Lemmara logoLemmaraWorked solution

4024/22 May/Jun 2020, Question 10: Trigonometry in 3D: Cuboids

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 11 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Rectangular-based pyramid ABCDE has base ABCD, apex E above centre F, with AD = 10, DC = 6, EC = 12; find height, volume, angle DEC and triangle DEC area.

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

Part (a) · 4 marks

Show that the vertical height EF is 10.5 cm to 1 decimal place.

  1. 1AC² = 10² + 6² = 136, so AC = √136
  2. 2FC = 1/2 × √136 = √34
  3. 3EF² + FC² = EC²: EF² = 12² − 34 = 110
  4. 4EF = √110 = 10.488… = 10.5 cm (1 d.p.)

Answer10.49 or 10.488…

Part (b) · 2 marks

Find the volume of the pyramid.

  1. 1Diagonal AC = √(10² + 6²) = √136 = 11.66… cm
  2. 2F is the midpoint of AC, so FC = 11.66… ÷ 2 = 5.83… cm
  3. 3Height EF = √(12² − 5.83…²) = √(144 − 34) = √110 = 10.49… cm
  4. 4Volume = 1/3 × 6 × 10 × 10.49… = 209.76… ≈ 210 cm³

Answer210 or 209.7 to 209.8

Part (c) · 3 marks

Calculate angle DEC.

  1. 1E is directly above F, the centre of the base, so ED = EC = 12 cm
  2. 2Cosine rule in triangle DEC: cos DEC = (12² + 12² − 6²)/(2 × 12 × 12)
  3. 3cos DEC = 252/288 = 0.875
  4. 4Angle DEC = 28.955… = 29.0°

Answer29.0 or 28.95 to 28.96

Part (d) · 2 marks

Calculate the area of triangle DEC.

  1. 1DE = EC = 12 cm, so triangle DEC is isosceles with base DC = 6 cm.
  2. 2The height from E meets DC at its midpoint, 3 cm from D.
  3. 3Height = √(12² − 3²) = √135 = 11.62… cm
  4. 4Area = 1/2 × 6 × 11.62… = 34.86 ≈ 34.9 cm²

Answer34.9 or 34.85 to 34.86

Stuck on a step? Read the free lesson: Trigonometry in 3D: Cuboids

Practise more trigonometry in 3d: cuboids questions

Real 4024 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on trigonometry in 3d: cuboids

Also searched as: 4024/22/M/J/20 · 4024_s20_qp_22 · 4024 s20 22 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.