on the rim of the wheel (the rider). Roll the wheel away from the starting point (see figure below) and stop the wheel in at least 16 different positions, including points at which the rider is at the top of the wheel and bottom of the wheel.
At each position, measure the distance from the starting point to the bottom of the wheel (x) and the height (h) of the rider. Care should be taken to make sure that the wheel rolls in a straight line, and that the wheel makes at least one complete rotation. Repeat part 1 three more times, using different starting positions for the rider each time.
Create a table of values and plot the graph of the distance (x) vs. height (h) for each set of data. Which variable should be on the horizontal axis?
Is the graph sinusoidal? Draw a smooth curve through the points.
Using the general equation of the sine function, y = asin [b(x – c)] + d, list as many characteristics as possible (e.g. maxima, minima, symmetry, period, amplitude, phase shift, vertical translation, etc.)
Determine the values of a, b, c, and d in the equation
y = asin [b(x – c)] + d that models the data and matches the graph.
Recall
a = max - min
2
d = max + min
2
Period = 360°
|b|
Phase Shift: c > 0 → shift "c" units to the right.
c < 0 → shift "|c|" units to the left.
Once you have a, b, and d, substitute a point on the curve into the equation to get a value for c.
Determine the radius of the wheel. How does this value relate to the equation?
Find the circumference of the wheel. What is the significance of this value?
Describe how this model of a Ferris wheel differs from a real Ferris wheel. For this model, what does the value of d correspond to? For a real Ferris wheel, would the value of d correspond to the same thing? (Hint: A real Ferris wheel doesn't touch the ground!)
What is the significance of the value of c?
Using the same data, model the movement of the rider using the general cosine function y = acos [b(x – c)] + d. Answer questions 6 to 10 again, using this new equation for reference.
Compare your equations and graphs of the sine and cosine functions, stating any similarities and differences.
for the 2nd part, recall that sin(x) = cos(90°-x) ...
To answer these questions, we'll need to follow the given instructions step by step. Let's break it down:
1. Start by rolling the wheel away from the starting point and stopping in at least 16 different positions, measuring the distance (x) from the starting point to the bottom of the wheel, and the height (h) of the rider at each position. Ensure the wheel rolls in a straight line and completes at least one full rotation. Repeat this process three more times, each time starting from a different position.
2. Create a table of values with the distance (x) as the horizontal axis and height (h) as the vertical axis. Using these values, plot the graph of distance vs. height for each set of data.
3. Determine which variable should be on the horizontal axis. In this case, the distance (x) should be on the horizontal axis.
4. Examine the graph to determine if it is sinusoidal. A sinusoidal graph has a smooth curve that repeats itself. Draw a smooth curve through the points on the graph to see if it resembles a sine wave.
5. Using the general equation of a sine function, y = asin[b(x – c)] + d, list as many characteristics as possible such as maxima, minima, symmetry, period, amplitude, phase shift, vertical translation, etc. Analyze the graph to determine these characteristics.
6. Determine the values of a, b, c, and d in the equation y = asin[b(x – c)] + d that match the data and graph. Use the given formulas:
- Amplitude (a) can be calculated as the difference between the maximum and minimum values divided by 2.
- Vertical translation (d) can be calculated as the sum of the maximum and minimum values divided by 2.
- Period can be determined as 360 degrees divided by |b|.
- Phase shift (c) can be found based on the starting position of the wheel.
7. Determine the radius of the wheel. The value of the radius relates to the equation through the amplitude (a) value. Recall that the amplitude of a sine function corresponds to half the distance between the maximum and minimum values, which is equivalent to half the radius of a circle.
8. Find the circumference of the wheel. The significance of this value is that it represents the total distance traveled by the wheel in one complete rotation.
9. Describe how this model of a Ferris wheel differs from a real Ferris wheel. In this model, the value of d corresponds to the height of the rider when they are at the bottom of the wheel. However, in a real Ferris wheel, the value of d would represent the height of the center of the wheel rather than the ground.
10. Explain the significance of the value of c. The value of c in the equation represents the phase shift or how much the graph is horizontally shifted. A positive value of c shifts the graph to the right, while a negative value shifts it to the left.
11. Repeat Steps 5-10 using the cosine function y = acos[b(x – c)] + d to model the movement of the rider. Compare the equations and graphs of the sine and cosine functions to identify any similarities and differences.
Remember, it is important to analyze the data, graphs, and equations carefully to answer each question accurately.