WMA11/01 Jan 2022, Question 5: Radians, Arcs and Sectors
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks
The question, in short
Use a semicircle of radius 3 m with chord BD parallel to diameter AE and angle BOA 0.7 radians to work out angle BOD, then the shaded flowerbed's area and perimeter.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 1 mark
Show the angle BOD equals 1.742 radians to 4 significant figures, using the given 0.7 radian angle at B.
- 1BD is parallel to AE, so by symmetry angle DOE = angle BOA = 0.7
- 2Angles on the straight line AOE add to π
- 3Angle BOD = π − 2 × 0.7 = 1.7416…
- 4Angle BOD = 1.742 radians (4 s.f.)
AnswerShown: by symmetry angle DOE = 0.7, so angle BOD = π − 2 × 0.7 = 1.7416… = 1.742 (4 s.f.)
Part (b) · 3 marks
Calculate the shaded region's area between chord BD and arc BCD, in square metres to one decimal place.
- 1Area of sector BOD = ½ × 3² × 1.742 = 7.839
- 2Area of triangle BOD = ½ × 3² × sin 1.742 = 4.434
- 3Area of R = 7.839 − 4.434 = 3.405
- 4Area ≈ 3.4 m²
Answer3.4 m²
Part (c) · 3 marks
Calculate the flowerbed's perimeter, combining chord BD with arc BCD, in metres to one decimal place.
- 1Arc BCD = 3 × 1.742 = 5.226
- 2BD = √(3² + 3² − 2 × 3 × 3 cos 1.742) = 4.589
- 3Perimeter = 5.226 + 4.589 = 9.815
- 4Perimeter ≈ 9.8 m
Answer9.8 m
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