The pressure increases by `1.0xx10^(4)N//m^(2)` for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex glass cube of edge length `1.0xx10^(-2)m` at the ocean's surface decrease by `1.0xx10^(-10)m^(3)` ? Bulk modulus of Pyrex glass is `2.6xx10^(10)N//m^(3)`

Options

A. `10.24xx10^(3)` inch
B. 260 m
C. 853 feet
D. 0.161 mile

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We need to find the depth at which the volume of a Pyrex glass cube decreases by \(1.0 \times 10^{-10} \, m^3\). The cube has an edge length of \(1.0 \times 10^{-2} \, m\), and the bulk modulus of Pyrex glass is given as \(2.6 \times 10^{10} \, N/m^2\). The pressure increases by \(1.0 \times 10^{4} \, N/m^2\) for every meter of depth.

### Step 2: Calculate the Original Volume of the Cube
The volume \(V\) of a cube is given by:
\[ V = \text{edge length}^3 \]
Substituting the edge length:
\[ V = (1.0 \times 10^{-2} \, m)^3 = 1.0 \times 10^{-6} \, m^3 \]

### Step 3: Use the Bulk Modulus Formula
The bulk modulus \(K\) is defined as:
\[ K = -\frac{P}{\Delta V / V} \]
Where:
- \(P\) is the change in pressure,
- \(\Delta V\) is the change in volume,
- \(V\) is the original volume.

Rearranging gives:
\[ P = K \cdot \frac{\Delta V}{V} \]

### Step 4: Substitute Known Values
We know:
- \(\Delta V = 1.0 \times 10^{-10} \, m^3\),
- \(V = 1.0 \times 10^{-6} \, m^3\),
- \(K = 2.6 \times 10^{10} \, N/m^2\).

Substituting these values into the equation:
\[ P = 2.6 \times 10^{10} \cdot \frac{1.0 \times 10^{-10}}{1.0 \times 10^{-6}} \]

### Step 5: Calculate Pressure Change
Calculating the fraction:
\[ \frac{1.0 \times 10^{-10}}{1.0 \times 10^{-6}} = 1.0 \times 10^{-4} \]
Now substituting back:
\[ P = 2.6 \times 10^{10} \cdot 1.0 \times 10^{-4} = 2.6 \times 10^{6} \, N/m^2 \]

### Step 6: Relate Pressure Change to Depth
Since the pressure increases by \(1.0 \times 10^{4} \, N/m^2\) for every meter of depth, we can find the depth \(x\) using:
\[ P = 1.0 \times 10^{4} \cdot x \]
Setting the two expressions for pressure equal:
\[ 2.6 \times 10^{6} = 1.0 \times 10^{4} \cdot x \]

### Step 7: Solve for Depth
Rearranging gives:
\[ x = \frac{2.6 \times 10^{6}}{1.0 \times 10^{4}} = 260 \, m \]

### Conclusion
The depth at which the volume of the Pyrex glass cube decreases by \(1.0 \times 10^{-10} \, m^3\) is \(260 \, m\).

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