document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=39353>Question 39353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello,\r<BR>\n" );
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document.write( "First and foremost I would like to thank you for your assistance.  I am a parent of a high school student and have not taken Alegbra in a years.  I would appreciate any assistance you can provide.  \r<BR>\n" );
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document.write( "Thanks,<BR>\n" );
document.write( "Steve\r<BR>\n" );
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document.write( "Problem 1:<BR>\n" );
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document.write( "Use the table shown below.  Assume the data can be represented as a linear relationship.<BR>\n" );
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document.write( "a) Using the two points defined by data from 2000 and 2005, develop a linear equation for the data.<BR>\n" );
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document.write( "b) Using the equation found in a), predict the tax revenue for 2006.<BR>\n" );
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document.write( "c) Use the calculator to determine the \"best fit\" equation for the data.<BR>\n" );
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document.write( "d) Using the equation found in c), predict the tax revenue for 2006.<BR>\n" );
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document.write( "e) Using the equation found in c), estimate tax revenue for the missing 2002 data<BR>\n" );
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document.write( "Year     Sales Tax Revenue <BR>\n" );
document.write( "2000     $19500<BR>\n" );
document.write( "2001     $21000<BR>\n" );
document.write( "2003     $24000<BR>\n" );
document.write( "2004     $27300<BR>\n" );
document.write( "2005     $31000<BR>\n" );
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document.write( "Note:  data is missing for 2002, the county clerk embezzled the funds for that year.<BR>\n" );
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document.write( "Problem #2<BR>\n" );
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document.write( "A rocket is shot from the top of an oceanside cliff that is 60ft high.  The original angle is 45 degrees and the original velocity is 50ft per sec.  The formula for the rocket's motion is:<BR>\n" );
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document.write( "y = f(x) = -0.0128x(squared) + x + 60<BR>\n" );
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document.write( "The variables x and y are both measured in feet.  The variable x represents horizontal displacement (distance from the base of the cliff), and y represents the height of the rocket above the ground.<BR>\n" );
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document.write( "a) What is the maximum height attained by the rocket?  How far has the rocket travelled horizontally at this point?<BR>\n" );
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document.write( "b) At waht point(s) (after launch) is the rocket 70 feet high?  (Horizontal displacement)<BR>\n" );
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document.write( "c) How far away from the base of the cliff does the rocket land? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #24805 by </B><A HREF=/tutors/aboutme.mpl?userid=Nate><B>Nate(3500)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Nate><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=24805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Year Sales Tax Revenue <BR>\n" );
document.write( "2000 $19500<BR>\n" );
document.write( "2001 $21000<BR>\n" );
document.write( "2003 $24000<BR>\n" );
document.write( "2004 $27300<BR>\n" );
document.write( "2005 $31000<BR>\n" );
document.write( "a) Using the two points defined by data from 2000 and 2005, develop a linear equation for the data.<BR>\n" );
document.write( "2000 $19500 at this time (0 years from 2000)<BR>\n" );
document.write( "2005 $31000 at this time (5 years from 2000)<BR>\n" );
document.write( "y = ((31000-19500)/5)x + 19500<BR>\n" );
document.write( "y = 2300x + 19500<BR>\n" );
document.write( "b) Using the equation found in a), predict the tax revenue for 2006.<BR>\n" );
document.write( "y = 2300(6) + 19500<BR>\n" );
document.write( "y = 13800 + 19500 = 33300<BR>\n" );
document.write( "c) Use the calculator to determine the \"best fit\" equation for the data.<BR>\n" );
document.write( "I haven't done any equations dealing with the \"best fit\" line, but I will try my best. It seems right.<BR>\n" );
document.write( "y = (31000-((19500 + 21000 + 24000 + 27300 + 31000)/5))x + 19500<BR>\n" );
document.write( "y = 2205x + 19500<BR>\n" );
document.write( "d) Using the equation found in c), predict the tax revenue for 2006.<BR>\n" );
document.write( "y = 2205(6) + 19500 = 32730<BR>\n" );
document.write( "e) Using the equation found in c), estimate tax revenue for the missing 2002 data<BR>\n" );
document.write( "y = 2205x(2) + 19500 = 23910<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-0.0128x%5E2+%2B+x+%2B+60 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-0.0128x%5E2+%2B+x+%2B+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a) What is the maximum height attained by the rocket? How far has the rocket travelled horizontally at this point?<BR>\n" );
document.write( "The highest point would be at the vertex.<BR>\n" );
document.write( "((-b/2a),f(x))<BR>\n" );
document.write( "10000/256 = 5000/128 = 2500/64 = 1250/32 = 625/16<BR>\n" );
document.write( "f(x) = -0.0128(625/16)^2 + (625/16) + 60<BR>\n" );
document.write( "v(39.0625,79.53125) Max height: 39.0625ft.  Distance traveled: 79.53125ft.<BR>\n" );
document.write( "b) At waht point(s) (after launch) is the rocket 70 feet high? (Horizontal displacement)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-0.0128x%5E2+%2B+x+%2B+60 ALIGN=MIDDLE  ALT=\"f%28x%29+=+-0.0128x%5E2+%2B+x+%2B+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70+=+-0.0128x%5E2+%2B+x+%2B+60 ALIGN=MIDDLE  ALT=\"70+=+-0.0128x%5E2+%2B+x+%2B+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-0.0128x%5E2+%2B+x+-+10 ALIGN=MIDDLE  ALT=\"0+=+-0.0128x%5E2+%2B+x+-+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the quadratic formula to determine the root.<BR>\n" );
document.write( "<TABLE  BORDER=1>\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=-.0128x%5E2%2B1x%2B-10+=+0 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B-10+=+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=%281%29%5E2-4%2A-.0128%2A-10=0.488 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%281%29%5E2-4%2A-.0128%2A-10=0.488\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=0.488 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%2B-sqrt%28+0.488+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-1%2B-sqrt%28+0.488+%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-%281%29%2Bsqrt%28+0.488+%29%29%2F2%5C-.0128+=+11.7746106303547 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%281%29%2Bsqrt%28+0.488+%29%29%2F2%5C-.0128+=+11.7746106303547\"   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-%281%29-sqrt%28+0.488+%29%29%2F2%5C-.0128+=+66.3503893696453 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%281%29-sqrt%28+0.488+%29%29%2F2%5C-.0128+=+66.3503893696453\"   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=-.0128x%5E2%2B1x%2B-10 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B-10\"   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=-.0128x%5E2%2B1x%2B-10+=+-.0128%28x-11.7746106303547%29%2A%28x-66.3503893696453%29 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B-10+=+-.0128%28x-11.7746106303547%29%2A%28x-66.3503893696453%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 11.7746106303547, 66.3503893696453.\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+-.0128%2Ax%5E2%2B1%2Ax%2B-10+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-.0128%2Ax%5E2%2B1%2Ax%2B-10+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "At about 11.8 ft. horizontally or about 66.4 ft.<BR>\n" );
document.write( "c) How far away from the base of the cliff does the rocket land?<BR>\n" );
document.write( "Again, use quadratic formula to determine the roots.<BR>\n" );
document.write( "<TABLE  BORDER=1>\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=-.0128x%5E2%2B1x%2B60+=+0 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B60+=+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=%281%29%5E2-4%2A-.0128%2A60=4.072 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%281%29%5E2-4%2A-.0128%2A60=4.072\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=4.072 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%2B-sqrt%28+4.072+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-1%2B-sqrt%28+4.072+%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-%281%29%2Bsqrt%28+4.072+%29%29%2F2%5C-.0128+=+-39.762489097684 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%281%29%2Bsqrt%28+4.072+%29%29%2F2%5C-.0128+=+-39.762489097684\"   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-%281%29-sqrt%28+4.072+%29%29%2F2%5C-.0128+=+117.887489097684 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%281%29-sqrt%28+4.072+%29%29%2F2%5C-.0128+=+117.887489097684\"   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=-.0128x%5E2%2B1x%2B60 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B60\"   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=-.0128x%5E2%2B1x%2B60+=+-.0128%28x--39.762489097684%29%2A%28x-117.887489097684%29 ALIGN=MIDDLE  ALT=\"-.0128x%5E2%2B1x%2B60+=+-.0128%28x--39.762489097684%29%2A%28x-117.887489097684%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -39.762489097684, 117.887489097684.\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+-.0128%2Ax%5E2%2B1%2Ax%2B60+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-.0128%2Ax%5E2%2B1%2Ax%2B60+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "At about 117.9 ft. horizontally.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );