Application of trig. functions question?

Question

Say you have a roller coaster 250 ft horizontally, and it reaches the max. twice and its a sine graph. Say the amplitude is 200ft and the midline is 100ft.

In this question, are you allowed to replace 250ft with 2pi? (the period length of a sin or cos) How would you find the height at 1/8 of the distance?

Answer 1

The sine function has a general form y = Asin(Bx - C) + D. The A drives the amplitude, the B deals with the period, the C deals with the horizontal shift of the graph, and the D deals with the vertical shift. We have a lot of information given already - let's try to make sense of it all.

We know that the roller coaster stretches 250 feet, and it peaks twice. That implies that there are two complete sine waves here, so one sine wave completes at 125 feet (so you're really using 125 as your period, not 250). The period factor B is found by B = 2pi/P, so we're looking at B = 2pi/125.

We also know that the roller coaster peaks at 200 feet, so A = 200. The midline is at 100 feet, so D = 100. There is no mention of a horizontal shift, so we can assume that C = 0.

Now we can write our expression for the roller coaster:

y = 200sin(2*pi*x/125) + 100

Now just for clarification, if you meant that the amplitude is 200 feet from the midline (that's the way it's normally stated anyway), then the above equation is correct. But if you mean the amplitude to be the distance above the ground, then A has to be 100 instead, so that at x = 31.25 feet, the roller coaster peaks at 200 feet. Using this assumption, picture a roller coaster that starts at 100 feet up when you board it, and then climbs to 200 feet, and then drops to the ground and does it all over again. If we stick to the equation above, then you're starting at 100 feet, climbing to 300 feet, and going underground 100 feet before climbing again and starting over. So depending on which version of a roller coaster we're after, A is either 200 or 100.

Now, to find the height after 1/8 of the distance, just plug in to x and solve for y. The total distance is 250 ft, so 1/8 of that would be 31.25 feet. Plugging in to our original equation (the one where A = 200), we have:

y = 200sin(2*pi*x/125) + 100
y = 200sin(2*pi*31.25/125) + 100
y = 200sin(pi/2) + 100
y = 200 + 100
y = 300 feet

Again, make adjustments for the interpretation of "amplitude" of the roller coaster and you'll either get this or 200 feet.