For a constant hydraulic stress on an object, the fractional change in the object's volume `((triangleV)/(V))` and its bulk modulus (b) are related as

Options

A. `(triangleV)/(V)propB`
B. `(triangleV)/(V)prop(1)/(B)`
C. `(triangleV)/(V)propB^2`
D. `(triangleV)/(V)propB^-2`

Explanation

To solve the problem, we need to establish the relationship between the fractional change in volume \(\frac{\Delta V}{V}\) and the bulk modulus \(B\) when a constant hydraulic stress is applied to an object.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Bulk Modulus (B)**: It is defined as the measure of a substance's resistance to uniform compression. Mathematically, it is given by:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
where \(\Delta P\) is the change in pressure and \(\frac{\Delta V}{V}\) is the fractional change in volume.

2. **Rearranging the Formula**:
- From the definition of bulk modulus, we can rearrange the equation to express the fractional change in volume:
\[ \frac{\Delta V}{V} = -\frac{\Delta P}{B} \]

3. **Considering Constant Hydraulic Stress**:
- When a constant hydraulic stress is applied, \(\Delta P\) remains constant. Thus, we can denote this constant pressure change as \(P\).
- Therefore, the equation simplifies to:
\[ \frac{\Delta V}{V} = -\frac{P}{B} \]

4. **Interpreting the Relationship**:
- The negative sign indicates that an increase in pressure (hydraulic stress) results in a decrease in volume, which is expected.
- This relationship shows that the fractional change in volume \(\frac{\Delta V}{V}\) is inversely proportional to the bulk modulus \(B\).

5. **Conclusion**:
- We conclude that the fractional change in volume is inversely proportional to the bulk modulus:
\[ \frac{\Delta V}{V} \propto -\frac{1}{B} \]
- Therefore, if the bulk modulus increases, the fractional change in volume decreases, and vice versa.

### Final Answer:
The relationship between the fractional change in volume \(\frac{\Delta V}{V}\) and the bulk modulus \(B\) is given by:
\[ \frac{\Delta V}{V} = -\frac{P}{B} \]
This indicates that \(\frac{\Delta V}{V}\) is inversely proportional to \(B\).