Question 1177187: 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.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down how to model Owen's trampoline jumping with a sinusoidal function.
**Understanding the Information**
* **Minimum Height:** 2 cm (This is the vertical shift or midline of the function)
* **Maximum Height:** 200 cm (This helps us find the amplitude)
* **Period:** 10 seconds (Time for one complete jump cycle)
* **First Maximum:** 6 seconds (This indicates a horizontal shift or phase shift)
**Building the Sinusoidal Function**
We'll use the general form of a sinusoidal function:
```
h(t) = A * sin(B(t - C)) + D
```
Where:
* **h(t):** Height of Owen at time t
* **A:** Amplitude (half the difference between maximum and minimum height)
* **B:** Determines the period (Period = 2π/B)
* **C:** Horizontal shift (phase shift)
* **D:** Vertical shift (midline)
**Calculations**
1. **Amplitude (A):**
* A = (Maximum Height - Minimum Height) / 2
* A = (200 cm - 2 cm) / 2 = 99 cm
2. **Period (B):**
* Period = 10 seconds
* B = 2π / Period = 2π / 10 = π/5
3. **Horizontal Shift (C):**
* Owen reaches his first maximum at 6 seconds. Since the sine function starts at its midline and increases, we need to shift it to the right by 6 seconds.
* C = 6 seconds
4. **Vertical Shift (D):**
* D = Minimum Height + Amplitude
* D = 2 cm + 99 cm = 101 cm
**Equation of the Sinusoidal Function**
Putting it all together, the equation that models Owen's jumping is:
```
h(t) = 99 * sin((π/5)(t - 6)) + 101
```
**Interpretation**
* This function models Owen's height (in cm) above the ground at any time t (in seconds).
* The amplitude of 99 cm represents how high Owen jumps above the midline.
* The period of 10 seconds represents the time it takes for one complete jump cycle.
* The horizontal shift of 6 seconds represents the time it takes for Owen to reach his first maximum height.
* The vertical shift of 101 cm represents the average height of Owen's jumps.
Let me know if you'd like to explore any variations of this scenario or have any other questions!
Answer by ikleyn(52834)  (Show Source):
You can put this solution on YOUR website! .
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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The answer in the post by @CPhill is INCORRECT.
The function h(t) = 99*sin((π/5)(t-6)) + 101 in the post by @CPhill
does not satisfy the condition " Owen reaches his first maximum at 6 seconds ".
A correct answer, satisfying all problem's conditions, is
h(t) = 99*cos((π/5)(t-6)) + 101.
Although this formula contains the cosine function, nevertheless,
it belongs to the class of so called " sinusoidal functions ".
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Regarding the post by @CPhill . . .
Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.
The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.
Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.
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