 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=776331\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The table lists a readings for three years.\r<BR>\n" );
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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​ 1962, 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" );
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document.write( "Year.<BR>\n" );
document.write( "1962<BR>\n" );
document.write( "1982<BR>\n" );
document.write( "2002<BR>\n" );
document.write( "CO2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=319 ALIGN=MIDDLE  ALT=\"319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=340 ALIGN=MIDDLE  ALT=\"340\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=368 ALIGN=MIDDLE  ALT=\"368\"   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;\" >\r<BR>\n" );
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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​ 1962, 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=319 ALIGN=MIDDLE  ALT=\"319\"   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=319=a%2A0%5E2%2Bb%2A0%2Bc ALIGN=MIDDLE  ALT=\"319=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=319%29 ALIGN=MIDDLE  ALT=\"highlight%28c=319%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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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 1982, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=340 ALIGN=MIDDLE  ALT=\"C=340\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=20 ALIGN=MIDDLE  ALT=\"t=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=319 ALIGN=MIDDLE  ALT=\"c=319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=340=a%2A20%5E2%2Bb%2A20%2B319 ALIGN=MIDDLE  ALT=\"340=a%2A20%5E2%2Bb%2A20%2B319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400a%2B20b=340-319 ALIGN=MIDDLE  ALT=\"400a%2B20b=340-319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400a%2B20b=21 ALIGN=MIDDLE  ALT=\"400a%2B20b=21\"   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=21%2F20-20a ALIGN=MIDDLE  ALT=\"b=21%2F20-20a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq.1\r<BR>\n" );
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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 2002, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=368 ALIGN=MIDDLE  ALT=\"C=368\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=40 ALIGN=MIDDLE  ALT=\"t=40\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=319 ALIGN=MIDDLE  ALT=\"c=319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=368=a%2A40%5E2%2Bb%2A40%2B319 ALIGN=MIDDLE  ALT=\"368=a%2A40%5E2%2Bb%2A40%2B319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1600a%2B40b=368-319 ALIGN=MIDDLE  ALT=\"1600a%2B40b=368-319\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1600a%2B40b=49 ALIGN=MIDDLE  ALT=\"1600a%2B40b=49\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40b=49-1600a ALIGN=MIDDLE  ALT=\"40b=49-1600a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=+49%2F40-40a ALIGN=MIDDLE  ALT=\"b=+49%2F40-40a\"   DISABLED_style= \"max-width:90%; height:auto;\" >........eq.2\r<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=21%2F20-20a=49%2F40-40a ALIGN=MIDDLE  ALT=\"21%2F20-20a=49%2F40-40a\"   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=40a-20a=49%2F40-21%2F20 ALIGN=MIDDLE  ALT=\"40a-20a=49%2F40-21%2F20\"   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=20a=7%2F40 ALIGN=MIDDLE  ALT=\"20a=7%2F40\"   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%2F800%29 ALIGN=MIDDLE  ALT=\"highlight%28a=7%2F800%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\n" );
document.write( "go to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=21%2F20-20a ALIGN=MIDDLE  ALT=\"b=21%2F20-20a\"   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=21%2F20-20%287%2F800%29 ALIGN=MIDDLE  ALT=\"b=21%2F20-20%287%2F800%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=7%2F8%29 ALIGN=MIDDLE  ALT=\"highlight%28b=7%2F8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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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%2F800%29t%5E2%2B%287%2F8%29t%2B319%29 ALIGN=MIDDLE  ALT=\"highlight%28C=%287%2F800%29t%5E2%2B%287%2F8%29t%2B319%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<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 1962 level.\r<BR>\n" );
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document.write( "1962 level is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=319 ALIGN=MIDDLE  ALT=\"319\"   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%2A319=638 ALIGN=MIDDLE  ALT=\"C=2%2A319=638\"   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=638=%287%2F800%29t%5E2%2B%287%2F8%29t%2B319 ALIGN=MIDDLE  ALT=\"638=%287%2F800%29t%5E2%2B%287%2F8%29t%2B319\"   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%2F800%29t%5E2%2B%287%2F8%29t%2B319-638=0 ALIGN=MIDDLE  ALT=\"%287%2F800%29t%5E2%2B%287%2F8%29t%2B319-638=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%2F800%29t%5E2%2B%287%2F8%29t-319=0 ALIGN=MIDDLE  ALT=\"%287%2F800%29t%5E2%2B%287%2F8%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+=+30+sqrt%28303%2F7%29+-+50 ALIGN=MIDDLE  ALT=\"t+=+30+sqrt%28303%2F7%29+-+50\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> exact solution  (only positive root, we will disregard negative solution)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+147.38 ALIGN=MIDDLE  ALT=\"t+=+147.38\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> approximately\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=147.38 ALIGN=MIDDLE  ALT=\"147.38\"   DISABLED_style= \"max-width:90%; height:auto;\" > years from 1962 is: year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2109 ALIGN=MIDDLE  ALT=\"2109\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the amount of carbon dioxide in the atmosphere will double from its 1962 level.\r<BR>\n" );
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document.write( "check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319 ALIGN=MIDDLE  ALT=\"638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319\"   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=638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319 ALIGN=MIDDLE  ALT=\"638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319\"   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=638=190.05%2B128.95%2B319 ALIGN=MIDDLE  ALT=\"638=190.05%2B128.95%2B319\"   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=638=638 ALIGN=MIDDLE  ALT=\"638=638\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );