In a liquid with density `900 kg//m^(3)`, lonfitudinal waves with frequency `250 H_(Z)` are found to have wavelength `8.0 m`. Calculate the bulk modulus of the liquid.

Explanation

To calculate the bulk modulus of the liquid, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Density of the liquid, \( \rho = 900 \, \text{kg/m}^3 \)
- Frequency of the wave, \( f = 250 \, \text{Hz} \)
- Wavelength of the wave, \( \lambda = 8.0 \, \text{m} \)

2. **Calculate the Velocity of the Wave**:
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = f \cdot \lambda \]
Substituting the values:
\[ v = 250 \, \text{Hz} \times 8.0 \, \text{m} = 2000 \, \text{m/s} \]

3. **Use the Relationship Between Bulk Modulus, Density, and Velocity**:
The bulk modulus \( B \) can be calculated using the formula:
\[ B = \rho \cdot v^2 \]
where \( \rho \) is the density and \( v \) is the velocity of the wave.

4. **Substitute the Values to Find the Bulk Modulus**:
Now substitute the values of \( \rho \) and \( v \):
\[ B = 900 \, \text{kg/m}^3 \cdot (2000 \, \text{m/s})^2 \]
\[ B = 900 \cdot 4,000,000 \, \text{m}^2/\text{s}^2 \]
\[ B = 3,600,000,000 \, \text{N/m}^2 \]
\[ B = 3.6 \times 10^9 \, \text{N/m}^2 \]

5. **Final Answer**:
The bulk modulus of the liquid is \( 3.6 \times 10^9 \, \text{N/m}^2 \).