Bulk modulus of water is `22 xx 10^(9) N//m^(2).` If the average depth of an ocean is 2600 m, then fractional `((triangleV)/(V))` compression in volume of water at the bottom of ocean will be:

Options

A. 0.0113
B. 0.0226
C. 0.0056
D. 0.03

Explanation

To solve the problem of finding the fractional change in volume of water at the bottom of the ocean, we will follow these steps:

### Step 1: Understand the Bulk Modulus Formula
The bulk modulus \( K \) is defined as:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
where:
- \( K \) is the bulk modulus,
- \( P \) is the pressure applied,
- \( \frac{\Delta V}{V} \) is the fractional change in volume.

### Step 2: Calculate the Pressure at the Depth of the Ocean
To find the pressure \( P \) at a depth \( h \) in a fluid, we use the formula:
\[ P = \rho g h \]
where:
- \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the depth (given as \( 2600 \, \text{m} \)).

Substituting the values:
\[ P = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 2600 \, \text{m} \]
Calculating this gives:
\[ P = 1000 \times 9.81 \times 2600 = 2.553 \times 10^7 \, \text{N/m}^2 \]

### Step 3: Rearranging the Bulk Modulus Formula
From the bulk modulus formula, we can rearrange it to find \( \frac{\Delta V}{V} \):
\[ \frac{\Delta V}{V} = -\frac{P}{K} \]

### Step 4: Substitute the Values
Now we can substitute the values of \( P \) and \( K \) into the equation. The bulk modulus \( K \) of water is given as \( 22 \times 10^9 \, \text{N/m}^2 \).
\[ \frac{\Delta V}{V} = -\frac{2.553 \times 10^7}{22 \times 10^9} \]

### Step 5: Calculate the Fractional Change in Volume
Calculating the above expression:
\[ \frac{\Delta V}{V} = -\frac{2.553 \times 10^7}{22 \times 10^9} = -0.00116 \]
This means:
\[ \frac{\Delta V}{V} \approx -0.00116 \]

### Step 6: Convert to Percentage
To express this as a percentage, we multiply by 100:
\[ \frac{\Delta V}{V} \times 100 \approx -0.116\% \]

### Final Answer
The fractional change in volume of water at the bottom of the ocean is approximately \( 0.116\% \).

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