As you ride a bicycle, the distance between your foot and the pavement varies sinusoidally with the horizontal distance the bicycle has gone. Suppose that you start with your right foot somewhere between a high point and a low point, and push down. When you have gone 7 meters, your right foot first reaches its lowest point, 11 centimeters above the pavement. The high point are 45 centimeters above the pavement. The bicycle moves a horizontal distance of 20 meters for each complete revolution of the pedals.
a.)Sketch and label a graph showing exactly two periods of your data beginning at t=0.
b.) Determine a function for your data using the cosine function
c.)Determine a function for your data using the sine function
a.)
The graph would look like a sinusoidal wave, with the x-axis representing the horizontal distance the bicycle has gone and the y-axis representing the distance between the foot and the pavement. The graph would start at (0, 45 cm) and end at (20 m, 45 cm).
b.)
The cosine function for this data would be y = 45 cos(2πx/20) + 11, where x is the horizontal distance the bicycle has gone and y is the distance between the foot and the pavement.
c.)
The sine function for this data would be y = 45 sin(2πx/20) + 11, where x is the horizontal distance the bicycle has gone and y is the distance between the foot and the pavement.
a.) In order to sketch the graph showing two periods of the given data, we need to understand the nature of the sinusoidal function that represents the distance between the foot and the pavement.
From the given information, we know that the distance between the foot and the pavement varies sinusoidally and reaches its lowest point at 7 meters, where it is 11 centimeters above the pavement. It also reaches its highest point at some interval where it is 45 centimeters above the pavement.
A sinusoidal function can be represented as:
y = A*cos(B(x - C)) + D
where:
A represents the amplitude of the graph (the maximum distance between the foot and the pavement),
B represents the frequency (number of cycles per unit distance),
C represents the horizontal shift of the graph,
D represents the vertical shift of the graph.
In this case, the amplitude is (45 cm - 11 cm)/2 = 17 cm, as the maximum distance from the high point to the low point is 45 cm - 11 cm = 34 cm, and we take half of that value. The frequency is given as 1 complete revolution per 20 meters, which can be converted to 1 cycle per 20 meters, or 1 cycle per 2π radians. Therefore, B = 2π/20. The horizontal shift, C, is 0, as the given information does not mention any horizontal shift. Lastly, the vertical shift, D, is 28 cm, as the highest point is 45 cm and the lowest point is 11 cm above the pavement, meaning the average height is (45 cm + 11 cm)/2 = 28 cm.
b.) Using the cosine function, we can write the function as:
y = 17*cos(2π/20*x) + 28
c.) Using the sine function, we can write the function as:
y = 17*sin(2π/20*x) + 28
Note: The vertical shift, D, is positive in this case because the foot initially starts between the high and low point, closer to the high point.
a.) To sketch the graph, we need to understand the characteristics of the data. The given information tells us that the distance between the foot and the pavement varies sinusoidally with the horizontal distance the bicycle has traveled. We are given the amplitude, the lowest point, and the highest point.
The amplitude of the graph is half the difference between the highest and lowest points, which is (45 - 11) / 2 = 17 centimeters. This means the graph oscillates between 17 cm above and 17 cm below the midline.
Since the bicycle moves a horizontal distance of 20 meters for each complete revolution of the pedals, we know that each period of the cosine function corresponds to a distance of 20 meters. Therefore, we can divide the horizontal distance into four equal parts, each representing 5 meters.
With this information, we can sketch the graph starting from the midline and moving upwards and downwards, reaching the highest and lowest points at the appropriate distances.
The graph would look like a wave, oscillating between 17 cm above and 17 cm below the midline over a distance of 20 meters. It would repeat this pattern for each period. Label the x-axis as horizontal distance (in meters) and the y-axis as foot-pavement distance (in centimeters).
b.) To determine a function for the data using the cosine function, we can use the general equation:
y = A * cos(B(x - C)) + D,
where A represents the amplitude, B represents the frequency, C represents the horizontal shift, and D represents the vertical shift.
In this case, we know the amplitude is 17 cm, so A = 17.
The frequency of the cosine function is the number of cycles (periods) that occur over a given distance. Since each period corresponds to a distance of 20 meters, we can calculate the frequency using the formula:
frequency = 2 * π / period,
where period is the length of one cycle. In this case, the period is 20 meters, so the frequency is:
frequency = 2 * π / 20 = π / 10.
So, B = π / 10.
The horizontal shift, C, indicates at what horizontal distance the graph starts. Since the graph starts at t = 0, C = 0.
The vertical shift, D, indicates the midline of the graph. In this case, the midline is 11 cm above the pavement. However, the cosine function has a midline at y = 0. To adjust for this, we need to shift the graph vertically by 11 cm downwards. Therefore, D = -11.
Putting all the values together, the function for the data using the cosine function is:
y = 17 * cos((π/10)x) - 11.
c.) To determine a function for the data using the sine function, we can use a similar approach as in part b.
The equation for a sine function is:
y = A * sin(B(x - C)) + D.
Using the same values for A, B, and C as in part b, we can focus on the vertical shift, D.
In this case, the sine function has a midline at y = 0, so the vertical shift, D, will be the same as the midline of the data.
The midline of the data is 11 cm above the pavement. Therefore, D = 11.
Putting all the values together, the function for the data using the sine function is:
y = 17 * sin((π/10)x) + 11.