Owen is jumping on a trampoline. When his feet hit the deck of the trampoline, the material depresses to a minimum height of 2cm. On average, Owen is reaching a maximum height of 200cm every 10 seconds. Determine the equation of a sinusoidal function that would model this situation, assuming Owen reaches his first maximum at 6 seconds.
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To determine the equation of a sinusoidal function that models this situation, we need to consider the amplitude, period, horizontal shift, and vertical shift of the function.
1. Amplitude:
The amplitude is the distance from the middle of the maximum height to the minimum height. In this case, the maximum height is 200cm, and the minimum height is 2cm. Therefore, the amplitude is half the difference between these two values, which is (200 - 2) / 2 = 98cm.
2. Period:
The period is the time it takes for one complete cycle of the function. In this case, Owen reaches his maximum height every 10 seconds. Therefore, the period is 10.
3. Horizontal Shift:
The horizontal shift is the amount of time it takes for the function to start repeating itself. We are given that Owen reaches his first maximum at 6 seconds. This means that the function is shifted 6 units to the right.
4. Vertical Shift:
The vertical shift represents the change in the vertical position of the function. In this case, Owen's feet depress to a minimum height of 2cm. Therefore, the function is shifted 2 units upwards.
Putting these factors together, the equation of the sinusoidal function that models this situation is:
f(t) = A * sin(B(t - C)) + D
where:
- A is the amplitude (98cm in this case)
- B is the frequency (2π divided by the period, which is 2π/10 = π/5)
- C is the horizontal shift (6 units to the right)
- D is the vertical shift (2 units upwards)
Therefore, the equation becomes:
f(t) = 98 * sin((π/5)(t - 6)) + 2
To determine the equation of a sinusoidal function that models Owen's jumping on a trampoline, we can use the standard form of a sinusoidal function:
y = A sin(B(x - C)) + D
Where:
- A represents the amplitude (half the vertical distance between the maximum and minimum points)
- B represents the period (the horizontal length it takes for one complete cycle)
- C represents the phase shift (the horizontal shift of the graph)
- D represents the vertical shift (the vertical displacement of the graph)
In this case, we are given the following information:
- The minimum height of the trampoline deck is 2cm, which corresponds to the graph's minimum point.
- The maximum height Owen reaches is 200cm, which corresponds to the graph's maximum point.
- The period of the function is 10 seconds since it takes Owen 10 seconds to complete one full cycle.
- Owen reaches his first maximum height at 6 seconds, which indicates a phase shift.
Let's use this information to determine the values of A, B, C, and D.
Amplitude (A): The amplitude is half the vertical distance between the maximum and minimum points. In this case, it is (200 - 2)/2 = 199/2 = 99.5cm.
Period (B): The period is the length it takes for one complete cycle. In this case, it is 10 seconds.
Phase shift (C): The phase shift indicates the horizontal shift of the graph. Owen reaches his first maximum at 6 seconds, so C = 6.
Vertical shift (D): The vertical shift represents the vertical displacement of the graph. In this case, the minimum height of the trampoline deck is 2cm, so D = 2.
Now we can plug these values into the equation to get the final result:
y = (99.5)sin((2π/10)(x - 6)) + 2
Simplifying further, we have:
y = 99.5sin((π/5)(x - 6)) + 2
Therefore, the equation of the sinusoidal function that models Owen's jumping on the trampoline is y = 99.5sin((π/5)(x - 6)) + 2.
2cm*200cm/10seconds*6seconds
=240seconds