The density of water at the surface of ocean is `rho` . If the bulk modulus of water is `B`, then the density of ocean water at depth, when the pressure at a depth is `alphap_(0)` and `p_(0)` is the atmospheric pressure is

Options

A. `(rhoK)/(k-nP_(0))`
B. `(rhoK)/(K +nP_(0))`
C. `(rhoK)/(K-(n-1)P_(0))`
D. `(rhoK)/((K+(n-1)P_(0)))`

Explanation

To find the density of ocean water at a certain depth where the pressure is given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Density of water at the surface: \( \rho \)
- Bulk modulus of water: \( B \)
- Pressure at depth: \( \alpha P_0 \) (where \( P_0 \) is atmospheric pressure)

2. **Understand the Relationship**:
- The bulk modulus \( B \) is defined as:
\[ B = -V \frac{dP}{dV} \]
- This means that an increase in pressure causes a decrease in volume, which in turn increases density.

3. **Relate Initial and Final States**:
- Let \( V_1 \) be the initial volume and \( V_2 \) be the final volume at depth.
- The initial density \( \rho_1 \) is \( \rho \) and the final density \( \rho_2 \) can be expressed as:
\[ \rho_2 = \frac{\rho_1 V_1}{V_2} \]

4. **Calculate Change in Pressure**:
- The change in pressure \( \Delta P \) when going from the surface to depth is:
\[ \Delta P = \alpha P_0 - P_0 = (\alpha - 1) P_0 \]

5. **Express Change in Volume**:
- Using the definition of bulk modulus:
\[ B = V_1 \frac{(\alpha - 1) P_0}{\Delta V} \]
- Rearranging gives:
\[ \Delta V = V_1 \frac{(\alpha - 1) P_0}{B} \]

6. **Relate Final Volume to Initial Volume**:
- The final volume \( V_2 \) can be expressed as:
\[ V_2 = V_1 - \Delta V = V_1 - V_1 \frac{(\alpha - 1) P_0}{B} \]
- Simplifying gives:
\[ V_2 = V_1 \left(1 - \frac{(\alpha - 1) P_0}{B}\right) \]

7. **Substitute Back to Find Density**:
- Now substituting \( V_2 \) back into the density equation:
\[ \rho_2 = \frac{\rho V_1}{V_1 \left(1 - \frac{(\alpha - 1) P_0}{B}\right)} = \frac{\rho}{1 - \frac{(\alpha - 1) P_0}{B}} \]

8. **Final Expression**:
- Therefore, the density of ocean water at depth is:
\[ \rho_2 = \frac{\rho B}{B - (\alpha - 1) P_0} \]