document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209012>Question 1209012</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If an archer shoots an arrow straight upward with an initial velocity of 160 ft/sec from a height of 8 ft, then its height above the ground in feet at time t in seconds is given by the function   h(t) = -16t^2 +160t + 8 \r<BR>\n" );
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document.write( "1. What shape would this function make when graphed? How do you know? \r<BR>\n" );
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document.write( "2. Will this shape have an obvious minimum value, an obvious maximum value, neither, or both? How do you know? \r<BR>\n" );
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document.write( "3. Where in the graph would you find the “optimal” value (either highest or lowest value)? Find, through calculation, the t-coordinate for that “optimal” value. \r<BR>\n" );
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document.write( "4. Find, through calculation, the h(t)-coordinate for the “optimal” value? \r<BR>\n" );
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document.write( "5. Interpret the results from Step 3 and Step 4. (I.e., what do those numbers mean?) \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847582 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52864)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "If an archer shoots an arrow straight upward with an initial velocity of 160 ft/sec <BR>\n" );
document.write( "from a height of 8 ft, then its height above the ground in feet at time t in seconds <BR>\n" );
document.write( "is given by the function h(t) = -16t^2 +160t + 8\r<BR>\n" );
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document.write( "1. What shape would this function make when graphed? How do you know?\r<BR>\n" );
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document.write( "2. Will this shape have an obvious minimum value, an obvious maximum value, neither, or both? How do you know?\r<BR>\n" );
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document.write( "3. Where in the graph would you find the “optimal” value (either highest or lowest value)? Find, through calculation, the t-coordinate for that “optimal” value.\r<BR>\n" );
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document.write( "4. Find, through calculation, the h-coordinate for the “optimal” value?\r<BR>\n" );
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document.write( "5. Interpret the results from Step 3 and Step 4. (I.e., what do those numbers mean?)<BR>\n" );
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document.write( "(1)  The shape is a parabola. I know it because the plot of every quadratic function \r\n" );
document.write( "     is a parabola.\r\n" );
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document.write( "(2)  This parabola has an obvious maximum. I know it, because the leading coefficient at t^2\r\n" );
document.write( "     is negative, and every quadratic function with the negative leading coefficient represents\r\n" );
document.write( "     a downward parabola, which has a maximum.\r\n" );
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document.write( "(3)  For the general formal quadratic function f(t) = at^2 + bt + c, the \"t-coordinate\" of its \r\n" );
document.write( "     optimum value is  t = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  which in this given case is  t = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-160%2F%282%2A%28-16%29%29 ALIGN=MIDDLE  ALT=\"-160%2F%282%2A%28-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=160%2F32 ALIGN=MIDDLE  ALT=\"160%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5 seconds.\r\n" );
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document.write( "(4)  The h-coordinate of the optimum is  h(5) = -16*5^2 + 160*5 + 8 = -400 + 800 + 8 = 408 ft.\r\n" );
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document.write( "(5)  These numbers mean that the arrow will reach its maximum height of 408 ft at t= 5 seconds \r\n" );
document.write( "     after the shoot.\r\n" );
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document.write( "Solved completely.\r<BR>\n" );
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