Figure 2 shows a plan view of a garden - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 4

To find the area of the garden, we need to calculate the area of triangle ABE and the area of sector BCDE.

1. Area of Triangle ABE: The area of a triangle is given by the formula:

   $ext{Area} = \frac{1}{2} \times AB \times AE \times \sin(\angle ABE$

   Here, we need to find AE first. Using the cosine rule for triangle ABE:

   $AE = \sqrt{AB^2 + BE^2 - 2 \times AB \times BE \times \cos(\angle ABE)}$

   Substituting the values:

   • AB = 23 m, AE = 12 m,
   • $\angle ABE = 0.64$ radians

   Next, we can calculate the area:

   $\text{Area}_{ABE} = \frac{1}{2} \times 23 \times 12 \times \sin(0.64) \approx 82.412...$

   This leads to: $\text{Area}_{ABE} \approx 82.4 \text{ m}^2$

2. Area of Sector BCDE: The area of a sector is given by:

   $\text{Area} = \frac{1}{2} r^2 \theta$

   Where $r = 12$ m and $\theta = 0.64$ radians:

   $\text{Area}_{BCDE} = \frac{1}{2} \times 12^2 \times 0.64 \approx 46.08$

3. Total Area of the Garden:

   $\text{Total Area} = \text{Area}_{ABE} + \text{Area}_{BCDE} \approx 82.4 + 46.08 \approx 128.48$

Rounding to 1 decimal place gives:

Final Area: Approximately 128.5 m².