Design Roller Coaster

Question

Please Helpp! I need to design a roller coaster

Rollercoaster Criteria

A roller coaster is the graph of a function R(x) with domain [0,200] such that:

the roller coaster starts on the ground: R(0) = 0.

The graph of the roller coaster must be a function. There are no loops.

The length of the rollercoaster is 200m long 0 ≤ x ≤200

the maximum height of the roller coaster is 75 meters: R(x) ≤ 75 for all x ∈ [0,200].

the roller coaster does not go underground: R(x) ≥ 0 for all x ∈ [0,200].

the ride is smooth: R(x) is continuous everywhere on its domain.

the angle of steepest descent for the roller coaster is never more than 80 degrees (4π/9 radians).

At least three equations of different types of functions must be used to define the roller coaster : Logarithmic, exponential, trigonometric, rational, polynomial

Submission

A report that outlines the design of your rollercoaster.

Your rollercoaster must be defined Algebraically and Graphically.

All calculations should be neatly presented by hand or typed.

1. Graphical Component (

Meet the Criteria outlined above

Draw your graph in Desmos (or GeoGebra or similar graphing software).

The starting point for your roller coaster should be the origin

A good starting point is the Desmos calculator below

2. Algebraic component

The equation for the graph of the roller coaster is clearly listed as a piecewise function.

Identify the sections of your function that are increasing and decreasing.

Identify the Domain of your Function

Identify the Range of your Function

Identify the coordinates of local maxima and minima. These turning points may be located graphically. Calculations to identify the turning points are not required

3. Calculations are required to show:

The maximum height is 75m

The maximum angle in any section of descent is less than 80 degrees

The function is continuous. This involves substituting x values into the function, a sample calculation is shown below.

Answer 1

Hi Nyugen N.,

So sketch what you think might be an "exciting" roller coaster course. Then go back into your "grab bag" of possible functions, to select types that might provide that.

I suggest that roller coaster rides are more exciting fast, so boost your ride up to the 75 m initially, then "let it go" from there. sin, cos, rational functions (of x^-values) and polynomials provide vertical wiggles; the other functions are useful for matching the slopes between segments, perhaps (that prevents physical destruction of the car and the track!).

How do you match slopes between segments? You could combine functions as "wedges", e.g.: x/200*sin(x) + (200-x)/200*exp(-x^2) would combine an increasing "wedge" of sine function with a decreasing "wedge" of exp(-x^2) function, for example. (But that doesn't meet the design criteria, it's just for an example of type). You can see that this summed function would have the slope of the exp(-x^2) at x=0, and the slope of the sin(x) at x=200. If you then do a following "wedge" with sin(x) and some other function, in the same way, that would automatically have a non-jolting transition at x=200, since the total function would be just sin(x) at that point.

-- Cheers

-- Cheers, -- Mr. d.