A cube of edge 20 cm is completely immersed in a rectangular vessel containing water. If the dimensions of the base of the vessel are 20 cm by 40 cm, the rise in water level will be

### Concept & Displacement of Fluid Archimedes' principle of volume displacement states that when a solid object is completely immersed in a liquid, it displaces a volume of liquid equal to its own volume. The displaced liquid raises the water level forming a "water cuboid". $$V_{\text{immersed object}} = \text{Base Area of Vessel} \times \text{Rise in Height}$$ ### Step-by-Step Solution 1. **Calculate the volume of the immersed cube:** - Edge of the cube, $a = 20 \text{ cm}$. - Volume of the cube = $a^3 = 20 \times 20 \times 20 = 8000 \text{ cm}^3$. 2. **Set up the displacement equation:** - Let the rise in the water level be $h \text{ cm}$. - The volume of the displaced water forms a cuboid with the vessel's base. - Volume of displaced water = $\text{Length} \times \text{Breadth} \times h = 20 \times 40 \times h = 800h \text{ cm}^3$. 3. **Equate the volumes:** - $800h = 8000$ 4. **Solve for $h$:** - $h = \frac{8000}{800} = 10 \text{ cm}$. ### Exam Strategy & Shortcut Skip computing full volumes if factors cancel out. Write the equation directly: $20 \times 20 \times 20 = 20 \times 40 \times h$ Cancel the common $20$: $400 = 40 \times h \implies h = 10$. ### Common Pitfall A common mistake is adding the initial volume of water in the vessel to the calculation, which is unnecessary since we only care about the *change* (rise) in the water volume. ### Final Answer Therefore, the correct answer is **10 cm**.