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0580/41 May/Jun 2023, Question 2: Choosing the Right Trigonometric Ratio

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 16 marks

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The question, in short

Triangular prism ABCDEF, length 15 cm, AB = 6.4 cm, EB = 5.7 cm, volume 145 cm³; work through angle, lengths, distance, angle with base and mass.

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

Part (a) · 3 marks

Show that angle EBA is 32°, rounded to the nearest degree, using the given dimensions and volume.

  1. 1Area of triangle ABE = 145 ÷ 15 = 9.666… cm²
  2. 21/2 × 6.4 × 5.7 × sin EBA = 9.666…
  3. 3sin EBA = 9.666… ÷ 18.24 = 0.52997…
  4. 4Angle EBA = sin⁻¹(0.52997…) = 32.00… = 32° to the nearest degree

Answersin EBA = 145/(1/2 × 6.4 × 5.7 × 15), giving 32.00

Part (b) · 3 marks

Find the length of EA in the triangular end of the prism.

  1. 1EA² = 6.4² + 5.7² − 2 × 6.4 × 5.7 × cos 32°
  2. 2EA² = 73.45 − 61.87… = 11.57…
  3. 3EA = √11.57… = 3.402… ≈ 3.40 cm

Answer3.40 or 3.402 to 3.403

Part (c) · 3 marks

Calculate the shortest distance from point E to line AB.

  1. 1The shortest distance is the perpendicular from E to AB
  2. 2h = 5.7 × sin 32°
  3. 3h = 3.020… ≈ 3.02 cm

Answer3.02 or 3.020 to 3.021

Part (d) · 4 marks

Calculate the angle that line BF makes with the rectangular base ABCD.

  1. 1The angle is between BF and its projection on the base; the height of F above the base is 3.020… cm
  2. 2BF = √(15² + 5.7²) = √257.49 = 16.046…
  3. 3sin θ = 3.020… ÷ 16.046… = 0.1882…
  4. 4θ = 10.84…° ≈ 10.8°

Answer10.8 or 10.9 or 10.84 to 10.85...

Part (e) · 3 marks

Using density 938 kg/m³ and the prism volume, calculate the mass in grams.

  1. 1Volume = 145 cm³ = 145 ÷ 1 000 000 m³
  2. 2Mass = 938 × 145 ÷ 1 000 000 = 0.13601 kg
  3. 3= 136.01… g ≈ 136 g

Answer136 or 136.0...

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