document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1021026>Question 1021026</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with this problem. Please help me and show all work. Thanks\r<BR>\n" );
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document.write( "A person standing close to the edge on top of a 144-foot building throws a ball vertically upward. The quadratic function h(t)=&#8722;16t^2+128t+144<BR>\n" );
document.write( "h(t)=-16t^2+128t+144 models the ball's height about the ground, h(t)<BR>\n" );
document.write( " in feet, t seconds after it was thrown.\r<BR>\n" );
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document.write( " a) What is the maximum height of the ball?\r<BR>\n" );
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document.write( "   _________feet\r<BR>\n" );
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document.write( " b) How many seconds does it take until the ball hits the ground?\r<BR>\n" );
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document.write( "   _________ seconds </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #636839 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=636839\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> WITHOUT ALGEBRA 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B128t%2B144 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B128t%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28t%5E2%2B128t%2F%28-16%29%2B144%2F%28-16%29%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28t%5E2%2B128t%2F%28-16%29%2B144%2F%28-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28t%5E2-8t-9%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28t%5E2-8t-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28t%5E2-8t%2B16-16-9%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28t%5E2-8t%2B16-16-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28%28t%5E2-8t%2B16%29-25%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28%28t%5E2-8t%2B16%29-25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28%28t-4%29%5E2-25%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28%28t-4%29%5E2-25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28t-4%29%5E2-16%2A%28-25%29 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28t-4%29%5E2-16%2A%28-25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16%28t-4%29%5E2%2B400 ALIGN=MIDDLE  ALT=\"h%28t%29=-16%28t-4%29%5E2%2B400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "a) From the last equation above we realize that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29%3C=400 ALIGN=MIDDLE  ALT=\"h%28t%29%3C=400\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the maximum height of the ball is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28400%29 ALIGN=MIDDLE  ALT=\"highlight%28400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet. <BR>\n" );
document.write( " <BR>\n" );
document.write( "b) The ball hits the ground when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B128t%2B144=0 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B128t%2B144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E0 ALIGN=MIDDLE  ALT=\"t%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we need a positive solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16%28%28t-4%29%5E2-25%29=0 ALIGN=MIDDLE  ALT=\"-16%28%28t-4%29%5E2-25%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16%28%28t-4%29%5E2-25%29=0 ALIGN=MIDDLE  ALT=\"-16%28%28t-4%29%5E2-25%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t-4%29%5E2-25=0 ALIGN=MIDDLE  ALT=\"%28t-4%29%5E2-25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t-4%29%5E2=25 ALIGN=MIDDLE  ALT=\"%28t-4%29%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t-4=5%2C%22or%22%2Ct-4=-5%29 ALIGN=MIDDLE  ALT=\"system%28t-4=5%2C%22or%22%2Ct-4=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-4=5 ALIGN=MIDDLE  ALT=\"t-4=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4%2B5 ALIGN=MIDDLE  ALT=\"t=4%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=9 ALIGN=MIDDLE  ALT=\"t=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only option with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E0 ALIGN=MIDDLE  ALT=\"t%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the ball hits the ground <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%289%29 ALIGN=MIDDLE  ALT=\"highlight%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after it is thrown upwards.\r<BR>\n" );
document.write( "\n" );
document.write( "WITH FORMULAS:<BR>\n" );
document.write( "a) 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;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "has a maximum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%282%2Aa%29\"   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=h%28t%29=-16t%5E2%2B128t%2B144 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B128t%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" > is such a function with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-16%3C0 ALIGN=MIDDLE  ALT=\"a=-16%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=128 ALIGN=MIDDLE  ALT=\"b=128\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so there is a maximum for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-128%29%2F%282%2A%28-16%29%29=%28-128%29%2F%28-32%29=4 ALIGN=MIDDLE  ALT=\"t=%28-128%29%2F%282%2A%28-16%29%29=%28-128%29%2F%28-32%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that maximum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%284%29=-16%2A4%5E2%2B128%2A4%2B144=-16%2A16%2B256%2B144=-256%2B512%2B144=highlight%28400%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) The ball hits the ground when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B128t%2B144=0 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B128t%2B144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E0 ALIGN=MIDDLE  ALT=\"t%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, we are looking for a positive solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t%5E2%2B128t%2B144=0 ALIGN=MIDDLE  ALT=\"-16t%5E2%2B128t%2B144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%5E2-128t-144=0 ALIGN=MIDDLE  ALT=\"16t%5E2-128t-144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%5E2%2F16-128t%2F16-144%2F16=0 ALIGN=MIDDLE  ALT=\"16t%5E2%2F16-128t%2F16-144%2F16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=9 ALIGN=MIDDLE  ALT=\"t=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=-1 ALIGN=MIDDLE  ALT=\"t=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the ball hits the ground <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%289%29 ALIGN=MIDDLE  ALT=\"highlight%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after it is thrown upwards.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be solved by factoring, by \"completing the square\", and by using the quadratic formula.<BR>\n" );
document.write( "By factoring:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%2B1%29%28t-9%29=0 ALIGN=MIDDLE  ALT=\"%28t%2B1%29%28t-9%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t%2B1=0%2C%22or%22%2Ct-9=0%29 ALIGN=MIDDLE  ALT=\"system%28t%2B1=0%2C%22or%22%2Ct-9=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=-1%2C%22or%22%2Ct=9%29 ALIGN=MIDDLE  ALT=\"system%28t=-1%2C%22or%22%2Ct=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By using the quadratic formula:<BR>\n" );
document.write( "The quadratic formula says that any solution(s) for<BR>\n" );
document.write( "<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;\" > , if there is any, are given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so any solution(s) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=x%2Ca=1%2Cb=-8%2Cc=-1%29 ALIGN=MIDDLE  ALT=\"system%28t=x%2Ca=1%2Cb=-8%2Cc=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , would be given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-%28-8%29+%2B-+sqrt%28%28-8%29%5E2-4%2A1%2A%28-9%29%29%29%2F%282%2A1%29=%288+%2B-+sqrt%2864%2B36%29%29%2F2=%288+%2B-+sqrt%28100%29%29%2F2=%288+%2B-+10%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=%288%2B10%29%2F2=18%2F2=9%2C%22or%22%2Ct=%288-10%29%2F2=%28-2%29%2F2=-1%29 ALIGN=MIDDLE  ALT=\"system%28t=%288%2B10%29%2F2=18%2F2=9%2C%22or%22%2Ct=%288-10%29%2F2=%28-2%29%2F2=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Completing the square is pretty much what was done at the top:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t-9=0 ALIGN=MIDDLE  ALT=\"t%5E2-8t-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-8t%2B16=9%2B16 ALIGN=MIDDLE  ALT=\"t%5E2-8t%2B16=9%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t-4%29%5E2=25 ALIGN=MIDDLE  ALT=\"%28t-4%29%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t-4=5%2C%22or%22%2Ct-4=-5%29 ALIGN=MIDDLE  ALT=\"system%28t-4=5%2C%22or%22%2Ct-4=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=5%2B4=9%2C%22or%22%2Ct=-5%2B4=-1%29 ALIGN=MIDDLE  ALT=\"system%28t=5%2B4=9%2C%22or%22%2Ct=-5%2B4=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );