document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460148>Question 460148</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good Morning all,\r<BR>\n" );
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document.write( "Use of the internet in a country is given by the function f(x)=0.485x^2 - 1.694x + 0.315, where the output is in millions of users. In this formula x=6 corresponds to 1996, x=7 corresponds to 1997, and so on until x=20 corresponds to 2010. <BR>\n" );
document.write( "Estimate when the number of internet users in the country reached 144 million?<BR>\n" );
document.write( "What year?\r<BR>\n" );
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document.write( "Thank you in advance. \r<BR>\n" );
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document.write( "Minka </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #315634 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Use of the internet in a country is given by the function f(x)=0.485x^2 - 1.694x + 0.315, where the output is in millions of users. In this formula x=6 corresponds to 1996, x=7 corresponds to 1997, and so on until x=20 corresponds to 2010.<BR>\n" );
document.write( "Estimate when the number of internet users in the country reached 144 million?<BR>\n" );
document.write( "Given:<BR>\n" );
document.write( "f(x)=0.485x^2 - 1.694x + 0.315<BR>\n" );
document.write( "Set f(x) to 144 and solve for x:<BR>\n" );
document.write( "144 = 0.485x^2 - 1.694x + 0.315<BR>\n" );
document.write( "0 = 0.485x^2 - 1.694x - 143.685<BR>\n" );
document.write( "applying the quadratic formula, we get:<BR>\n" );
document.write( "x = {19.05, -15.55}<BR>\n" );
document.write( "we can toss out the negative solution leaving:<BR>\n" );
document.write( "x = 19 (years from 1990<BR>\n" );
document.write( ".<BR>\n" );
document.write( "calculating the year:<BR>\n" );
document.write( "1990+ 19 = 2009<BR>\n" );
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document.write( "Details of quadratic to follow:<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.485x%5E2%2B-1.694x%2B-143.685+=+0 ALIGN=MIDDLE  ALT=\"0.485x%5E2%2B-1.694x%2B-143.685+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-1.694%29%5E2-4%2A0.485%2A-143.685=281.618536 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-1.694%29%5E2-4%2A0.485%2A-143.685=281.618536\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=281.618536 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--1.694%2B-sqrt%28+281.618536+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--1.694%2B-sqrt%28+281.618536+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-1.694%29%2Bsqrt%28+281.618536+%29%29%2F2%5C0.485+=+19.0469008812773 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-1.694%29%2Bsqrt%28+281.618536+%29%29%2F2%5C0.485+=+19.0469008812773\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-1.694%29-sqrt%28+281.618536+%29%29%2F2%5C0.485+=+-15.5541173761227 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-1.694%29-sqrt%28+281.618536+%29%29%2F2%5C0.485+=+-15.5541173761227\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.485x%5E2%2B-1.694x%2B-143.685 ALIGN=MIDDLE  ALT=\"0.485x%5E2%2B-1.694x%2B-143.685\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.485x%5E2%2B-1.694x%2B-143.685+=+0.485%28x-19.0469008812773%29%2A%28x--15.5541173761227%29 ALIGN=MIDDLE  ALT=\"0.485x%5E2%2B-1.694x%2B-143.685+=+0.485%28x-19.0469008812773%29%2A%28x--15.5541173761227%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 19.0469008812773, -15.5541173761227.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+0.485%2Ax%5E2%2B-1.694%2Ax%2B-143.685+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+0.485%2Ax%5E2%2B-1.694%2Ax%2B-143.685+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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