document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=971165>Question 971165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I shot a rocket from the roof of a second story house. The rocket propelled upward with an initial velocity of 44ft/sec and it's height after t seconds is given by <BR>\n" );
document.write( "H (t)=-16t2+44t\r<BR>\n" );
document.write( "\n" );
document.write( "How long will it take to reach its highest point and how high will it go. Assume that each floor of the house requires 12 feet of vertical space </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #593740 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=593740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your height function is incorrect.  The function you provided is the height ABOVE the roof of the house. The correct function is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ H(t)\ =\ -16t^2\ +\ v_ot\ +\ h_o\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v_o\"> is the initial velocity and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h_o\"> is the initial height of the projectile when it was fired.  Since each floor of the hous measures 12 feet, one must presume that this value is equal to 24 feet.  Your function is therefore:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ H(t)\ =\ -16t^2\ +\ 44t\ +\ 24\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<u>Algebra method</u>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The height function is a quadratic polynomial that has a graph that is a parabola.  Since the lead coefficient is negative, the parabola opens downward and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large H(t)\"> coordinate of the vertex represents the maximum height of the projectile and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> coordinate of the vertex represents the time at which the projectile reaches that maximum height.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The coordinates of the vertex of \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "are given by\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v = \frac{-b}{2a}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_v = f(x_v) = f\left(\frac{-b}{2a}\right)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So for your situation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t_{max} = \frac{-44}{2(-16)}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can do your own arithmetic to find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t_{max}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ H(t_{max})\ =\ -16(t_{max})^2\ +\ 44t_{max}\ +\ 24\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Just substitute and calculate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<u>Calculus Method</u>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The local maximum is the point where the first derivative is equal to zero, so\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ H(t)\ =\ -16t^2\ +\ 44t\ +\ 24\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dH}{dt}\ =\ -32t\ +\ 44\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -32t\ +\ 44\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ H(t)\ =\ -16t^2\ +\ 44t\ +\ 24\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Just substitute and calculate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );