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WMA11/01 Oct 2022, Question 8: Triangles and Sectors in Radians

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 14 marks

The question, in short

For a pond plan made of major sector AOBX (radius 3 m, angle 2π/3) joined to quadrilateral AOBC with BC = 8 m and angle BCA = π/6, find areas, perimeter, AB and total surface area.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (ai) · 3 marks

Calculate the exact area of the major sector AOBX using radius 3 m and angle 2π/3 radians.

  1. 1Angle of sector AOBX = 2π − 2π/3 = 4π/3
  2. 2Area = ½ × 3² × 4π/3
  3. 3= 6π m²

Answer6π m²

Part (aii) · 2 marks

Calculate the exact perimeter of the major sector AOBX, including its two radii and arc.

  1. 1Arc length = 3 × 4π/3 = 4π
  2. 2Perimeter = 4π + 3 + 3 = 4π + 6 m

Answer4π + 6 m

Part (b) · 2 marks

Calculate the exact area of triangle AOB with OA = OB = 3 m and angle AOB = 2π/3.

  1. 1Area = ½ × 3 × 3 × sin(2π/3)
  2. 2= (9/2)(√3/2) = 9√3/4 m²

Answer9√3/4 m²

Part (c) · 2 marks

Show, using triangle AOB, that the chord length AB equals 3√3 m.

  1. 1AB² = 3² + 3² − 2(3)(3)cos(2π/3)
  2. 2= 18 − 18(−½) = 27
  3. 3AB = √27 = 3√3 m

AnswerShown: AB² = 9 + 9 − 18cos(2π/3) = 27, so AB = 3√3 m

Part (d) · 5 marks

Find the pond's total surface area in m² to 2 significant figures, combining sector and quadrilateral areas.

  1. 1Sine rule in ABC: sin BAC/8 = sin(π/6)/(3√3) ⇒ sin BAC = 4√3/9, BAC = 0.8785…
  2. 2Angle ABC = π − π/6 − 0.8785… = 1.739…
  3. 3Area ABC = ½ × 3√3 × 8 × sin(1.739…) = 20.49…
  4. 4Total = 6π + 9√3/4 + 20.49… = 18.85 + 3.90 + 20.49
  5. 5= 43.2… ≈ 43 m²

Answer43 m²

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Also searched as: WMA11 October 2022 · WMA11/01 · wma11 oct22 · IAS Maths Pure Mathematics 1 October 2022 · question 8.
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