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. `(Delta V)/(V) prop B`
B. `(Delta V)/(V) prop (1)/(B)`
C. `(Delta V)/(V) prop B^(2)`
D. `(Delta V)/(V) prop B^(-2)`

Explanation

To solve the problem regarding the relationship between the fractional change in volume \(\left(\frac{\Delta V}{V}\right)\) and the bulk modulus \(B\) for a constant hydraulic stress, we can follow these steps:

### Step 1: Understand the Definition of Bulk Modulus
The bulk modulus \(B\) is defined as the ratio of the change in pressure \(\Delta P\) to the fractional change in volume \(\frac{\Delta V}{V}\):
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
Here, \(\Delta P\) is the change in pressure applied to the object, and \(\Delta V\) is the change in volume.

### Step 2: Rearranging the Bulk Modulus Equation
We can rearrange the equation to express the fractional change in volume in terms of the bulk modulus:
\[ \frac{\Delta V}{V} = -\frac{\Delta P}{B} \]

### Step 3: Analyzing the Relationship
From the rearranged equation, we see that the fractional change in volume \(\frac{\Delta V}{V}\) is inversely proportional to the bulk modulus \(B\) when the change in pressure \(\Delta P\) is constant:
\[ \frac{\Delta V}{V} \propto -\frac{1}{B} \]

### Step 4: Expressing the Relationship
This can be expressed in a more simplified form:
\[ \frac{\Delta V}{V} \propto \frac{1}{B} \]
This indicates that as the bulk modulus increases, the fractional change in volume decreases for a constant hydraulic stress.

### Final Conclusion
Thus, we conclude that the relationship between the fractional change in volume and the bulk modulus for a constant hydraulic stress is:
\[ \frac{\Delta V}{V} \propto \frac{1}{B} \]