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The problem presented is about calculating the area of the patio to determine the number of paving slabs required, selecting the most cost-effective option from the available packs.

### Step-by-Step Solution:

1. **Calculate the total area of the patio**:
    - The patio consists of a **rectangle** and a **semi-circle**.
   
    **For the rectangle**:
    - The length is 8 meters, and the width is 4 meters.
    - The area of the rectangle is \( \text{Area} = 8 \times 4 = 32 \, \text{m}^2 \).
   
    **For the semi-circle**:
    - The diameter is given as 2.5 meters, so the radius \( r \) is \( \frac{2.5}{2} = 1.25 \, \text{m} \).
    - The area of a full circle is \( \pi r^2 = \pi (1.25)^2 = \pi \times 1.5625 = 4.91 \, \text{m}^2 \).
    - Since it's a semi-circle, divide the area by 2: 
      \[
      \text{Semi-circle area} = \frac{4.91}{2} = 2.455 \, \text{m}^2
      \]

    **Total area of the patio**:
    \[
    \text{Total area} = 32 \, \text{m}^2 + 2.455 \, \text{m}^2 = 34.455 \, \text{m}^2
    \]

2. **Determine the number of packs needed**:
   - **Pack A** covers 1 m² and costs £20.
   - **Pack B** covers 5 m² and costs £75.
   - **Pack C** covers 10 m² and costs £140.

   The builder needs to cover 34.455 m². Let's find the best option:

   **Pack C (10 m², £140)**:
   \[
   \frac{34.455}{10} \approx 3.445
   \]
   This means 4 packs are required (since you can't order a fraction of a pack), costing \( 4 \times 140 = 560 \, \text{pounds} \).

   **Pack B (5 m², £75)**:
   \[
   \frac{34.455}{5} \approx 6.891
   \]
   This means 7 packs are required, costing \( 7 \times 75 = 525 \, \text{pounds} \).

   **Pack A (1 m², £20)**:
   \[
   \frac{34.455}{1} \approx 34.455
   \]
   This means 35 packs are required, costing \( 35 \times 20 = 700 \, \text{pounds} \).

3. **Conclusion**:
   The cheapest option is to order **7 packs of Pack B**, which will cover 35 m² at a cost of **£525**.

Would you like more details or have any further questions?

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Here are five related questions:
1. How would the cost change if the patio's size increased by 10%?
2. What if the builder could mix and match different packs—what would be the optimal combination?
3. If the price of Pack C drops to £130, would it be the better choice?
4. How would the total cost change if there was a 5% discount on all packs?
5. What is the formula for calculating the area of a sector, and how does it apply to semi-circles?

**Tip**: Always round up when calculating quantities for coverage to ensure you don't fall short of material.

A builder needs to order slabs for a patio. The patio consists of a rectangle and a semi-circle with specified dimensions. The goal is to find the cheapest option among three packs of paving slabs, each covering different areas and costs. What should the builder order?

Calculate the most cost-effective paving slab pack for a patio consisting of a rectangle and a semi-circle. Learn to determine the area using geometric formulas and analyze the costs of different packs to find the optimal solution. Suitable for Grades 7-9.

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