document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=914493>Question 914493</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The path of a roller coaster is to be modelled by a piecewise function made of cubic functions<BR>\n" );
document.write( "It must have the following features<BR>\n" );
document.write( "-The roller coaster is a straight line roller coaster<BR>\n" );
document.write( "-Its total horizontal length is less the 200m<BR>\n" );
document.write( "-The track starts at 75m above the ground<BR>\n" );
document.write( "-The height must not reach greater than 75m at any given point<BR>\n" );
document.write( "-No ascent or descent should be greater than 80 degrees to the horizontal<BR>\n" );
document.write( "-The beginning and end of the roller coaster must have a zero incline<BR>\n" );
document.write( "-It must not go below ground<BR>\n" );
document.write( "-There must be a minimum of two descents </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #555476 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=555476\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I do not know what strategy is best or what strategy is expected.<BR>\n" );
document.write( "A straight line roller coaster can be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=f%28x%29 ALIGN=MIDDLE  ALT=\"y=f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= height above the ground, and<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;\" >= horizontal distance from the starting point.<BR>\n" );
document.write( "Making it a piecewise function made of cubic functions helps somewhat.<BR>\n" );
document.write( "The requirement for minimum (ground level, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=0 ALIGN=MIDDLE  ALT=\"y%3C=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "maximum height (no more than 75m, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=75 ALIGN=MIDDLE  ALT=\"y%3C=75\"   DISABLED_style= \"max-width:90%; height:auto;\" > ), and<BR>\n" );
document.write( "maximum slope (no more than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80%5Eo ALIGN=MIDDLE  ALT=\"80%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the horizontal, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28dy%2Fdx%29%3Ctan%2880%5Eo%29 ALIGN=MIDDLE  ALT=\"abs%28dy%2Fdx%29%3Ctan%2880%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "suggest that you are supposed to use calculus.<BR>\n" );
document.write( "Cubic functions can have one maximum and one minimum,<BR>\n" );
document.write( "as seen in the graphs of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E3-3x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E3-3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which slopes down from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=0 ALIGN=MIDDLE  ALT=\"f%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=-1 ALIGN=MIDDLE  ALT=\"f%281%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.5%2C1.5%2C-1.5%2C0.5%2C2x%5E3-3x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-0.5%2C1.5%2C-1.5%2C0.5%2C2x%5E3-3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-2x%5E3%2B3x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=-2x%5E3%2B3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.5%2C1.5%2C-0.5%2C1.5%2C-2x%5E3%2B3x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-0.5%2C1.5%2C-0.5%2C1.5%2C-2x%5E3%2B3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which does the opposite.<BR>\n" );
document.write( "Calculating the derivatives shows that the function's maxima and minima area at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In fact, I found those functions by starting with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and setting up equations making the derivatives zero at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and equations making the maximum and minimum zero, -1, or 1 as needed. (Luckily, the system of 4 linear equations that results is easy, because you immediately get <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=0 ALIGN=MIDDLE  ALT=\"d=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If we take the interval between maximum and minimum in those functions, dilate it as needed, and shift it up as needed, we can get the three pieces needed for the piecewise function.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to end at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=end%3C=200 ALIGN=MIDDLE  ALT=\"x=end%3C=200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A first piece would be an initial downslope (descent) from 75m to a minimum at or above ground.<BR>\n" );
document.write( "The second piece would be an upslope to a maximum no greater than 75m.<BR>\n" );
document.write( "The third piece would be a second downslope (descent) to a minimum at ground level.<BR>\n" );
document.write( "The beginning and end of each piece (and the whole ride) have a slope of zero by design.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The maximum limit for the slope is no problem.<BR>\n" );
document.write( "The second derivative of the function is zero for the point of maximum slope.<BR>\n" );
document.write( "For either of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E3%2Bbx%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E3%2Bbx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > functions above, the first and second derivatives are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=3ax%5E2%2B2bx ALIGN=MIDDLE  ALT=\"df%2Fdx=3ax%5E2%2B2bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2f%2Fdx%5E2=6ax%2B2b ALIGN=MIDDLE  ALT=\"d%5E2f%2Fdx%5E2=6ax%2B2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The maximum (or minimum) slope occurs when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2f%2Fdx%5E2=6ax%2B2b=0 ALIGN=MIDDLE  ALT=\"d%5E2f%2Fdx%5E2=6ax%2B2b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6ax=-2b ALIGN=MIDDLE  ALT=\"6ax=-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F3a ALIGN=MIDDLE  ALT=\"x=-b%2F3a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "At that point the slope is the value of the derivative,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%28-b%2F3a%29%5E2%2B2b%28-b%2F3a%29=b%5E2%2F3a-2b%5E2%2F3a=-b%5E2%2F3a ALIGN=MIDDLE  ALT=\"3a%28-b%2F3a%29%5E2%2B2b%28-b%2F3a%29=b%5E2%2F3a-2b%5E2%2F3a=-b%5E2%2F3a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The absolute value of that derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2F%283%2Aabs%28a%29%29=3%5E2%2F%283%2A2%29=3%2F2 ALIGN=MIDDLE  ALT=\"b%5E2%2F%283%2Aabs%28a%29%29=3%5E2%2F%283%2A2%29=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The maximum slope will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2=1.5 ALIGN=MIDDLE  ALT=\"3%2F2=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > adjusted by the dilation factors used.<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2880%5Eo%29=5.67 ALIGN=MIDDLE  ALT=\"tan%2880%5Eo%29=5.67\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded down), we have plenty of wiggle room.<BR>\n" );
document.write( "Having the roller coaster go from 75m to ground in 20m is a bit extreme,<BR>\n" );
document.write( "but it would mean a maximum downslope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5%2A%2875%2F20%29=5.625 ALIGN=MIDDLE  ALT=\"1.5%2A%2875%2F20%29=5.625\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and it would still be permissible.<BR>\n" );
document.write( " <BR>\n" );
document.write( "There are a lot of choices.<BR>\n" );
document.write( "You can make the first descent to go from 75m to 25m, or from 75 m to the ground, for example.<BR>\n" );
document.write( "That first descent could happen over a horizontal distance of 20 meters (from x=0 to x=20),<BR>\n" );
document.write( "or it could happen over 50 meters (from x=0 to x=50).<BR>\n" );
document.write( "Similarly, the first upslope could go up to a height of 25 meters, 50 meters, or even an unrealistic 75 meters.<BR>\n" );
document.write( "There is also a choice on how to get to the equations for each piece.<BR>\n" );
document.write( "You could start from scratch, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E3%2Bbx%5E2%2Bcx%2Bd ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E3%2Bbx%5E2%2Bcx%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "setting values for the function and derivative at the beginning and end of each piece, and find the coefficients a, b, c, and d.<BR>\n" );
document.write( "You could also use the functions given above, and just transform them by dilating and shifting them as needed.<BR>\n" );
document.write( " <BR>\n" );
document.write( "One possibility:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=red%280.0008x%5E3-0.06x%5E2%2B75%29 ALIGN=MIDDLE  ALT=\"f%28x%29=red%280.0008x%5E3-0.06x%5E2%2B75%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%280%3C=x%3C=50%29 ALIGN=MIDDLE  ALT=\"red%280%3C=x%3C=50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first descent,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=green%28-0.0004x%5E3%2B0.09x%5E2-6x%2B150%29 ALIGN=MIDDLE  ALT=\"f%28x%29=green%28-0.0004x%5E3%2B0.09x%5E2-6x%2B150%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%2850%3Cx%3C=100%29 ALIGN=MIDDLE  ALT=\"green%2850%3Cx%3C=100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the upslope, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=blue%280.0001x%5E3-0.045x%5E2%2B6x-200%29 ALIGN=MIDDLE  ALT=\"f%28x%29=blue%280.0001x%5E3-0.045x%5E2%2B6x-200%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28100%3Cx%3C=200%29 ALIGN=MIDDLE  ALT=\"blue%28100%3Cx%3C=200%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second descent.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-25%2C225%2C-10%2C90%2C0.0008x%5E3-0.06x%5E2%2B75%2C-0.0004x%5E3%2B0.09x%5E2-6x%2B150%2C0.0001x%5E3-0.045x%5E2%2B6x-200%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );