document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115426>Question 1115426</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A flare is shot upward from the edge of a 88 foot tall cliff with an initial velocity of 82 ft/sec.  What is the maximum height above the ground this flare will go in feet? \r<BR>\n" );
document.write( "\n" );
document.write( "Round your answer to 1 decimal place. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #730286 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=730286\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classical physics problem, and classical physics should be intuitive, because we see it in action in the world around us.<BR>\n" );
document.write( "Alternatively, we can memorize \"formulas\" for every \"type\" of problem without really understanding.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE SIMPLE/FIFTH-GRADER SOLUTION:<BR>\n" );
document.write( "Somewhere in the physics book, it is stated that the acceleration of gravity increases the downward speed of an object by 32 feet per second every second:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=32 ALIGN=MIDDLE  ALT=\"g=32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ft%2Fs%5E2 ALIGN=MIDDLE  ALT=\"ft%2Fs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with s as the symbol for second).<BR>\n" );
document.write( "So, it will take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=82%2F32=2.5625 ALIGN=MIDDLE  ALT=\"82%2F32=2.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds for gravity to get that initial upwards velocity down to zero, before the flare remain star having increasing downwards velocity.<BR>\n" );
document.write( "During those initial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.5625 ALIGN=MIDDLE  ALT=\"2.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds the upwards velocity decreased steadily/linearly from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2282+ft+%2F+s%22 ALIGN=MIDDLE  ALT=\"%2282+ft+%2F+s%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%220+ft+%2F+s%22 ALIGN=MIDDLE  ALT=\"%220+ft+%2F+s%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C4%2C-10%2C90%2Cgrid%280%29%2C%0D%0Aline%280%2C82%2C2.5625%2C0%29%2Clocate%282.5%2C7%2Ctime%29%2C%0D%0Alocate%283%2C7%2C%22%28+seconds+%29%22%29%2Clocate%280.1%2C88%2Cupwards%29%2C%0D%0Alocate%281.15%2C88%2Cvelocity%29%2Clocate%280.5%2C83%2C%22%28+ft+%2F+s+%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "During that time the average upwards velocity (in ft/s) was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2882%2B0%29%2F2=41 ALIGN=MIDDLE  ALT=\"%2882%2B0%29%2F2=41\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the upwards distance covered (in feet) was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=41%2A2.5625=105.0625 ALIGN=MIDDLE  ALT=\"41%2A2.5625=105.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can round that to 1 decimal place to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=105.1ft ALIGN=MIDDLE  ALT=\"105.1ft\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As the flare started from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=88ft ALIGN=MIDDLE  ALT=\"88ft\"   DISABLED_style= \"max-width:90%; height:auto;\" > above ground,<BR>\n" );
document.write( "the maximum height above the ground this flare will go (in feet) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=88%2B105.1=highlight%28193.1%29 ALIGN=MIDDLE  ALT=\"88%2B105.1=highlight%28193.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH FORMULAS (maybe your teacher wants that):<BR>\n" );
document.write( " <BR>\n" );
document.write( "The basic formulas for an object moving in one direction with an \"initial velocity\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D ALIGN=MIDDLE  ALT=\"v%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a constant acceleration <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=v%5B0%5D%2Bat ALIGN=MIDDLE  ALT=\"v=v%5B0%5D%2Bat\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=v%5B0%5Dt%2B%281%2F2%29at%5E2 ALIGN=MIDDLE  ALT=\"d=v%5B0%5Dt%2B%281%2F2%29at%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bavg%5D=%28v%2Bv%5B0%5D%29%2F2 ALIGN=MIDDLE  ALT=\"v%5Bavg%5D=%28v%2Bv%5B0%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (pick one).<BR>\n" );
document.write( "In those formulas <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > is distance covered,<BR>\n" );
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 time since a time, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "that we can conveniently say it is the start of the observation,<BR>\n" );
document.write( "an \"initial velocity\" is the velocity at the start of the observation,<BR>\n" );
document.write( "when the velocity was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D ALIGN=MIDDLE  ALT=\"v%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From those two formulas, and your algebra skills,<BR>\n" );
document.write( "you can get any other formula you will be given.<BR>\n" );
document.write( " <BR>\n" );
document.write( "In particular, for projectiles shot upwards, from somewhere om planet Earth,<BR>\n" );
document.write( "the height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > above the ground is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=h%5B0%5D%2Bv%7B0%5Dt-16t%5E2 ALIGN=MIDDLE  ALT=\"h=h%5B0%5D%2Bv%7B0%5Dt-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B0%5D ALIGN=MIDDLE  ALT=\"h%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial height above the ground, and <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 time since \"launch\".<BR>\n" );
document.write( "plugging into that formula the numbers from the problem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=88%2B82t-16t%5E2 ALIGN=MIDDLE  ALT=\"h=88%2B82t-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In algebra class, you would probably write it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B82t%2B88 ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B82t%2B88\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because we like to order polynomial terms from highest to lowest exponent.<BR>\n" );
document.write( "In algebra 2 courses is the US, you are likely to be told that<BR>\n" );
document.write( "a quadratic function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a maximum (if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%222+a%22 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%222+a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > . If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > it would be a minimum)<BR>\n" );
document.write( "(another formula, but you do not need to memorize it if your algebra skills include \"completing the square\").<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B82t%2B88 ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B82t%2B88\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > (instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-16 ALIGN=MIDDLE  ALT=\"a=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=82 ALIGN=MIDDLE  ALT=\"b=82\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so it has a maximum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-b%29%2F%222+a%22=%28-82%29%2F%282%2A%28-16%29%29=%28-82%29%2F%28-32%29=5.5625 ALIGN=MIDDLE  ALT=\"t=%28-b%29%2F%222+a%22=%28-82%29%2F%282%2A%28-16%29%29=%28-82%29%2F%28-32%29=5.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and for that value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in seconds, the height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16%2A2.5625%5E2%2B82%2A2.5625%2B88=-106.0625%2B210.125%2B88=193.0625 ALIGN=MIDDLE  ALT=\"h=-16%2A2.5625%5E2%2B82%2A2.5625%2B88=-106.0625%2B210.125%2B88=193.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Rounding to 1 decimal place, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28h=193.1%29 ALIGN=MIDDLE  ALT=\"highlight%28h=193.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );