Find the sine waves and points representing the following platforms:

• The platform moves between (2,5) and (12,5) with a pulsation of 2 Pi /10

The platform moves horizontally between the points \((2, 5)\) and \((12, 5)\) with a pulsation of \( \omega = \frac{2\pi}{10} \).

### Step 1: Understanding the problem
The platform is moving horizontally along the x-axis while staying at \( y = 5 \). The x-coordinates oscillate between 2 and 12, which suggests a sinusoidal motion for the x-position over time \( t \). The midpoint of this motion is the average of the x-coordinates, and the amplitude is half the distance between them.

### Step 2: Determining the parameters of the sine wave

- **Midpoint of motion** (the equilibrium position) is the average of 2 and 12:
  \[
  \text{Midpoint} = \frac{2 + 12}{2} = 7
  \]

- **Amplitude** (half the distance between the two extreme points):
  \[
  \text{Amplitude} = \frac{12 - 2}{2} = 5
  \]

- **Pulsation** \( \omega = \frac{2\pi}{10} \) gives the angular frequency of the sine wave. This tells us the period of oscillation is \( T = \frac{2\pi}{\omega} = 10 \) units of time.

### Step 3: Writing the sine function

The general form of the sine function for horizontal motion is:
\[
x(t) = \text{Midpoint} + \text{Amplitude} \cdot \sin(\omega t)
\]

Substitute the values:
\[
x(t) = 7 + 5 \sin\left( \frac{2\pi}{10} t \right)
\]

Since the vertical position is constant at \( y = 5 \), the complete parametric equations for the platform's motion are:
\[
x(t) = 7 + 5 \sin\left( \frac{2\pi}{10} t \right)
\]
\[
y(t) = 5
\]

### Step 4: Points representing the platform

The position of the platform at any time \( t \) is given by the coordinates \( (x(t), y(t)) \), where \( x(t) \) follows the sinusoidal motion as described, and \( y(t) \) is fixed at 5.

### Summary
The sine wave that describes the platform's horizontal motion is:
\[
x(t) = 7 + 5 \sin\left( \frac{2\pi}{10} t \right)
\]
and the vertical position is constant at \( y = 5 \).

### Do you want details or have any questions?

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Here are some related questions to expand your understanding:
1. How does changing the amplitude affect the platform's motion?
2. What happens if the pulsation \( \omega \) is doubled?
3. Can you find the velocity of the platform as a function of time?
4. What would the sine wave look like if the platform moved vertically instead of horizontally?
5. How would you modify the equation if the platform started from the point \((12, 5)\) at \(t = 0\)?

**Tip:** The angular frequency \( \omega \) is inversely related to the period \( T \). Increasing \( \omega \) makes the wave oscillate faster!

Find the sine waves and points representing the following platforms: The platform moves between (2,5) and (12,5) with a pulsation of 2 Pi /10.

Explore the motion of a platform oscillating horizontally between points (2,5) and (12,5) with a pulsation of 2π/10. This problem covers trigonometric concepts such as sine waves, parametric equations, and periodic motion. Suitable for high school students in Grades 9-12.

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