The equation `y=5sin((pix)/(25))cos(450t)` represents the stationary wave in a vibrating sonometer wire, where `x,y` are in cm and t in sec. The distances of `2nd` and `3rd` nodes from one end are (in cm).

Options

A. `50,75`
B. `25,50`
C. `15,50`
D. `20,50`

Explanation

To find the distances of the 2nd and 3rd nodes from one end in the stationary wave represented by the equation \( y = 5 \sin\left(\frac{\pi x}{25}\right) \cos(450t) \), we will follow these steps:

### Step 1: Understand the standing wave equation
The standing wave equation is given as:
\[ y = 5 \sin\left(\frac{\pi x}{25}\right) \cos(450t) \]
Here, the term \( 5 \sin\left(\frac{\pi x}{25}\right) \) represents the amplitude of the wave, which varies with position \( x \).

### Step 2: Identify the condition for nodes
Nodes are points where the amplitude of the wave is zero. This occurs when:
\[ \sin\left(\frac{\pi x}{25}\right) = 0 \]

### Step 3: Solve for \( x \)
The sine function is zero at integer multiples of \( \pi \):
\[ \frac{\pi x}{25} = n\pi \quad \text{where } n = 0, 1, 2, 3, \ldots \]
Dividing both sides by \( \pi \):
\[ \frac{x}{25} = n \quad \Rightarrow \quad x = 25n \]

### Step 4: Calculate the positions of the nodes
Now, we can find the positions of the nodes by substituting values of \( n \):
- For \( n = 0 \) (1st node):
\[ x = 25 \times 0 = 0 \, \text{cm} \]
- For \( n = 1 \) (2nd node):
\[ x = 25 \times 1 = 25 \, \text{cm} \]
- For \( n = 2 \) (3rd node):
\[ x = 25 \times 2 = 50 \, \text{cm} \]

### Final Answer
The distances of the 2nd and 3rd nodes from one end are:
- 2nd node: 25 cm
- 3rd node: 50 cm