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4PM1/02 Jun 2025, Question 6: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

Cylinder with radius r and height h has total surface area 700π cm². Use this to derive a volume formula, then use calculus to find r and h giving maximum volume.

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

Part (a) · 4 marks

Starting from the total surface area 700π, eliminate h to show the volume equals πr(350 − r²).

  1. 1700π = 2πr² + 2πrh
  2. 2h = (350 − r²)/r
  3. 3V = πr²h = πr²(350 − r²)/r
  4. 4V = πr(350 − r²)

AnswerV = πr(350 − r²) shown

Part (b) · 5 marks

Differentiate the volume with respect to r, find r in cm to 3 significant figures for maximum V, and justify it is a maximum.

  1. 1V = 350πr − πr³
  2. 2dV/dr = 350π − 3πr² = 0
  3. 3r² = 350/3, r = 10.8
  4. 4d²V/dr² = −6πr < 0 for r > 0, so V is a maximum

Answerr = 10.8 (maximum since d²V/dr² = −6πr < 0)

Part (c) · 1 mark

Using the r found in part b, calculate the height h in cm to 3 significant figures.

  1. 1h = (350 − r²)/r with r² = 350/3
  2. 2h = (700/3)/√(350/3) = 21.6

Answerh = 21.6 cm

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