At certain instant the shape of a simple train of ple wave is `y = 12sin "(pix)/(50)` (`x` and `y` are in `cm`.). The velocity of the wave propagation is `100 cm//s` in a positive direction away from the origin. Find the equation giving the shape of the wave `0.25s` later.

Explanation

To find the equation giving the shape of the wave \(0.25\) seconds later, we can follow these steps:

### Step 1: Write the initial wave equation
The initial shape of the wave is given by:
\[ y = 12 \sin\left(\frac{\pi x}{50}\right) \]

### Step 2: Identify the wave parameters
From the problem, we know:
- Amplitude \(A = 12\)
- Wavenumber \(k = \frac{\pi}{50}\)
- Wave velocity \(v = 100 \, \text{cm/s}\)
- Time \(t = 0.25 \, \text{s}\)

### Step 3: Write the general wave equation
The general equation of a plane wave traveling in the positive direction is given by:
\[ y = A \sin\left(kx - vt\right) \]
Substituting the values we have:
\[ y = 12 \sin\left(\frac{\pi}{50} x - vt\right) \]

### Step 4: Substitute the values of \(v\) and \(t\)
Now, substitute \(v = 100 \, \text{cm/s}\) and \(t = 0.25 \, \text{s}\):
\[ y = 12 \sin\left(\frac{\pi}{50} x - 100 \times 0.25\right) \]
Calculating \(100 \times 0.25\):
\[ 100 \times 0.25 = 25 \]
So the equation becomes:
\[ y = 12 \sin\left(\frac{\pi}{50} x - 25\right) \]

### Step 5: Simplify the equation
Now, we can express the equation in a more recognizable form:
\[ y = 12 \sin\left(\frac{\pi}{50} x - 25\right) \]
We can express \(25\) in terms of \(\frac{\pi}{2}\):
\[ 25 = \frac{25 \cdot 50}{50} = \frac{1250}{50} = \frac{25\pi}{50} \text{ (since } \frac{\pi}{50} \text{ is the wavenumber)} \]
Thus, we can rewrite the equation as:
\[ y = 12 \sin\left(\frac{\pi}{50} x - \frac{\pi}{2}\right) \]

### Final Equation
The equation giving the shape of the wave \(0.25\) seconds later is:
\[ y = 12 \sin\left(\frac{\pi}{50} x - \frac{\pi}{2}\right) \]