https://file.aiquickdraw.com/mathbot/file/1730043354953az46e7vx.jpg

Let's break down each problem and calculate the volumes as requested.

### Part 1: Volume of the Building
The building consists of two main sections:
1. A **square** base of \(22 \text{ m} \times 22 \text{ m}\),
2. A **cylindrical** section with a radius of \(11 \text{ m}\) (half of the diameter shown).

The building's height is given as \(43 \text{ m}\).

#### Step 1: Volume of the Square Prism
The volume \( V \) of a prism is given by:
\[
V = \text{base area} \times \text{height}
\]

For the square section:
- Base area of the square = \(22 \times 22 = 484 \, \text{m}^2\).
- Height = \(43 \, \text{m}\).

So, the volume of the square prism:
\[
V_{\text{square}} = 484 \times 43 = 20812 \, \text{m}^3
\]

#### Step 2: Volume of the Cylinder
For the cylindrical section:
- Radius \( r = 11 \, \text{m} \),
- Height \( h = 43 \, \text{m}\),
- Volume of a cylinder is given by \( V = \pi r^2 h \).

So,
\[
V_{\text{cylinder}} = \pi \times (11)^2 \times 43
\]
\[
V_{\text{cylinder}} \approx 16334.1 \, \text{m}^3
\]

#### Step 3: Total Volume of the Building
The total volume is the sum of the square and cylindrical sections:
\[
V_{\text{total}} = V_{\text{square}} + V_{\text{cylinder}} = 20812 + 16334.1 \approx 37146 \, \text{m}^3
\]

### Part 2: Volume of the Greenhouse
The greenhouse is made of a prism with a trapezoidal face as the cross-section.

#### Step 1: Area of the Trapezoidal Cross-Section
The trapezoid has:
- A top side length of \(2 \, \text{m}\),
- A bottom side length of \(4 \, \text{m}\),
- A height of \(3.2 \, \text{m}\) (sum of \(0.8 + 2.4\)).

The area \( A \) of a trapezoid is given by:
\[
A = \frac{1}{2} \times (\text{top} + \text{bottom}) \times \text{height}
\]
\[
A = \frac{1}{2} \times (2 + 4) \times 3.2
\]
\[
A = \frac{1}{2} \times 6 \times 3.2 = 9.6 \, \text{m}^2
\]

#### Step 2: Volume of the Greenhouse
The length of the greenhouse is \(14 \, \text{m}\).

So, the volume \( V \) of the prism is:
\[
V = \text{cross-sectional area} \times \text{length}
\]
\[
V = 9.6 \times 14 = 134.4 \, \text{m}^3
\]

### Answers
1. **Volume of the Building:** \( \approx 37146 \, \text{m}^3 \)
2. **Volume of the Greenhouse:** \( \approx 134 \, \text{m}^3 \) (to the nearest cubic meter)

Would you like further explanations or additional questions on related concepts?

### Related Questions
1. How is the volume of a cylinder calculated in general?
2. What are other methods to find the volume of irregularly shaped buildings?
3. Why is the trapezoidal cross-sectional area important in calculating prism volumes?
4. How would the calculations change if the height of the building or greenhouse were doubled?
5. Can we use integral calculus to find the volume of these shapes in a different way?

#### Tip
For composite shapes, calculate the volume of each part individually before summing them up. This simplifies handling complex structures.

Work out the volume of the building and the greenhouse shown in the diagram.

Solve a problem involving volume calculations of a building and a greenhouse with composite shapes. Learn the steps for finding volumes of square prisms, cylinders, and trapezoidal prisms. Suitable for students in Grades 10-12.

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