Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12 m and centre B. The points A, B and C lie on a straight line with AB = 23 m and BC = 12 m. Given that the size of angle ABE is exactly 0.64 radians, find (a) the area of the garden, giving your answer in m², to 1 decimal place. (b) the perimeter of the garden, giving your answer in metres, to 1 decimal place.

A-Level Edexcel Maths Pure Applications of Differentiation: Figure 2 shows a plan view of a

Figure 2 shows a plan view of a garden - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Question 9

Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12 m and centre B.... show full transcript

Worked Solution & Example Answer:Figure 2 shows a plan view of a garden - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Step 1

Find the area of the garden

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Answer

To find the area of the garden, we combine the area of triangle ABE and the area of the sector BCDE.

Area of triangle ABE: The formula is given by: $ext{Area} = \frac{1}{2} ab \sin(C)$ Here, ( a = 23 ) m, ( b = 12 ) m, and ( C = 0.64 ) radians.

$ext{Area}_{ABE} = \frac{1}{2} \times 23 \times 12 \times \sin(0.64) \approx 82.4 \text{ m}^2$
Area of sector BCDE: The formula for the area of a sector is: $ext{Area}_{sector} = \frac{1}{2} r^2 \theta$ Here, ( r = 12 ) m, and ( \theta = 0.64 ) radians.

$ext{Area}_{sector} = \frac{1}{2} \times 12^2 \times 0.64 \approx 38.4 \text{ m}^2$
Total Area:

$\text{Total Area} = \text{Area}_{ABE} + \text{Area}_{sector} \approx 82.4 + 38.4 = 120.8 \text{ m}^2$

Rounding to 1 decimal place: 120.8 m².

Step 2

Find the perimeter of the garden

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Answer

To find the perimeter of the garden, we must sum the lengths of all sides.

Perimeter Calculation: The perimeter includes the following segments:
- AB = 23 m
- BC = 12 m
- CD (arc length)
To find the arc length, use the formula: $ext{Arc Length} = r \theta$ Where ( r = 12 ) m and ( \theta = 0.64 ) radians.

$ext{Arc Length} = 12 \times 0.64 = 7.68 \text{ m}$
1. Total Perimeter:
$ext{Perimeter} = AB + BC + CD = 23 + 12 + 7.68 = 42.68 ext{ m}$

Rounding to 1 decimal place: 42.7 m.

Figure 2 shows a plan view of a garden - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Worked Solution & Example Answer:Figure 2 shows a plan view of a garden - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Find the area of the garden

Find the perimeter of the garden

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