document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316415>Question 316415</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The kinetic energy, K, of a moving particle acted on by various forces depends quadratically on the velocity, x, of the particle. The particle's energy is 210 ergs when its velocity is 5 meters per second;860 ergs at 10 meters per second; and 2010 ergs at 15 meters per second. What is the quadratic function K(x) that fits this data? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226625 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=226625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The kinetic energy, K, of a moving particle acted on by various forces depends quadratically on the velocity, x, of the particle.<BR>\n" );
document.write( " The particle's energy is 210 ergs when its velocity is 5 meters per second;<BR>\n" );
document.write( "860 ergs at 10 meters per second; and <BR>\n" );
document.write( "2010 ergs at 15 meters per second.<BR>\n" );
document.write( " What is the quadratic function K(x) that fits this data?<BR>\n" );
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document.write( "Using the form K(x) = ax^2 + bx, find a & b by elimination<BR>\n" );
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document.write( "x=5; K(x)=210<BR>\n" );
document.write( "5^2a + 5b = 210<BR>\n" );
document.write( "25a + 5b = 210<BR>\n" );
document.write( "Simplify, divide by 5<BR>\n" );
document.write( "5a + b = 42<BR>\n" );
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document.write( "x=15; K(x)=2010<BR>\n" );
document.write( "15^2a + 15b = 2010<BR>\n" );
document.write( "225a + 15b = 2010<BR>\n" );
document.write( "Simplify, divide by 5<BR>\n" );
document.write( "45 + 3b = 402<BR>\n" );
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document.write( "Multiply the simplified version of the 1st equation by 3, subtract from above<BR>\n" );
document.write( "45a + 3b = 402<BR>\n" );
document.write( "15a + 3b = 126<BR>\n" );
document.write( "--------------subtraction eliminates b, find a<BR>\n" );
document.write( "30a = 276<BR>\n" );
document.write( "a = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=276%2F30 ALIGN=MIDDLE  ALT=\"276%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a = 9.2<BR>\n" );
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document.write( "Find b using 5a + b= 42<BR>\n" );
document.write( "5(9.2) + b = 42<BR>\n" );
document.write( "b = 42 - 46<BR>\n" );
document.write( "b = -4<BR>\n" );
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document.write( "The quadratic function: k(x) = 9.2x^2 - 4x<BR>\n" );
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document.write( "It fits exactly for x=5 and x=15<BR>\n" );
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document.write( "Check the fit using x=10;<BR>\n" );
document.write( "k(x) = 9.2(10^2) - 4(10)<BR>\n" );
document.write( "k(x) = 920 - 40<BR>\n" );
document.write( "k(x) = 880 a slight difference, should be 860<BR>\n" );
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document.write( "Graphically; k(x) on the y axis<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+20%2C+-1000%2C+2500%2C+9.2x%5E2-4x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+20%2C+-1000%2C+2500%2C+9.2x%5E2-4x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );