We are tasked with calculating the total cost of the edging for a pond shaped as a sector of a circle.

Given:

• The sector has a central angle of $210^\circ$ and two radii of 3.5 meters.
• The edging needs to be applied to the perimeter of the pond, which consists of the arc and the two straight edges (the radii).
• Edging is sold in lengths of 1.75 meters, with each length costing £2.99.

Step-by-Step Solution:

1. Calculate the length of the arc: The formula for the arc length of a circle is: $L = 2 \pi r \times \left(\frac{\theta}{360}\right)$ where:

   • $r = 3.5 \, \text{m}$ (radius),
   • $\theta = 210^\circ$ (central angle in degrees),
   • $\pi \approx 3.1416$.

   Substituting the values: $L = 2 \times 3.1416 \times 3.5 \times \left(\frac{210}{360}\right)$ $L \approx 12.5664 \times 0.5833 = 7.33 \, \text{meters}$

2. Calculate the total perimeter: The total perimeter includes the arc length and the two radii: $\text{Perimeter} = \text{Arc length} + 2 \times \text{Radius}$ Substituting: $\text{Perimeter} = 7.33 + 2 \times 3.5 = 7.33 + 7 = 14.33 \, \text{meters}$

3. Determine the number of edging pieces required: Each piece of edging is 1.75 meters long. The total number of pieces required is: $\text{Number of pieces} = \frac{\text{Total perimeter}}{\text{Length of one piece}}$ Substituting: $\text{Number of pieces} = \frac{14.33}{1.75} \approx 8.19$ Since Toby cannot buy a fraction of a piece, he will need to buy 9 pieces.

4. Calculate the total cost: Each piece of edging costs £2.99. Therefore, the total cost is: $\text{Total cost} = 9 \times 2.99 = 26.91 \, \text{pounds}$

Final Answer:

The total cost of edging Toby needs to buy is £26.91.

Would you like further clarification or details on any of the steps? Here are some related questions to expand on this problem:

1. How is the arc length formula derived from the circumference of a circle?
2. What would happen if the radius was doubled? How would it affect the cost?
3. Could you calculate the total perimeter for a different angle, such as 180°?
4. How would the cost change if the edging was sold in lengths of 2 meters instead of 1.75 meters?
5. Can you explain how the central angle affects the perimeter of the sector?

Tip: Always remember to round up when dealing with real-world problems that involve buying whole units, like pieces of edging.