Question 914493: The path of a roller coaster is to be modelled by a piecewise function made of cubic functions
It must have the following features
-The roller coaster is a straight line roller coaster
-Its total horizontal length is less the 200m
-The track starts at 75m above the ground
-The height must not reach greater than 75m at any given point
-No ascent or descent should be greater than 80 degrees to the horizontal
-The beginning and end of the roller coaster must have a zero incline
-It must not go below ground
-There must be a minimum of two descents
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! I do not know what strategy is best or what strategy is expected.
A straight line roller coaster can be expressed as  , where
 = height above the ground, and
 = horizontal distance from the starting point.
Making it a piecewise function made of cubic functions helps somewhat.
The requirement for minimum (ground level, so  ),
maximum height (no more than 75m, so  ), and
maximum slope (no more than  to the horizontal, so  )
suggest that you are supposed to use calculus.
Cubic functions can have one maximum and one minimum,
as seen in the graphs of  , which slopes down from  to  
and   , which does the opposite.
Calculating the derivatives shows that the function's maxima and minima area at  and  .
In fact, I found those functions by starting with  ,
and setting up equations making the derivatives zero at and 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  and  .
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.
to end at 
A first piece would be an initial downslope (descent) from 75m to a minimum at or above ground.
The second piece would be an upslope to a maximum no greater than 75m.
The third piece would be a second downslope (descent) to a minimum at ground level.
The beginning and end of each piece (and the whole ride) have a slope of zero by design.
The maximum limit for the slope is no problem.
The second derivative of the function is zero for the point of maximum slope.
For either of the  functions above, the first and second derivatives are
 and  .
The maximum (or minimum) slope occurs when
 ---> ---> .
At that point the slope is the value of the derivative,
 .
The absolute value of that derivative is  .
The maximum slope will be  adjusted by the dilation factors used.
Since  (rounded down), we have plenty of wiggle room.
Having the roller coaster go from 75m to ground in 20m is a bit extreme,
but it would mean a maximum downslope of  ,
and it would still be permissible.
There are a lot of choices.
You can make the first descent to go from 75m to 25m, or from 75 m to the ground, for example.
That first descent could happen over a horizontal distance of 20 meters (from x=0 to x=20),
or it could happen over 50 meters (from x=0 to x=50).
Similarly, the first upslope could go up to a height of 25 meters, 50 meters, or even an unrealistic 75 meters.
There is also a choice on how to get to the equations for each piece.
You could start from scratch, with ,
setting values for the function and derivative at the beginning and end of each piece, and find the coefficients a, b, c, and d.
You could also use the functions given above, and just transform them by dilating and shifting them as needed.
One possibility:
 for  for the first descent,
 for  for the upslope, and
 for  for the second descent.
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