Question 1177187
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!