The density of ice is 0.92 g/`cm^(3)` and the density of seawater is 1.03 g/`cm^3`. A large iceberg floats in Arctic waters. What fraction of the volume of the iceberg is exposed?

Options

A. `0.080` %
B. `11` %
C. 0.89
D. 0.92

Answer

To solve the problem of finding the fraction of the volume of the iceberg that is exposed above the water, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Densities**:
- Density of ice, \( \rho_{\text{ice}} = 0.92 \, \text{g/cm}^3 \)
- Density of seawater, \( \rho_{\text{water}} = 1.03 \, \text{g/cm}^3 \)

2. **Understand the Concept of Buoyancy**:
- An iceberg floats due to the buoyant force, which is equal to the weight of the water displaced by the submerged part of the iceberg.
- The weight of the iceberg can be expressed as \( mg \), where \( m \) is the mass of the iceberg.

3. **Set Up the Equation for Buoyancy**:
- The buoyant force is given by:
\[ F_b = V_{\text{submerged}} \cdot \rho_{\text{water}} \cdot g \]
- Where \( V_{\text{submerged}} \) is the volume of the iceberg submerged in water.

4. **Relate Mass and Volume**:
- The mass of the iceberg can also be expressed in terms of its volume and density:
\[ m = V_{\text{total}} \cdot \rho_{\text{ice}} \]
- Where \( V_{\text{total}} \) is the total volume of the iceberg.

5. **Equate the Forces**:
- Since the iceberg is floating, the weight of the iceberg equals the buoyant force:
\[ V_{\text{total}} \cdot \rho_{\text{ice}} \cdot g = V_{\text{submerged}} \cdot \rho_{\text{water}} \cdot g \]
- The \( g \) cancels out from both sides:
\[ V_{\text{total}} \cdot \rho_{\text{ice}} = V_{\text{submerged}} \cdot \rho_{\text{water}} \]

6. **Express the Submerged Volume**:
- Rearranging the equation gives:
\[ \frac{V_{\text{submerged}}}{V_{\text{total}}} = \frac{\rho_{\text{ice}}}{\rho_{\text{water}}} \]

7. **Calculate the Fraction**:
- Substitute the known densities:
\[ \frac{V_{\text{submerged}}}{V_{\text{total}}} = \frac{0.92}{1.03} \]

8. **Calculate the Value**:
- Performing the division:
\[ \frac{0.92}{1.03} \approx 0.8932 \]
- This means approximately 89.32% of the iceberg is submerged in water.

9. **Determine the Exposed Volume**:
- The fraction of the iceberg that is exposed is:
\[ \text{Exposed fraction} = 1 - \frac{V_{\text{submerged}}}{V_{\text{total}}} \approx 1 - 0.8932 \approx 0.1068 \]
- Converting to percentage:
\[ \text{Exposed percentage} \approx 10.68\% \]

### Final Answer:
The fraction of the volume of the iceberg that is exposed is approximately **10.68%**.