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4MA1/2HR Jun 2022, Question 24: Frustum of a Cone

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 6 marks

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The question, in short

Use similar cones and a hemisphere joined to a frustum or small cone to relate volumes of solids A and B, then find h in terms of k and r given k > ∛7.

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

Solution · 6 marks

  1. 1Volume of hemisphere = (2/3)πr³
  2. 2Volume of small cone = (1/3)πr²h
  3. 3Volume of large cone = (1/3)π(kr)²(kh) = (1/3)πk³r²h
  4. 4Volume of frustum = (1/3)πk³r²h − (1/3)πr²h = (1/3)πr²h(k³ − 1)
  5. 5Volume of A = (1/3)πr²h(k³ − 1) + (2/3)πr³
  6. 6Volume of B = (1/3)πr²h + (2/3)πr³
  7. 7Volume of A = 6 × volume of B:
  8. 8(1/3)πr²h(k³ − 1) + (2/3)πr³ = 6[(1/3)πr²h + (2/3)πr³]
  9. 9Multiply both sides by 3 ÷ π:
  10. 10r²h(k³ − 1) + 2r³ = 6r²h + 12r³
  11. 11r²hk³ − r²h − 6r²h = 12r³ − 2r³
  12. 12r²h(k³ − 7) = 10r³
  13. 13Divide both sides by r²:
  14. 14h(k³ − 7) = 10r
  15. 15h = 10r/(k³ − 7)

Answerh = 10r/(k³ − 7)

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