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0580/43 May/Jun 2022, Question 4: Choosing the Right Trigonometric Ratio

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 11 marks

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The question, in short

For a regular 12-sided polygon with side 6 cm, prove an interior angle, find the circumradius, then circle and segment measures, and the volume of a 2 cm long prism.

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

Part (a) · 1 mark

Show that an interior angle of the regular 12-sided polygon equals 150°.

  1. 1Exterior angle = 360 ÷ 12 = 30°
  2. 2Interior angle = 180 − 30 = 150°

Answer((12 − 2) × 180)/12 = 150 or 180 − 360/12 = 150

Part (bi) · 3 marks

Using chord AB = 6 cm and centre O, show the circle radius AO is 11.6 cm to 1 decimal place.

  1. 1Angle AOB = 360 ÷ 12 = 30°, so angle OAB = 150 ÷ 2 = 75°
  2. 2Half of AB = 3 cm: cos 75 = 3/AO
  3. 3AO = 3 ÷ cos 75 = 11.59… = 11.6 cm (1 d.p.)

Answer11.59…

Part (biia) · 2 marks

Calculate the circumference of the circle.

  1. 1Circumference = 2 × π × 11.59…
  2. 2= 72.82… ≈ 72.8 cm

Answer72.8 or 72.9 or 72.82 to 72.89…

Part (biib) · 2 marks

Calculate the perimeter of the shaded minor segment cut off by chord AB.

  1. 1Arc AB = 72.82… ÷ 12 = 6.069… cm
  2. 2Perimeter = 6 + 6.069… = 12.06… ≈ 12.1 cm

Answer12.1 or 12.06 to 12.08

Part (c) · 3 marks

Find the volume of the prism whose cross-section is the 12-sided polygon and whose length is 2 cm.

  1. 1Area of triangle OAB = 1/2 × 6 × 11.59… × sin 75 = 33.58… cm²
  2. 2Cross-section area = 12 × 33.58… = 403.06… cm²
  3. 3Volume = 403.06… × 2 = 806.1… ≈ 806 cm³

Answer806 or 807 or 805.9 to 807.4

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Also searched as: 0580/43/M/J/22 · 0580_s22_qp_43 · 0580 s22 43 · question 4.
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