document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1068621>Question 1068621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ball is thrown vertically upward. After t seconds, its height h (in feet) is given by the function <BR>\n" );
document.write( "h(t) = 60t - 16t^2\r<BR>\n" );
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document.write( "What is the maximum height that the ball will reach?\r<BR>\n" );
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document.write( "Do not round your answer.\r<BR>\n" );
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document.write( "Height: _____ feet\r<BR>\n" );
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document.write( "Please show steps to answer this problem. Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #683861 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=683861\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=60t-16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29=60t-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which could be written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B60t ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B60t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "is a quadratic function<BR>\n" );
document.write( "(a polynomial with degree 2),<BR>\n" );
document.write( "and graphs as a parabola.<BR>\n" );
document.write( "Your teacher probably wants to remember all the names and formulas.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH FORMULAS:<BR>\n" );
document.write( "You have probably studied quadratic functions of the form<BR>\n" );
document.write( "<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;\" > and were told that they have<BR>\n" );
document.write( "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;\" > , or a minimum is <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;\" ><BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F%222a%22 ALIGN=MIDDLE  ALT=\"x=-b%2F%222a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That point in the graph is also called the vertex of the parabola.<BR>\n" );
document.write( "In this case, the variable is called <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;\" > ,<BR>\n" );
document.write( "and the coefficients are<BR>\n" );
document.write( "<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;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=60 ALIGN=MIDDLE  ALT=\"b=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=0 ALIGN=MIDDLE  ALT=\"c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the maximum happens at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=-b%2F%222a%22=-60%2F%282%2A%28-16%29%29=-60%2F%28-32%29=15%2F8=1.875 ALIGN=MIDDLE  ALT=\"t=-b%2F%222a%22=-60%2F%282%2A%28-16%29%29=-60%2F%28-32%29=15%2F8=1.875\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the maximum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2815%2F8%29=-16%2A%2815%2F8%29%5E2%2B60%2A%2815%2F8%29=highlight%2856.25%29 ALIGN=MIDDLE  ALT=\"h%2815%2F8%29=-16%2A%2815%2F8%29%5E2%2B60%2A%2815%2F8%29=highlight%2856.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH EASE (AND THINKING):<BR>\n" );
document.write( "Those parabolas are symmetrical,<BR>\n" );
document.write( "so if a parabola has two zeros,<BR>\n" );
document.write( "the maximum or minimum happens halfway between the zeros.<BR>\n" );
document.write( "This parabola must have two zeros<BR>\n" );
document.write( "and a maximum in between because the ball<BR>\n" );
document.write( "was thrown up from the ground (how did they do that?)<BR>\n" );
document.write( "gets to some maximum height,<BR>\n" );
document.write( "and falls back to the ground on the ground.<BR>\n" );
document.write( "Let me find the zeros by factoring.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60t-16t%5E2=0 ALIGN=MIDDLE  ALT=\"60t-16t%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2860-16t%29=0 ALIGN=MIDDLE  ALT=\"t%2860-16t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the zeros are <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;\" > and<BR>\n" );
document.write( "(from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60-16t=0 ALIGN=MIDDLE  ALT=\"60-16t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60=16t ALIGN=MIDDLE  ALT=\"60=16t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=60%2F16 ALIGN=MIDDLE  ALT=\"t=60%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=15%2F4 ALIGN=MIDDLE  ALT=\"t=15%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The maximum happens halfway between <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=15%2F4 ALIGN=MIDDLE  ALT=\"t=15%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, it happens at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%281%2F2%29%2815%2F4%29=15%2F8 ALIGN=MIDDLE  ALT=\"t=%281%2F2%29%2815%2F4%29=15%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The maximum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2815%2F8%29 ALIGN=MIDDLE  ALT=\"h%2815%2F8%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=60t-16t%5E2=t%2860-16t%29 ALIGN=MIDDLE  ALT=\"h%28t%29=60t-16t%5E2=t%2860-16t%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%2815%2F8%29=%2815%2F8%29%2860-16%2815%2F8%29%29=%2815%2F8%29%2860-30%29=%2815%2F8%29%2830%29=15%2A15%2F4=225%2F4=highlight%2856.25%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Either way, the maximum height is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2856.25feet%29 ALIGN=MIDDLE  ALT=\"highlight%2856.25feet%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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