4PM1/01 Nov 2023, Question 6: Trigonometry in 3D: Pyramids and Cones
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks
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The question, in short
Right pyramid VABCD has square base side x and slant edges x, with V above centre O; show VO, find angles AVC and between planes VAB and VDC, then x for volume 200 cm³.
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 VO equals (√2/2)x cm using the base diagonal and a right triangle.
- 1AC = √(x² + x²) = √2 x
- 2AO = (√2/2)x
- 3VO = √(x² − ((√2/2)x)²) = √(x²/2)
- 4VO = (√2/2)x
AnswerVO = (√2/2)x shown
Part (b) · 2 marks
Find the angle AVC in degrees using triangle AVC with sides x, x and diagonal AC.
- 1Triangle AVC has VA = VC = x and AC = √2 x
- 2x² + x² = (√2 x)², so angle AVC = 90°
Answer90°
Part (c) · 3 marks
Find the angle between planes VAB and VDC, to one decimal place.
- 1Let M, N be midpoints of AB and DC: required angle is MVN
- 2VM = VN = √(x² − (x/2)²) = (√3/2)x, MN = x
- 3cos MVN = (3x²/4 + 3x²/4 − x²)/(2 × 3x²/4) = 1/3
- 4Angle MVN = 70.5°
Answer70.5°
Part (d) · 3 marks
Given volume 200 cm³, find x to 3 significant figures using the pyramid volume formula.
- 1Volume = (1/3) × x² × (√2/2)x = (√2/6)x³
- 2(√2/6)x³ = 200
- 3x³ = 1200/√2 = 848.5…
- 4x = 9.47
Answerx = 9.47
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