A duck is floating on a lake with 25 % of its volume beneath the water . What is the average density of the duck ?

Options

A. `750 kg//m^(3)`
B. `130 kg//m^(3)`
C. `500 kg//m^(3)`
D. `250 kg//m^(3)`

Answer

To find the average density of the duck that is floating on a lake with 25% of its volume submerged, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that the duck is floating, which means the buoyant force acting on it is equal to its weight.
- The volume of the duck that is submerged is 25% of its total volume.

2. **Define Variables**:
- Let \( V \) be the total volume of the duck.
- The volume submerged in water is \( \frac{25}{100} V = \frac{V}{4} \).

3. **Buoyant Force Calculation**:
- The buoyant force \( F_B \) can be expressed as:
\[ F_B = V_{submerged} \cdot \rho_{water} \cdot g \]
- Here, \( V_{submerged} = \frac{V}{4} \), \( \rho_{water} \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)), and \( g \) is the acceleration due to gravity.

4. **Weight of the Duck**:
- The weight of the duck \( W \) is given by:
\[ W = m \cdot g \]
- Here, \( m \) is the mass of the duck.

5. **Equating Buoyant Force and Weight**:
- Since the duck is floating, we have:
\[ F_B = W \]
- Substituting the expressions for buoyant force and weight:
\[ \frac{V}{4} \cdot \rho_{water} \cdot g = m \cdot g \]
- We can cancel \( g \) from both sides:
\[ \frac{V}{4} \cdot \rho_{water} = m \]

6. **Density of the Duck**:
- The density of the duck \( \rho_{duck} \) is defined as:
\[ \rho_{duck} = \frac{m}{V} \]
- Substituting \( m \) from the previous equation:
\[ \rho_{duck} = \frac{\frac{V}{4} \cdot \rho_{water}}{V} \]
- Simplifying this gives:
\[ \rho_{duck} = \frac{\rho_{water}}{4} \]

7. **Calculating the Average Density**:
- Substituting the density of water:
\[ \rho_{duck} = \frac{1000 \, \text{kg/m}^3}{4} = 250 \, \text{kg/m}^3 \]

### Final Answer:
The average density of the duck is \( 250 \, \text{kg/m}^3 \).

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