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0580/41 Oct/Nov 2024, Question 6: The Sine Rule

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 11 marks

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The question, in short

Use the field diagram with diagonal AC, given side lengths and angles, to find AC, an obtuse angle, and the cost per hectare of triangle ACD sold for $41 500.

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

Part (a) · 3 marks

Using AD = 420 m, DC = 830 m and angle ADC = 106°, show that AC rounds to 1028 m.

  1. 1AC² = 420² + 830² − 2 × 420 × 830 × cos 106°
  2. 2AC² = 1 057 474.3…
  3. 3AC = 1028.3… = 1028 m (nearest metre)

Answer1028.3...

Part (b) · 4 marks

Given AB = 1150 m, angle ABC = 62° and AC from part a, calculate the obtuse angle ACB.

  1. 1sin ACB/1150 = sin 62°/1028
  2. 2sin ACB = 1150 × sin 62° ÷ 1028 = 0.9877…
  3. 3Acute angle = 81.02°, but angle ACB is obtuse
  4. 4Angle ACB = 180 − 81.02 = 98.98… ≈ 99°

Answer99 or 98.98 to 99.1

Part (c) · 4 marks

Find the area of triangle ACD and use the $41 500 price to calculate cost per hectare, to the nearest dollar.

  1. 1Area ACD = 1/2 × 420 × 830 × sin 106° = 167 547.9… m²
  2. 2= 167 547.9… ÷ 10 000 = 16.754… hectares
  3. 3Cost of 1 hectare = 41 500 ÷ 16.754… = 2476.9…
  4. 4= $2477

Answer2477

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