document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172759>Question 172759</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If an object is propelled upward from a 288 foot high tower at an initial velocity of 48 feet per second, then its height h(in feet) after t seconds is given by the equation h=-16t^2+48t+288. What is the maximum height reached by the object? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127594 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=127594\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are two ways to do this.\r<BR>\n" );
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document.write( "1. Algebra solution.  The vertex of a parabola described by <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 an x-coordinate at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a y-coordinate at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-b%2F2a%29 ALIGN=MIDDLE  ALT=\"f%28-b%2F2a%29\"   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;\" > then you have a concave up parabola and the vertex represents a minimum value of the function.  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;\" >, then it is concave down and the vertex represents a maximum.  The latter is the situation in the given problem.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B48t%2B288 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B48t%2B288\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That means the t-coordinate of the vertex is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-48%2F%282%28-16%29%29=%28-48%29%2F%28-32%29=3%2F2 ALIGN=MIDDLE  ALT=\"-48%2F%282%28-16%29%29=%28-48%29%2F%28-32%29=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Evaluating to determine the h-coordinate of the vertex which is also the maximum height:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%283%2F2%29=-16%283%2F2%29%5E2%2B48%283%2F2%29%2B288=-36%2B72%2B288=324 ALIGN=MIDDLE  ALT=\"h%283%2F2%29=-16%283%2F2%29%5E2%2B48%283%2F2%29%2B288=-36%2B72%2B288=324\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Maximum height:  324 feet\r<BR>\n" );
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document.write( "2. Calculus method.\r<BR>\n" );
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document.write( "A local extrema for a continuous differentiable function is located where the value of the first derivitive is equal to zero.  If the function is twice differentiable, the character of the extrema (maximum or minimum) can be determined by examining the sign of the second derivative evaluated at the extreme point.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B48t%2B288 ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B48t%2B288\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dh%28t%29%29%2Fdt=-32t%2B48 ALIGN=MIDDLE  ALT=\"%28dh%28t%29%29%2Fdt=-32t%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32t%2B48=0 ALIGN=MIDDLE  ALT=\"-32t%2B48=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-48%29%2F%28-32%29=3%2F2 ALIGN=MIDDLE  ALT=\"t=%28-48%29%2F%28-32%29=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (Look familiar?)\r<BR>\n" );
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document.write( "Evaluation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%283%2F2%29 ALIGN=MIDDLE  ALT=\"h%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as in the Algebra method above, so the value for maximum height is the same, namely 324 feet.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%5E2h%28t%29%29%2Fdt%5E2=-32 ALIGN=MIDDLE  ALT=\"%28d%5E2h%28t%29%29%2Fdt%5E2=-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is negative for all values of t in the domain of h(t), therefore the extreme point is a maximum.\r<BR>\n" );
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