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0580/42 May/Jun 2023, Question 10: Trigonometry in 3D: Cuboids

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 8 marks

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The question, in short

Diagonals of cuboids: find the base side x from a 3D diagonal of 28 cm and height 20 cm, then a lower-bound angle between a diagonal and the base using rounded lengths.

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Part (a) · 4 marks

Square-based cuboid, height 20 cm, space diagonal 28 cm: find the side x.

  1. 1AC² = AG² − CG² = 28² − 20² = 784 − 400 = 384
  2. 2AC² = x² + x², so 2x² = 384
  3. 3x² = 192
  4. 4x = √192 = 13.85… ≈ 13.9

Answer13.9 or 13.85 to 13.86

Part (b) · 4 marks

Lengths 30 cm and 37 cm to the nearest cm: lower bound of the angle between the diagonal and the base.

  1. 1The angle between KR and the base is angle RKM, with sin RKM = MR/KR
  2. 2Lower bound: smallest MR and largest KR
  3. 3sin RKM = (30 − 0.5)/(37 + 0.5) = 29.5/37.5
  4. 4Angle = sin⁻¹(0.7866…) = 51.87…° ≈ 51.9°

Answer51.9 or 51.87 to 51.88

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