To solve the problem step by step, we will follow these calculations:
### Step 1: Understand the given data
- Area of the base of the cuboid \( A = 4.5 \, \text{dm}^2 = 4.5 \times 100 \, \text{cm}^2 = 450 \, \text{cm}^2 \)
- Initial height of water in the vessel \( h_1 = 2 \, \text{cm} \)
- Mass of the wooden cube \( m = 4 \, \text{kg} \)
- Specific gravity of the wooden cube \( \text{SG} = 0.5 \)
### Step 2: Calculate the density of the wooden cube
Specific gravity is defined as the ratio of the density of the substance to the density of water. The density of water is \( 1 \, \text{g/cm}^3 \) or \( 1000 \, \text{kg/m}^3 \).
\[
\rho_{\text{wood}} = \text{SG} \times \rho_{\text{water}} = 0.5 \times 1000 \, \text{kg/m}^3 = 500 \, \text{kg/m}^3
\]
### Step 3: Calculate the volume of the wooden cube
Using the mass and density of the wooden cube, we can find its volume:
\[
V_{\text{wood}} = \frac{m}{\rho_{\text{wood}}} = \frac{4 \, \text{kg}}{500 \, \text{kg/m}^3} = 0.008 \, \text{m}^3 = 8000 \, \text{cm}^3
\]
### Step 4: Calculate the height of the water displaced by the wooden cube
The volume of water displaced by the wooden cube will be equal to the volume of the cube itself since it is submerged in water.
Using the area of the base of the cuboid:
\[
h_{\text{displaced}} = \frac{V_{\text{wood}}}{A} = \frac{8000 \, \text{cm}^3}{450 \, \text{cm}^2} \approx 17.78 \, \text{cm}
\]
### Step 5: Calculate the new height of water in the vessel
The new height of water after placing the wooden cube will be the sum of the initial height of the water and the height of the water displaced:
\[
h_{\text{new}} = h_1 + h_{\text{displaced}} = 2 \, \text{cm} + 17.78 \, \text{cm} \approx 19.78 \, \text{cm}
\]
### Step 6: Find the height of water level above the base of the wooden block
Since the wooden block is floating, the height of the water above the base of the wooden block will be the height of the new water level minus the height of the wooden block submerged in water.
To find the height of the wooden block submerged in water, we can use the buoyancy principle:
\[
\text{Weight of water displaced} = \text{Weight of wooden cube}
\]
\[
\rho_{\text{water}} \cdot V_{\text{displaced}} \cdot g = m \cdot g
\]
This means:
\[
V_{\text{displaced}} = \frac{m}{\rho_{\text{water}}} = \frac{4 \, \text{kg}}{1000 \, \text{kg/m}^3} = 0.004 \, \text{m}^3 = 4000 \, \text{cm}^3
\]
Now, we can find the height of the wooden block submerged in water:
\[
h_{\text{submerged}} = \frac{V_{\text{displaced}}}{A} = \frac{4000 \, \text{cm}^3}{450 \, \text{cm}^2} \approx 8.89 \, \text{cm}
\]
### Final Step: Calculate the height of water level above the base of the wooden block
\[
h_{\text{above}} = h_{\text{new}} - h_{\text{submerged}} = 19.78 \, \text{cm} - 8.89 \, \text{cm} \approx 10.89 \, \text{cm}
\]
Thus, the height of the water level above the base of the wooden block is approximately **10.89 cm**.
