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4MA1/1HR Jan 2019, Question 17: Volume and Surface Area of a Sphere

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 5 marks

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The question, in short

For a sphere and cylinder both with radius r, cylinder height 2r, find k in the cylinder surface area kπ, then compare two ratios.

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

Part (a) · 2 marks

Find k, in terms of r, for the cylinder's total surface area given as kπ cm².

  1. 1Two circular ends: 2 × πr² = 2πr²
  2. 2Curved surface: 2πr × 2r = 4πr²
  3. 3Total = 2πr² + 4πr² = 6πr² = kπ
  4. 4k = 6r²

Answer6r²

Part (b) · 3 marks

Show the ratio of cylinder to sphere surface areas equals the ratio of their volumes.

  1. 1Total surface area of the cylinder = 6πr²
  2. 2Surface area of the sphere = 4πr²
  3. 3surface area of cylinder : surface area of sphere = 6πr² : 4πr² = 6 : 4 = 3 : 2
  4. 4Volume of the cylinder = πr² × 2r = 2πr³
  5. 5Volume of the sphere = (4/3)πr³
  6. 6volume of cylinder : volume of sphere = 2πr³ : (4/3)πr³ = 2 : 4/3
  7. 7Multiplying both parts by 3 gives 6 : 4 = 3 : 2
  8. 8Both ratios are 3 : 2, so they are the same

AnswerBoth ratios are 3 : 2, since 6πr² : 4πr² = 3 : 2 and 2πr³ : (4/3)πr³ = 3 : 2

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