# Rollercoaster Calculus

1. Sep 7, 2006

### music_man05

The height of the tracks of a roller coaster varies sinusoidally with the horizontal distance from its starting point. The roller coaster tracks start from the lowest point of the sinusoidal curve and reach a maximum height of 66 meters. The tracks return to the surface of the park for the first time at 198 meters. What is the slope of the roller coaster track at 132 meters from the starting point?

slope is given by the derrivative.

y = sin x for the wave.

Therefore,

y= cos x

Anything beyond that is kinda fuzzy, any help would be appreciated.

2. Sep 7, 2006

### music_man05

Maybe if I throw some more thinking out there..I went this route....

since the max height is 66. the amplitude is 33. the wavelength is 198 =
lamda.

thus the equation for the wave is

y = amplitude sin (x/lamda * 2 pi)

y = 33 sin (x / 198 * 6.28 + 3 pi / 2 ) the 3 pi/2 is the minium point on
a sin curve.

the first derrivative would be :

y' = 33 cos ( x /198 * 6.28 + 3 pi / 2 ) * (6.28/198)

What do I solve for? is y' = 0?

3. Sep 7, 2006

### Andrew Mason

Height, y, is proportional to a sinusoidal function of x, not necessarily equal to sinx.

I think you have to assume that the bottom of the 'troughs' of this sine curve are at ground level. So ground level is, in effect, at maximum negative amplitude.

Write the general form of the equation for y (height above ground), which should contain constants for the amplitude (max. vertical distance from the middle of the sine curve), 'wavelength' and 'phase shift' of the sin curve.

How is the 66 metres related to the "amplitude"?

What does the 198 metre distance represent in terms of the 'wavelength'?

That should give you the equation for the track. Just plug in x=132 to determine y.

AM

4. Sep 7, 2006

### ertaius

I also have the...

I have the same type of problem, yet I still find myself confused. Are you talking about y=a*sin(b*x-c)+d. Where a, b, c, and d are the constants which modulate the sine curve? In this case my maximum height for the coaster is 70.6666666666667 meters(I'm not joking about the accuracy). So, I thought perhaps using the height for a, then for b I used 2pi divided by the horizontal distance, for c i used pi/2, and for d I used 1. Now if i remember correctly, taking the derivative would mean that constant d is completely arbitrary? Correct me if I'm wrong, I just need to know where do go from here to get the slope...or what I have to undo. Thanks for the help.

p.s. My other given numbers where horizontal distance of coaster =212m, and I'm supposed to find the slope at 141.3333333333337m( I seriously hate randomly generated numbers for problems).

5. Sep 7, 2006

### Andrew Mason

Use the chain rule to differentiate y = f(g(x)) where $f(x) = Asin(g(x))$ and $g(x) = \frac{2\pi}{\lambda}x + \phi$

AM