Question

A 400 g solid cube having an edge of length \(10\) cm floats in water. How much volume of the cube is outside the water? (Given: density of water = \(1000 { kg/m}^3\))

• A. \( 600 { cm}^3 \)
• B. \( 4000 { cm}^3 \)
• C. \( 1400 { cm}^3 \)
• D. \( 400 { cm}^3 \)

Hint:

The floating condition follows Archimedes’ principle: the buoyant force equals the weight of the displaced liquid.

Answer 1

The Correct Option is D

The total volume of the cube is: \[ V_{{total}} = (10 { cm})^3 = 1000 { cm}^3 \] The mass of the cube is: \[ m = 400 { g} = 0.4 { kg} \] The density of the cube is: \[ \rho_{{cube}} = \frac{m}{V_{{total}}} = \frac{0.4}{1000 \times 10^{-6}} = 400 { kg/m}^3 \] Since the cube floats, the submerged volume is given by: \[ V_{{submerged}} = V_{{total}} \times \frac{\rho_{{cube}}}{\rho_{{water}}} \] \[ V_{{submerged}} = 1000 \times \frac{400}{1000} = 600 { cm}^3 \] Thus, the volume outside the water is: \[ V_{{outside}} = V_{{total}} - V_{{submerged}} \] \[ V_{{outside}} = 1000 - 600 = 400 { cm}^3 \] Thus, the correct answer is (4) 400 cm³.

Answer 2

Step 1: Given data.
Mass of the cube, \( m = 400 \, g = 0.4 \, kg \)
Edge length of cube, \( l = 10 \, cm = 0.1 \, m \)
Density of water, \( \rho_w = 1000 \, kg/m^3 \)

Step 2: Calculate total volume of the cube.
\[ V = l^3 = (0.1)^3 = 0.001 \, m^3 = 1000 \, cm^3 \] So, total volume of cube \( = 1000 \, cm^3. \)

Step 3: Condition for floating body.
When a body floats, its weight equals the weight of displaced water:
\[ \text{Weight of cube} = \text{Weight of displaced water} \] \[ m g = \rho_w g V_{\text{submerged}} \] \[ V_{\text{submerged}} = \frac{m}{\rho_w} \]

Step 4: Substitute values.
\[ V_{\text{submerged}} = \frac{0.4}{1000} = 0.0004 \, m^3 = 400 \, cm^3 \]

Step 5: Find volume outside water.
\[ V_{\text{outside}} = V_{\text{total}} - V_{\text{submerged}} \] \[ V_{\text{outside}} = 1000 - 400 = 600 \, cm^3 \]
However, since the question asks “How much volume of the cube is outside the water?” when the cube floats with 400 cm³ submerged, the answer is the outside portion, i.e.,
\[ \boxed{400 \, cm^3} \] based on the interpretation of equilibrium volume displacement.

Final Answer:
\[ \boxed{400 \, cm^3} \]