A cube of density 250 kg/`m^2` floats in water, then what part of total volume of the cube outside the water?

Options

A. 0.75
B. 0.25
C. 0.333
D. 0.677

Answer

To solve the problem of determining what part of the total volume of a cube floats outside of water, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Density of the cube, \( \rho_{cube} = 250 \, \text{kg/m}^3 \)
- Density of water, \( \rho_{water} = 1000 \, \text{kg/m}^3 \)

2. **Use Archimedes' Principle**:
According to Archimedes' principle, the weight of the fluid displaced by the cube is equal to the weight of the cube itself when it is floating. Mathematically, this can be expressed as:
\[ \text{Weight of cube} = \text{Weight of displaced water} \]
\[ mg = \rho_{water} \cdot V_d \cdot g \]
where \( V_d \) is the volume of the displaced water.

3. **Express the Weight of the Cube**:
The weight of the cube can also be expressed in terms of its density and volume:
\[ mg = \rho_{cube} \cdot V \cdot g \]
where \( V \) is the total volume of the cube.

4. **Set the Two Expressions Equal**:
Since both expressions equal the weight of the cube, we can set them equal to each other:
\[ \rho_{cube} \cdot V = \rho_{water} \cdot V_d \]

5. **Relate the Volume of Displaced Water to the Total Volume**:
From the equation above, we can solve for \( V_d \):
\[ V_d = \frac{\rho_{cube}}{\rho_{water}} \cdot V \]
Substituting the known densities:
\[ V_d = \frac{250}{1000} \cdot V = \frac{1}{4} V \]

6. **Determine the Volume Outside the Water**:
The volume of the cube that is outside the water can be found by subtracting the volume of the displaced water from the total volume:
\[ V_{outside} = V - V_d = V - \frac{1}{4} V = \frac{3}{4} V \]

7. **Calculate the Fraction of Volume Outside**:
To find the fraction of the total volume that is outside the water, we divide the volume outside by the total volume:
\[ \text{Fraction outside} = \frac{V_{outside}}{V} = \frac{\frac{3}{4} V}{V} = \frac{3}{4} \]

8. **Convert to Decimal**:
The fraction \( \frac{3}{4} \) can also be expressed as a decimal:
\[ \frac{3}{4} = 0.75 \]

### Final Answer:
Therefore, the part of the total volume of the cube that is outside the water is \( 0.75 \) or \( 75\% \).

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