A cylindrical boiler, 2 m high, is 3.5 m in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid.

Answer

To find the volume of the cylindrical boiler with a hemispherical lid, we will calculate the volume of the cylinder and the volume of the hemisphere separately, and then add them together.

### Step 1: Identify the dimensions
- Height of the cylinder (h) = 2 m
- Diameter of the cylinder = 3.5 m
- Radius of the cylinder (r) = Diameter / 2 = 3.5 m / 2 = 1.75 m

### Step 2: Calculate the volume of the cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Substituting the values:
\[ V_{\text{cylinder}} = \pi (1.75)^2 (2) \]
Calculating \( (1.75)^2 \):
\[ (1.75)^2 = 3.0625 \]
Now substituting back:
\[ V_{\text{cylinder}} = \pi (3.0625)(2) = 6.125\pi \, \text{m}^3 \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is given by:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Substituting the radius:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (1.75)^3 \]
Calculating \( (1.75)^3 \):
\[ (1.75)^3 = 5.359375 \]
Now substituting back:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (5.359375) = \frac{10.71875}{3} \pi \, \text{m}^3 \approx 3.57291667\pi \, \text{m}^3 \]

### Step 4: Total volume
Now, we add the volumes of the cylinder and the hemisphere:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 6.125\pi + 3.57291667\pi \]
\[ V_{\text{total}} = (6.125 + 3.57291667)\pi = 9.69791667\pi \, \text{m}^3 \]

### Step 5: Approximate the value using \(\pi \approx 3.14\)
\[ V_{\text{total}} \approx 9.69791667 \times 3.14 \approx 30.5 \, \text{m}^3 \]

### Final Answer
The total volume of the interior of the cylindrical boiler including the hemispherical lid is approximately \(30.5 \, \text{m}^3\).

---