document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1068471>Question 1068471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please can somebody help me with this please it is due tomorrow!! Please!! \r<BR>\n" );
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document.write( "The segment of your coaster ride must have at least 3 relative maxima and/or<BR>\n" );
document.write( "minima.The coaster ride starts and ends at ground level.The ride dives	below the ground into a tunnel at least once,\r<BR>\n" );
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document.write( "1. Create a roller coaster function R(t), where t is measured in seconds and R(t),	in feet,<BR>\n" );
document.write( "measures the	height	 of the	 roller	 coaster at time t. R(t)	 must meet the minimum	<BR>\n" );
document.write( "Requirements listed above.	\r<BR>\n" );
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document.write( "2. Create a sketch of this function using technology.	\r<BR>\n" );
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document.write( "3. List ALL intervals where your graph is increasing	and/or	decreasing.\r<BR>\n" );
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document.write( "4. What are the average rates of change (average	speed)	of the roller coaster during each	<BR>\n" );
document.write( "Descent? Which interval is the	‘scariest’? Why?	\r<BR>\n" );
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document.write( "5. How	 long is the entire ride of the roller coaster segment	that you have created?\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #683702 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=683702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem cones up every year.<BR>\n" );
document.write( "You may be able to search this site<BR>\n" );
document.write( "(or the internet in general) for ideas.<BR>\n" );
document.write( " <BR>\n" );
document.write( "TIPS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28%22WARNING+%3A%22%29 ALIGN=MIDDLE  ALT=\"red%28%22WARNING+%3A%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > In the examples below I used t in minutes, and your problem requests that you use t in seconds. You will have to think on humour own (not a bad idea) because you will not be able to use my work, but the explanations still work.<BR>\n" );
document.write( "For parts 1 and 2:<BR>\n" );
document.write( "You cannot be expected to design a realistic roller coaster.<BR>\n" );
document.write( "Ideally, I would design one as a piece-wise function,<BR>\n" );
document.write( "but a polynomial would do for the expected unrealistic coaster design.<BR>\n" );
document.write( "The points at ground level would be zeros of the function,<BR>\n" );
document.write( "So, if you want the ride to be 5 minutes, with t in minutes,<BR>\n" );
document.write( "you need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%280%29=0 ALIGN=MIDDLE  ALT=\"R%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%285%29=0 ALIGN=MIDDLE  ALT=\"R%285%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That can be achieved by having<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29 ALIGN=MIDDLE  ALT=\"%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as factors<BR>\n" );
document.write( "of your polynomial function.<BR>\n" );
document.write( "You also need zeros at the point where you want the ride<BR>\n" );
document.write( "to dip below ground, and<BR>\n" );
document.write( "to resurface again.<BR>\n" );
document.write( "For example,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29=-t%28t-2%29%28t-3%29%28t-5%29 ALIGN=MIDDLE  ALT=\"R%28t%29=-t%28t-2%29%28t-3%29%28t-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=t%3C=5 ALIGN=MIDDLE  ALT=\"0%3C=t%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "would give you a 5 minute ride that<BR>\n" );
document.write( "starts and ends at ground level,<BR>\n" );
document.write( "dips below ground between 2 and 3 minutes into the ride,<BR>\n" );
document.write( "has a maximum somewhere between 0 and 2 minutes,<BR>\n" );
document.write( "a minimum somewhere between 2 and 3 minutes,<BR>\n" );
document.write( "and a maximum somewhere between 3 and 5 minutes.<BR>\n" );
document.write( "The graph of the function above looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C6%2C-5%2C15%2C-x%2A%28x-2%29%2A%28x-3%29%2A%28x-5%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C6%2C-5%2C15%2C-x%2A%28x-2%29%2A%28x-3%29%2A%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I gave it a minus sign because<BR>\n" );
document.write( "a degree 4 polynomial with a positive leading coefficient<BR>\n" );
document.write( "would be positive at its left end,<BR>\n" );
document.write( "but would turn negative at its smallest zero,<BR>\n" );
document.write( "and I did not want the ride to go underground right after it started.<BR>\n" );
document.write( "Use technology (software or a graphing calculator)<BR>\n" );
document.write( "to graph your first attempt function,<BR>\n" );
document.write( "and change the function,and re-graph until<BR>\n" );
document.write( "you are happy with the result.<BR>\n" );
document.write( "You can move the middle zeros around to change the shape.<BR>\n" );
document.write( "For example, using 2 and 3.5 as the middle zeros,<BR>\n" );
document.write( "You get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29=-t%28t-2%29%28t-3.5%29%28t-5%29 ALIGN=MIDDLE  ALT=\"R%28t%29=-t%28t-2%29%28t-3.5%29%28t-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C6%2C-5%2C15%2C-x%2A%28x-2%29%2A%28x-3.5%29%2A%28x-5%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C6%2C-5%2C15%2C-x%2A%28x-2%29%2A%28x-3.5%29%2A%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can make the ride shorter.<BR>\n" );
document.write( "You can have more zeros, and/or a polyomial of higher degree.<BR>\n" );
document.write( "You can add zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3C0 ALIGN=MIDDLE  ALT=\"t%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can add zeros after the end of the run.<BR>\n" );
document.write( "You can n make some zeros double zeros,<BR>\n" );
document.write( "as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29=t%28t-2%29%28t-3.5%29%28t-5%29%5E2 ALIGN=MIDDLE  ALT=\"R%28t%29=t%28t-2%29%28t-3.5%29%28t-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C6%2C-20%2C80%2Cx%2A%28x-2%29%2A%28x-3.5%29%2A%28x-5%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C6%2C-20%2C80%2Cx%2A%28x-2%29%2A%28x-3.5%29%2A%28x-5%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For part 3:<BR>\n" );
document.write( "Use your graph Hong,technology to find out the t values<BR>\n" );
document.write( "for the maxima and minima.<BR>\n" );
document.write( "In between those points,<BR>\n" );
document.write( "and between them and the beginning (t=0),<BR>\n" );
document.write( "and the end of the ride,<BR>\n" );
document.write( "the function/graph is increasing or decreasing.\r<BR>\n" );
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document.write( "For example, in the graph above,<BR>\n" );
document.write( "the maxima and minima happen at about<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0.646 ALIGN=MIDDLE  ALT=\"t=0.646\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2.624 ALIGN=MIDDLE  ALT=\"t=2.624\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4.126 ALIGN=MIDDLE  ALT=\"t=4.126\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The function<BR>\n" );
document.write( "increases in the internal [0, 0.646],<BR>\n" );
document.write( "decreases in [0.646,2.624],<BR>\n" );
document.write( "increases in [2.624,4.126], and<BR>\n" );
document.write( "decreases in [4.126,5].<BR>\n" );
document.write( "Part 4:<BR>\n" );
document.write( "For the graph above,<BR>\n" );
document.write( "in the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.624-0.646=1.978 ALIGN=MIDDLE  ALT=\"2.624-0.646=1.978\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes<BR>\n" );
document.write( "between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0.646 ALIGN=MIDDLE  ALT=\"t=0.646\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2.624 ALIGN=MIDDLE  ALT=\"t=2.624\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "The height goes<BR>\n" );
document.write( "from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%280.646%29=47.3251 ALIGN=MIDDLE  ALT=\"R%280.646%29=47.3251\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%282.624%29=-8.0794 ALIGN=MIDDLE  ALT=\"R%282.624%29=-8.0794\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is a drop of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=46.3251-%28-8.0974%29=55.4215 ALIGN=MIDDLE  ALT=\"46.3251-%28-8.0974%29=55.4215\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet<BR>\n" );
document.write( "in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.978 ALIGN=MIDDLE  ALT=\"1.978\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes for the first descent.<BR>\n" );
document.write( "That is an average rate of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=55.4125%2F1.978=28.02 ALIGN=MIDDLE  ALT=\"55.4125%2F1.978=28.02\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet per minute(rounded).<BR>\n" );
document.write( "The other descent goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%284.126%29=4.1946 ALIGN=MIDDLE  ALT=\"R%284.126%29=4.1946\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%280%29=0 ALIGN=MIDDLE  ALT=\"R%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "A drop of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.1946 ALIGN=MIDDLE  ALT=\"4.1946\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.874 ALIGN=MIDDLE  ALT=\"0.874\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes.<BR>\n" );
document.write( "That is an average rate of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.1946%2F0.874=4.80 ALIGN=MIDDLE  ALT=\"4.1946%2F0.874=4.80\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet per minute (rounded).<BR>\n" );
document.write( "Obviously the first descent is art a faster rate<BR>\n" );
document.write( "(and longer in time, to boot),<BR>\n" );
document.write( "so the first descent is scarier. </SPAN>\n" );
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