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The problem describes a warehouse with a roof that has a semi-circular cross-section and provides several measurements to calculate the surface area of the roof (Dachfläche) and the height of the building (Höhe).

### Known values:
- Length of the building (a) = 13.80 m
- Width of the building (b) = 9.58 m
- Volume of the building (V) = 1470.38 m³

The building is a combination of a rectangular base and a half-cylinder on top, as shown in the sketch.

### To solve:

1. **Height of the building (h)**: The highest point of the hall will be the radius \(r\) of the semi-circular roof plus the height of the rectangular part. However, based on the drawing, it seems that the semi-circular roof extends all the way from the floor to the top (meaning no additional rectangular section). So, the height of the hall will be twice the radius of the semi-circle. We can derive the radius by using the volume formula.
   
   The volume of this structure is composed of a rectangular base plus the half-cylinder on top. The volume \(V\) is:

   \[
   V = a \cdot \left( b \cdot h_{\text{rect}} + \frac{1}{2} \pi r^2 \right)
   \]

   where:
   - \(a\) is the length,
   - \(b\) is the width (which equals \(2r\), since the diameter of the semi-circle is the same as the width of the building),
   - \(h_{\text{rect}}\) is the height of the rectangular section (if any).

2. **Roof surface area (F)**: The roof is a semi-circular cylinder, so the surface area of the roof is:

   \[
   F = a \cdot \pi r
   \]

Let me first calculate the height \(h\) using the provided volume formula, then calculate the roof surface area \(F\).### Results:
1. The volume of the half-cylinder is approximately \(497.36 \, \text{m}^3\). This represents the volume of the semi-circular roof section.

2. The **roof surface area (Dachfläche)** is approximately:
   \[
   F \approx 207.67 \, \text{m}^2
   \]

3. The **height of the hall (Höhe)** is:
   \[
   h \approx 9.58 \, \text{m}
   \]
   (which is the diameter of the semi-circular roof section).

Let me know if you'd like more details or have further questions!

Here are 5 related questions for further practice:
1. How would the height of the building change if the width \(b\) increased?
2. What is the total surface area of the building, including the side walls?
3. How would you compute the volume if the roof were a full cylinder instead of a half-cylinder?
4. How does the roof surface area change if the length \(a\) is increased by 50%?
5. Can you calculate the surface area of the building's walls and roof combined?

**Tip**: When working with volumes and surface areas of combined shapes, breaking them down into familiar geometric components (e.g., rectangular prisms, cylinders, etc.) simplifies the calculations.

Eine Großmarkthalle habe ein gewölbtes Dach mit halbkreisförmigem Querschnitt. Die Längsseite der Halle habe eine Länge von a = 13,80 m, die andere Seite eine Länge von b = 9,58 m und das gesamte Volumen der Halle (inklusive Dachbereich) betrage V = 1470,38 m³. Wie groß ist die gewölbte Dachfläche (in m²) und wie hoch ist die Halle an der höchsten Stelle (in m)?

Learn how to calculate the roof surface area and height of a semi-circular warehouse using the given dimensions: length a = 13.80 m, width b = 9.58 m, and total volume V = 1470.38 m³. This problem involves geometric calculations for surface area and volume of solids. Suitable for Grades 10-12.

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