Toán tiếng Anh lớp 7

To solve the problem, we need to find out how much water must be poured into the cylindrical vessel so that the top face of the wooden cube floats level with the mouth of the vessel.

### Given data:
- **Base radius of the cylindrical vessel (r)** = 5 cm
- **Height of the cylindrical vessel (h)** = 20 cm
- **Side length of the cube (a)** = 6 cm
- **Volume of the cube (V_cube)** = \( a^3 = 6^3 = 216 \, \text{cm}^3 \)

### Step 1: Find the volume of water displaced by the cube
Since \(\frac{2}{3}\) of the cube's volume is submerged when it floats:
\[
V_{\text{submerged}} = \frac{2}{3} \times V_{\text{cube}} = \frac{2}{3} \times 216 = 144 \, \text{cm}^3
\]

### Step 2: Find the height of water that corresponds to the submerged volume
The volume of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
To find the height \(h_w\) of water needed to produce a volume of 144 cm³:
\[
144 = \pi (5^2) h_w
\]
Substituting \(\pi \approx 3.14\):
\[
144 = 3.14 \times 25 \times h_w
\]
\[
144 = 78.5 \times h_w
\]
\[
h_w = \frac{144}{78.5} \approx 1.83 \, \text{cm}
\]

### Step 3: Calculate the total height of water needed
For the cube to float with its top face level with the top of the vessel, we need to consider the additional height of the cube above the water level. The total height of water needed is:
\[
h_{\text{total}} = h_w + a = 1.83 + 6 = 7.83 \, \text{cm}
\]

### Step 4: Calculate the total volume of water that needs to be poured in
Now, we find the volume for this height in the cylinder:
\[
V_{\text{water}} = \pi r^2 h_{\text{total}} = 3.14 \times 25 \times 7.83
\]
Calculating this gives:
\[
V_{\text{water}} \approx 3.14 \times 25 \times 7.83 \approx 614.25 \, \text{cm}^3
\]

### Conclusion
The volume of water that must be poured into the vessel, so that the top face of the cube is level with the mouth of the vessel, is approximately **614.25 cm³**.