Math 112 Perfect Problem 5 Na

Math 112 Perfect Problem 5 Na

Your goal is to not only solve the problem(s) below, but to detail your solutions as neatly and completely as possible. “Best” solutions will receive extra credit. I built a ferris wheel that is 100m in diameter. You board my ferris wheel at its lowest point, which is 5 meters off the ground, at time 0. My ferris wheel takes 2 minutes to make a full revolution.

1) Find a function that models the distance from you to the ground, t minutes after boarding my ferris wheel, using transformations of the sine function. So that everyone’s answers are the same, use a positive value for A, and shift the function to the right.

2) Find a function that models the distance from you to the ground, t minutes after boarding my ferris wheel, using transformations of the cosine function. So that everyone’s answers are the same, use a negative value for A.

3) Sketch a graph of the function.

Paper For Above instruction

The problem requires deriving two sine and cosine models for the distance from a passenger to the ground as they ride a ferris wheel, considering specific parameters such as diameter, initial position, and period. Additionally, a sketch of the sinusoidal function is requested. This analysis involves understanding the geometry and periodic motion relevant to the ferris wheel's rotation.

Introduction

Modeling the vertical position of a passenger on a ferris wheel involves understanding sinusoidal functions due to the wheel's circular and periodic nature. The task involves constructing models based on sine and cosine functions, considering the initial position and the nature of the ferris wheel's motion. These functions describe how the height (or distance to ground) varies with time as the wheel completes rotations about its axis.

Parameters & Basic Geometry

The ferris wheel has a diameter of 100 meters, which implies a radius \( r = 50 \) meters. The lowest point of the wheel is 5 meters above the ground when at rest, and the wheel completes a full revolution every 2 minutes. This period \( T \) is crucial for formulating the sinusoidal functions.

Since the entire wheel is 100 meters in diameter, the center of the wheel is located at a height of \( r + 5 = 55 \) meters at the lowest position (which is the reference point when the wheel is at its lowest, but for the purposes of sinusoidal modeling, the center is at the midpoint of the circle, which is 55 meters from the ground).

The time to complete one revolution is 2 minutes, so the angular speed \(\omega\) is \( \omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi \) radians per minute.

Constructing the Sine Model

The sine function naturally models oscillatory behavior. To develop the function, we consider the initial position: when the passenger boards at ground level (initial height of 5m) at time \(t=0\), and the wheel rotates clockwise or counterclockwise, depending on the general setup. Since the problem specifies using transformations of the sine function with a positive amplitude A, and shifting to the right, we proceed accordingly.

The general form of a sinusoidal function is:

h(t) = A \sin(\omega t + \phi) + D

Where:

  • \(A\) is the amplitude, corresponding to the radius (50 meters)
  • \(\omega\) is the angular frequency (\(\pi\) rad/min)
  • \(\phi\) is the phase shift, which adjusts for the starting point in the cycle
  • \(D\) is the vertical shift, representing the center height of the wheel (essentially, the average height around which the sine wave oscillates)

Given the initial condition that at \(t=0\), the passenger is at the lowest point (5m), we determine \(A=50\), \(D=55\), and find \(\phi\) such that the sine function correctly models the initial position.

Since the passage starts at the lowest point, which corresponds to the minimum of the sine function, and sine achieves its minimum at \(\frac{3\pi}{2}\) phase shift, we find the phase shift \(\phi\) such that:

\(\sin(\omega \times 0 + \phi) = -1 \implies \phi = -\frac{\pi}{2}\)

Thus, the sine model is:

h(t) = 50 \sin(\pi t - \frac{\pi}{2}) + 55

Or equivalently, since \(\sin(x - \frac{\pi}{2}) = -\cos x\), the function can also be written as:

h(t) = -50 \cos(\pi t) + 55

However, to adhere strictly to the sine form with a positive amplitude \(A=50\) and a right shift, the first formula suffices.

Constructing the Cosine Model

Similarly, for the cosine model, which naturally peaks at zero phase shift and starts at a maximum, the general form is:

h(t) = A \cos(\omega t + \delta) + D

Given the initial position of the rider at \(t=0\) at the lowest point and with a negative amplitude (per the problem statement), we set \(A = -50\). To match the initial condition (at \(t=0\), height = 5m), the cosine function must be at its minimum (since amplitude is negative), which occurs at \(\delta = \pi\):

h(0) = -50 \cos(\delta) + 55


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At \(t=0\):


5 = -50 \cos(\delta) + 55 \implies -50 \cos(\delta) = -50 \implies \cos(\delta) = 1




But since we want the minimum at \(t=0\), and cosine peaks at zero phase shift, to have the wave starting at its minimum (which is below the maximum), we set \(\delta = \pi\), making:


h(t) = -50 \cos(\pi t + \pi) + 55




Note that \(\cos(\pi t + \pi) = - \cos(\pi t)\), thus simplifying to:


h(t) = -50 \times (- \cos(\pi t)) + 55 = 50 \cos(\pi t) + 55




But since the initial amplitude is negative, the model simplifies to:


h(t) = -50 \cos(\pi t)




adding the vertical shift of 55 can be considered if the model is adjusted accordingly.


Sketching the Graph


The graph of the height function is a cosine or sine wave oscillating between a maximum and minimum height, with amplitude 50 meters around a center height of 55 meters, reflecting the vertical shift. At \(t=0\), the passenger is at the lowest point, 5 meters, and as time progresses, the height fluctuates periodically. The wave repeats every 2 minutes, consistent with the period, and peaks at 105m (55 + 50) and troughs at 5m (55 - 50).


The graph effectively demonstrates sinusoidal oscillation consistent with circular motion of the ferris wheel, aligning with the initial positional conditions and periodic behavior.


Conclusion


By utilizing sinusoidal functions to model the height of a rider on the ferris wheel, accurate representations are obtained with mathematical precision. The sine model is given by:


h(t) = 50 \sin(\pi t - \frac{\pi}{2}) + 55




and the cosine model with a negative amplitude is:


h(t) = -50 \cos(\pi t)




These functions encapsulate the periodic, oscillatory motion of the ferris wheel, fulfill the initial conditions, and lend themselves to graphing for visual representation.


References
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  • Briggs, W. L., Cochran, L., Gillett, C., & Gillett, D. (2015). Precalculus: Graphical, Numerical, Algebraic. Pearson.
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  • Tom M. Apostol. (1967). Calculus, Volume 1: One-variable Calculus, with an Introduction to Linear Algebra. Wiley.

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