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 internal volume of the container in m?.

Answer

To find the internal volume of the metal container, which consists of a cylinder surmounted by a hemisphere, we will follow these steps:

### Step 1: Identify the dimensions
- The internal radius \( r \) of the cylinder (and hemisphere) is given as \( 3.5 \, \text{m} \).
- The internal height \( h \) of the cylinder is given as \( 7 \, \text{m} \).

### Step 2: Calculate the volume of the cylinder
The formula for the volume \( V_c \) of a cylinder is:
\[ V_c = \pi r^2 h \]
Substituting the values:
\[ V_c = \pi (3.5)^2 (7) \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting back:
\[ V_c = \pi (12.25) (7) = 85.75\pi \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume \( V_h \) of a hemisphere is:
\[ V_h = \frac{2}{3} \pi r^3 \]
Substituting the radius:
\[ V_h = \frac{2}{3} \pi (3.5)^3 \]
Calculating \( (3.5)^3 \):
\[ (3.5)^3 = 42.875 \]
Now substituting back:
\[ V_h = \frac{2}{3} \pi (42.875) = \frac{85.75}{3}\pi \]

### Step 4: Calculate the total volume of the container
The total volume \( V \) of the container is the sum of the volumes of the cylinder and the hemisphere:
\[ V = V_c + V_h \]
Substituting the values:
\[ V = 85.75\pi + \frac{85.75}{3}\pi \]
To combine these, we can find a common denominator:
\[ V = \frac{257.25}{3}\pi + \frac{85.75}{3}\pi = \frac{343}{3}\pi \]

### Step 5: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{343}{3} \times \frac{22}{7} \]
Calculating this:
\[ V = \frac{343 \times 22}{3 \times 7} = \frac{7566}{21} \]
Calculating \( \frac{7566}{21} \):
\[ V \approx 359.33 \, \text{m}^3 \]

### Final Answer
The internal volume of the container is approximately \( 359.33 \, \text{m}^3 \).

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