https://file.aiquickdraw.com/mathbot/file/1727347212265ifcsdk00.jpg

Let's go through the questions in the image step by step.

### 1.1: Beregning af terrassens areal (Calculating the terrace's area)
- **Terrace dimensions**: 6.30 meters long and 4.20 meters wide.
- **Area of the terrace**: 
  \[
  A = \text{length} \times \text{width} = 6.30 \, m \times 4.20 \, m = 26.46 \, m^2
  \]
This confirms that the terrace area is approximately 26.5 m².

### 1.2: Hvor mange sække med grus skal Lucas og hans far købe? (How many sacks of gravel should Lucas and his father buy?)
- The gravel layer is 15 cm thick, and they need to cover 26.46 m².
- To calculate the volume of gravel needed:
  \[
  V = \text{area} \times \text{depth} = 26.46 \, m^2 \times 0.15 \, m = 3.969 \, m^3
  \]
- Each sack contains 0.5 m³ of gravel, so the number of sacks needed:
  \[
  \text{Number of sacks} = \frac{3.969 \, m^3}{0.5 \, m^3} \approx 7.94
  \]
This rounds up to **8 sacks**.

### 1.3: Hvor mange hele og halve fliser? (How many whole and half tiles are needed?)
- **Tile dimensions**: Full tile = 21 cm × 14 cm, Half tile = 10.5 cm × 14 cm.
- To calculate the number of tiles, the terrace's area must be covered in tiles with the given pattern.
  Let me calculate this next.

### 1.4: Hvor stort skal terrassens procentvise fald være? (What is the terrace's percentage slope?)
- The terrace needs a slope of 2 cm per meter, meaning a 2 cm drop for every 100 cm (1 meter).
  The percentage slope is calculated as:
  \[
  \text{Slope percentage} = \frac{\text{fall}}{\text{run}} \times 100 = \frac{2 \, cm}{100 \, cm} \times 100 = 2\%
  \]
So, the slope should be **2%**.

### 1.5: Undersøg med beregning om Lucas' mor har ret. (Investigate if Lucas' mother is correct.)
- Lucas' mother says that a 2 cm per meter slope results in an angle \(v\) of 88°. To verify, we need to calculate the angle based on a 2 cm fall per meter:
  \[
  \tan(v) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2 \, cm}{100 \, cm}
  \]
  \[
  v = \tan^{-1}\left(\frac{2}{100}\right)
  \]
Let me compute the exact value for this angle next.

I'll now calculate the number of tiles in 1.3 and the angle for 1.5.### 1.3: Number of whole and half tiles
- For the terrace dimensions, approximately **30 full tiles** fit along both the length (6.30 m) and width (4.20 m).
- No additional half tiles are needed since the dimensions fit exactly with full tiles (no remaining space to fill).

### 1.5: Calculation of the angle \( v \)
- The angle corresponding to a 2 cm fall per meter (100 cm) is approximately **1.15°**.
- Therefore, Lucas' mother is not correct in saying that the angle is 88°. The actual angle is much smaller, only about **1.15°**.

Let me know if you need any more details, or if you have further questions.

Here are 5 related questions:
1. How would the required number of sacks change if the gravel layer were thicker?
2. How could you adjust the tile layout if the terrace dimensions were slightly smaller?
3. If the slope were increased to 3 cm per meter, how would the angle and percentage change?
4. What would be the impact on the number of tiles if the terrace width were extended by 0.5 m?
5. How many extra tiles should be bought in case of damage or errors during construction?

### Tip:
When planning for tiles, it's often wise to account for a 5-10% surplus to cover breakage or future repairs.

Lucas vil hjælpe sin far med at anlægge en terrasse med fliser. Terrassen skal have form som et rektangel. Den skal være 6,30 m lang og 4,20 m bred. Du skal vise med beregning, at terrassens areal bliver ca. 26,5 m². Hvor mange sække med grus skal Lucas og hans far købe? Undersøg, hvor mange hele og hvor mange halve fliser de skal bruge til terrassen. Hvor stort skal terrassens procentvise fald være?

Learn how to calculate the area of a terrace, determine the volume of gravel needed, and find the percentage slope required for proper drainage. This exercise involves geometric calculations, volume determination, and trigonometric analysis, suitable for students in Grades 8-10.

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