An cube is shifted to a depth of `100 m` is a lake. The change in volume is `0.1%`. The bulk modulus of the material is

Options

A. `10Pa`
B. `10^(4)Pa`
C. `10^(7)Pa`
D. `10^(9)Pa`

Explanation

To find the bulk modulus of the material when a cube is submerged to a depth of 100 m in a lake, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Pressure at Depth**:
The pressure at a depth in a fluid can be calculated using the formula:
\[ P = \rho g h \]
where:
- \(\rho\) is the density of the fluid (for water, \(\rho \approx 1000 \, \text{kg/m}^3\)),
- \(g\) is the acceleration due to gravity (\(g \approx 10 \, \text{m/s}^2\)),
- \(h\) is the depth (in this case, \(h = 100 \, \text{m}\)).

Plugging in the values:
\[ P = 1000 \, \text{kg/m}^3 \times 10 \, \text{m/s}^2 \times 100 \, \text{m} = 10^6 \, \text{Pa} \]

2. **Determine the Change in Volume**:
The change in volume is given as a percentage:
\[ \frac{\Delta V}{V} = 0.1\% = \frac{0.1}{100} = 0.001 \]

3. **Use the Bulk Modulus Formula**:
The bulk modulus \(B\) is defined as:
\[ B = -\frac{P}{\frac{\Delta V}{V}} \]
Substituting the values we have:
\[ B = -\frac{10^6 \, \text{Pa}}{0.001} \]

4. **Calculate the Bulk Modulus**:
\[ B = -10^6 \, \text{Pa} \times 1000 = 10^9 \, \text{Pa} \]

Thus, the bulk modulus of the material is:
\[ B = 10^9 \, \text{Pa} \]

### Final Answer:
The bulk modulus of the material is \(10^9 \, \text{Pa}\).

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