The wave-function for a certain standing wave on a string fixed at both ends is `y(x,t) = 0.5 sin (0.025pix) cos500t` where `x` and `y` are in centimeters and `t` is seconds. The shortest possible length of the string is :

Options

A. `126 cm`
B. `160 cm`
C. `40 cm`
D. `80 cm`

Explanation

To find the shortest possible length of the string given the wave function \( y(x,t) = 0.5 \sin(0.025 \pi x) \cos(500 t) \), we can follow these steps:

### Step 1: Identify the wave number \( k \)
The wave function can be expressed in the form:
\[ y(x,t) = A \sin(kx) \cos(\omega t) \]
By comparing this with the given wave function, we can see that:
\[ k = 0.025 \pi \]

### Step 2: Relate wave number to wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ 0.025 \pi = \frac{2\pi}{\lambda} \]

### Step 3: Solve for the wavelength \( \lambda \)
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{0.025 \pi} \]
The \( \pi \) cancels out:
\[ \lambda = \frac{2}{0.025} \]

### Step 4: Calculate \( \lambda \)
Calculating \( \lambda \):
\[ \lambda = \frac{2}{0.025} = 80 \text{ cm} \]

### Step 5: Find the shortest length of the string
For a standing wave fixed at both ends, the shortest length of the string \( L \) is given by:
\[ L = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ L = \frac{80}{2} = 40 \text{ cm} \]

### Conclusion
Thus, the shortest possible length of the string is:
\[ \boxed{40 \text{ cm}} \]