A cube of side 20 cm is floating on a liquid with 5 cm of the cube outside the liquid. If the density of liquid is `0.8gm//c c` then the mass of the cube is

Options

A. 4.2 kg
B. 4.8 kg
C. 5 kg
D. 5.2 kg

Explanation

To find the mass of the cube floating in the liquid, we can follow these steps:

### Step 1: Understand the problem
We have a cube with a side length of 20 cm, and it is floating in a liquid. The cube has 5 cm of its height above the liquid, which means that 15 cm of the cube is submerged in the liquid.

### Step 2: Calculate the submerged volume of the cube
The volume of the submerged part of the cube can be calculated using the formula for the volume of a cube:
\[ \text{Volume} = \text{side}^3 \]
However, since only part of the cube is submerged, we need to calculate the volume of the submerged part:
\[ \text{Submerged Height} = 20 \, \text{cm} - 5 \, \text{cm} = 15 \, \text{cm} \]
The volume of the submerged part is:
\[ \text{Volume}_{\text{submerged}} = \text{Area of base} \times \text{Submerged Height} = (20 \, \text{cm} \times 20 \, \text{cm}) \times 15 \, \text{cm} = 400 \, \text{cm}^2 \times 15 \, \text{cm} = 6000 \, \text{cm}^3 \]

### Step 3: Use Archimedes' principle
According to Archimedes' principle, the buoyant force (upward force) acting on the cube is equal to the weight of the liquid displaced by the submerged part of the cube. The buoyant force can be expressed as:
\[ F_b = \text{Volume}_{\text{submerged}} \times \text{Density}_{\text{liquid}} \times g \]
Since we are looking for mass, we can express weight as:
\[ F_b = m \cdot g \]
Thus, we can equate the two:
\[ \text{Volume}_{\text{submerged}} \times \text{Density}_{\text{liquid}} \times g = m \cdot g \]
The \(g\) cancels out:
\[ \text{Volume}_{\text{submerged}} \times \text{Density}_{\text{liquid}} = m \]

### Step 4: Substitute the values
Given that the density of the liquid is \(0.8 \, \text{g/cm}^3\) and the volume of the submerged part is \(6000 \, \text{cm}^3\):
\[ m = 6000 \, \text{cm}^3 \times 0.8 \, \text{g/cm}^3 = 4800 \, \text{g} \]

### Step 5: Convert grams to kilograms
To express the mass in kilograms:
\[ m = 4800 \, \text{g} = 4.8 \, \text{kg} \]

### Final Answer
The mass of the cube is \(4.8 \, \text{kg}\).

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