A cylindrical container with internal radius of its base 10 cm, contains water up to a height of 7 cm. Find the area of the wet surface of the cylinder.

Answer

To find the area of the wet surface of a cylindrical container filled with water, we need to calculate the curved surface area of the cylinder up to the height of the water and the area of the circular base that is in contact with the water.

### Step-by-step Solution:

1. **Identify the given values:**
- Internal radius of the base (r) = 10 cm
- Height of the water (h) = 7 cm

2. **Calculate the curved surface area (CSA) of the cylinder:**
The formula for the curved surface area of a cylinder is:
\[ \text{CSA} = 2 \pi r h \]
Substituting the given values:
\[ \text{CSA} = 2 \times \pi \times 10 \times 7 \]
We can use \(\pi \approx \frac{22}{7}\) for calculation:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 10 \times 7 \]

3. **Simplify the curved surface area calculation:**
\[ \text{CSA} = 2 \times \frac{22}{7} \times 10 \times 7 = 2 \times 22 \times 10 = 440 \text{ cm}^2 \]

4. **Calculate the area of the circular base:**
The formula for the area of a circle is:
\[ \text{Area of base} = \pi r^2 \]
Substituting the radius:
\[ \text{Area of base} = \pi \times (10)^2 = \pi \times 100 \]
Again using \(\pi \approx \frac{22}{7}\):
\[ \text{Area of base} = \frac{22}{7} \times 100 = \frac{2200}{7} \text{ cm}^2 \]

5. **Combine the areas to find the total wet surface area:**
The total wet surface area is the sum of the curved surface area and the area of the base:
\[ \text{Total wet surface area} = \text{CSA} + \text{Area of base} \]
\[ \text{Total wet surface area} = 440 + \frac{2200}{7} \]

6. **Convert 440 to a fraction with a common denominator:**
\[ 440 = \frac{440 \times 7}{7} = \frac{3080}{7} \]
Now add the two fractions:
\[ \text{Total wet surface area} = \frac{3080}{7} + \frac{2200}{7} = \frac{5280}{7} \]

7. **Calculate the final value:**
Now divide:
\[ \frac{5280}{7} \approx 754.2857 \text{ cm}^2 \]
Rounding to two decimal places gives:
\[ \text{Total wet surface area} \approx 754.29 \text{ cm}^2 \]

### Final Answer:
The area of the wet surface of the cylinder is approximately **754.29 cm²**.