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4PM1/02R Jan 2020, Question 4: Volume of a Pyramid

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 9 marks

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The question, in short

Triangular pyramid ABCX has equilateral base ABC with sides 8 cm and vertical edge AX = h cm. Volume is 48√3 cm³. Find h, angle BXC, and angle between plane BCX and base ABC.

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

Part (a) · 3 marks

Show that the vertical height h equals 9 using the given volume of 48√3 cm³ and the equilateral base.

  1. 1Base area = ½ × 8 × 8 × sin 60° = 16√3
  2. 2Volume = ⅓ × 16√3 × h = 48√3
  3. 3h = 9

Answerh = 9 shown

Part (b) · 3 marks

Calculate the size of angle BXC in degrees, giving the answer to one decimal place.

  1. 1BX = CX = √(9² + 8²) = √145
  2. 2cos BXC = (145 + 145 − 64)/(2 × 145) = 226/290
  3. 3∠BXC = 38.80… = 38.8°

Answer38.8°

Part (c) · 3 marks

Find the angle between plane BCX and the base ABC, in degrees to one decimal place.

  1. 1Let M be the midpoint of BC; AM = √(8² − 4²) = 4√3
  2. 2The angle is ∠XMA, with tan XMA = 9/(4√3)
  3. 3∠XMA = 52.41… = 52.4°

Answer52.4°

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