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4PM1/02R Jun 2025, Question 7: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 11 marks

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The question, in short

A cuboid x by 2x by y with volume 72 cm³: express its surface area in terms of x, then use differentiation to find and justify its minimum.

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

Part (a) · 4 marks

Eliminate y to get the surface area formula.

  1. 1Volume: 2x²y = 72, so y = 36/x²
  2. 2A = 2(2x·x) + 2(2x·y) + 2(x·y) = 4x² + 6xy
  3. 3A = 6x(36/x²) + 4x²
  4. 4A = 216/x + 4x²

AnswerA = 216/x + 4x² shown

Part (b) · 7 marks

Differentiate, set to zero, and check it is a minimum.

  1. 1dA/dx = −216/x² + 8x
  2. 2−216/x² + 8x = 0, so x³ = 27, x = 3
  3. 3A = 216/3 + 4(3²) = 108
  4. 4d²A/dx² = 432/x³ + 8
  5. 5At x = 3: 432/27 + 8 = 24 > 0, so minimum

AnswerMinimum A = 108 (at x = 3), since d²A/dx² = 24 > 0

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