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4PM1/01R Jun 2019, Question 9: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 10 marks

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The question, in short

A cuboid has sides x cm, 3x cm and h cm, volume 540 cm³ and surface area S cm²; derive a formula for S, then use calculus to minimise it.

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

Part (a) · 4 marks

Using the given volume, express h in terms of x and substitute into the surface area formula to obtain S = 6x² + 1440/x.

  1. 1Volume: 3x²h = 540 → h = 180/x²
  2. 2S = 2(3x² + 3xh + xh) = 6x² + 8xh
  3. 3S = 6x² + 8x(180/x²) = 6x² + 1440/x

AnswerS = 6x² + 1440/x shown

Part (b) · 5 marks

Differentiate S with respect to x, solve for the stationary x to 3 significant figures, and confirm it is a minimum.

  1. 1dS/dx = 12x − 1440x⁻²
  2. 212x − 1440/x² = 0 → x³ = 120
  3. 3x = 4.932… ≈ 4.93
  4. 4d²S/dx² = 12 + 2880/x³ > 0 for x > 0, so S is a minimum

Answerx = 4.93; d²S/dx² = 12 + 2880/x³ > 0 so minimum

Part (c) · 1 mark

Substitute the x found in part b into S and state the minimum surface area to 3 significant figures.

  1. 1S = 6(4.932…)² + 1440/4.932… = 437.9…
  2. 2S = 438 cm² (3 s.f.)

Answer438 cm²

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