document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=101390>Question 101390</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rock is thrown upward at an initial velocity of 30 ft/sec from a height of 20 feet.  What will its maximum height be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #73833 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=73833\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The equation you are looking for is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-g%2F2%29t%5E2+%2B+V%5Bo%5D%2At+%2B+H%5Bo%5D ALIGN=MIDDLE  ALT=\"H+=+%28-g%2F2%29t%5E2+%2B+V%5Bo%5D%2At+%2B+H%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In this equation, the letters represent the following quantities:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the height of the rock at t seconds after it is launched.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g ALIGN=MIDDLE  ALT=\"g\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the acceleration of an object due to gravity. In English units it is generally <BR>\n" );
document.write( "accepted to be 32 ft/sec^2<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bo%5D ALIGN=MIDDLE  ALT=\"V%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial velocity at which the is launched. In this problem it <BR>\n" );
document.write( "will have a positive sign because it is launched upward.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5Bo%5D ALIGN=MIDDLE  ALT=\"H%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the height above ground from which the is launched. Its sign will be positive<BR>\n" );
document.write( "because it is above ground level.<BR>\n" );
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document.write( "<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 the number of seconds after the rock is launched. <BR>\n" );
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document.write( "With this equation and the definitions of the variables in mind, we can substitute the<BR>\n" );
document.write( "given values into the equation. g is 32, the initial velocity is +30 ft/sec, and the initial<BR>\n" );
document.write( "height of launch is 20 ft. The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-%2832%2F2%29%2At%5E2+%2B+30t+%2B+20 ALIGN=MIDDLE  ALT=\"H+=+-%2832%2F2%29%2At%5E2+%2B+30t+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and you can divide the multiplier of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2 ALIGN=MIDDLE  ALT=\"t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term.  32 divided by 2 is 16 so the equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2+%2B+30t+%2B+20 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2+%2B+30t+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that this is a quadratic equation of the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you are familiar with the quadratic formula you know that when you set y equal to zero<BR>\n" );
document.write( "and in standard form the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the quadratic formula tells you that the values of x that make this happen are given<BR>\n" );
document.write( "by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28%28-b%2F%282%2Aa%29%29+%2B-+%28sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28%28-b%2F%282%2Aa%29%29+%2B-+%28sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ". <BR>\n" );
document.write( "You can find the \"peak\" of this curve in a couple of ways. It will occur when the two values<BR>\n" );
document.write( "of x are found and averaged. The average of the two values of x will be the value of x<BR>\n" );
document.write( "where the peak occurs. Then you can substitute this average value of x into the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and it will tell you the value of y at the peak of the graph.<BR>\n" );
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document.write( "The average value of x is also equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the first term in the solution<BR>\n" );
document.write( "to the quadratic equation.<BR>\n" );
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document.write( "Let's use this second method to find the value of t we are looking for. We'll begin by <BR>\n" );
document.write( "comparing our gravity equation to the equation of the quadratic form. That is: compare:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2+%2B+30t+%2B+20 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2+%2B+30t+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "a is the multiplier of the squared term and in our gravity equation, that multiplier<BR>\n" );
document.write( "is -16.<BR>\n" );
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document.write( "b is the multiplier of the x term and in our gravity equation the multiplier of the corresponding<BR>\n" );
document.write( "t term is +30.<BR>\n" );
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document.write( "Now we can substitute these values into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%2830%29%2F%282%2A%28-16%29%29+=+-30%2F%28-32%29+=+15%2F16 ALIGN=MIDDLE  ALT=\"-%2830%29%2F%282%2A%28-16%29%29+=+-30%2F%28-32%29+=+15%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F16 ALIGN=MIDDLE  ALT=\"15%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after launch the rock will be at maximum height. (In decimal form<BR>\n" );
document.write( "this is 0.9375 seconds after launch.)<BR>\n" );
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document.write( "Now if we return to the height equation and substitute 0.9375 seconds for t we can get the<BR>\n" );
document.write( "height at the peak of the path of the rock.<BR>\n" );
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document.write( "Start with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16t%5E2+%2B+30t+%2B+20 ALIGN=MIDDLE  ALT=\"H+=+-16t%5E2+%2B+30t+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substituting 0.9375 for t makes this equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16%2A%280.9375%29%5E2+%2B+30%2A%280.9375%29+%2B+20 ALIGN=MIDDLE  ALT=\"H+=+-16%2A%280.9375%29%5E2+%2B+30%2A%280.9375%29+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Put your calculator to work and you should get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-16%2A0.8789+%2B+28.125+%2B+20+=+-14.0625+%2B+28.125+%2B+20+=+34.0625 ALIGN=MIDDLE  ALT=\"H+=+-16%2A0.8789+%2B+28.125+%2B+20+=+-14.0625+%2B+28.125+%2B+20+=+34.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the rock rises to a maximum height of 34.0625 feet.<BR>\n" );
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document.write( "Lots of work to follow here. Hope this helps you to see how you might solve this problem.<BR>\n" );
document.write( ".\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );