A metal container in the form of a cylinder is surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7 m and the internal radius is 3-5 m. Calculate :
the total area of the internal surface, excluding the base,

Answer

To solve the problem, we need to calculate the total internal surface area of a metal container that consists of a cylinder surmounted by a hemisphere, excluding the base.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Internal height of the cylinder (h) = 7 m
- Internal radius of the cylinder (r) = 3.5 m

2. **Understand the Shapes Involved:**
- The container consists of a cylinder and a hemisphere on top.
- We need to find the curved surface area of the cylinder and the curved surface area of the hemisphere.

3. **Calculate the Curved Surface Area of the Cylinder:**
- The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA}_{\text{cylinder}} = 2 \pi r h \]
- Substituting the values:
\[ \text{CSA}_{\text{cylinder}} = 2 \pi (3.5) (7) \]

4. **Calculate the Curved Surface Area of the Hemisphere:**
- The formula for the curved surface area of a hemisphere is:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi r^2 \]
- Substituting the radius:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi (3.5)^2 \]

5. **Combine the Areas:**
- The total internal surface area excluding the base is:
\[ \text{Total Area} = \text{CSA}_{\text{cylinder}} + \text{CSA}_{\text{hemisphere}} \]

6. **Calculate Each Area:**
- For the cylinder:
\[ \text{CSA}_{\text{cylinder}} = 2 \pi (3.5) (7) = 49 \pi \]
- For the hemisphere:
\[ \text{CSA}_{\text{hemisphere}} = 2 \pi (3.5)^2 = 24.5 \pi \]

7. **Combine the Results:**
- Total Area:
\[ \text{Total Area} = 49 \pi + 24.5 \pi = 73.5 \pi \]

8. **Substitute \(\pi\) with \(\frac{22}{7}\) for Calculation:**
- Total Area:
\[ \text{Total Area} = 73.5 \times \frac{22}{7} \]
- Calculating:
\[ \text{Total Area} = 231 \, \text{m}^2 \]

### Final Answer:
The total internal surface area of the container, excluding the base, is **231 m²**.