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4PM1/01 Jun 2023, Question 5: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks

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The question, in short

Cuboid with width x cm, length 4x cm, height h cm, volume 75 cm³ and surface area S cm². Use calculus to find the minimum surface area and the x value giving it.

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

Part (a) · 4 marks

Show that the surface area simplifies to S = 8x² + 375/(2x).

  1. 1V = x × 4x × h = 4x²h = 75, so h = 75/(4x²)
  2. 2S = 2(4x² + xh + 4xh) = 8x² + 10xh
  3. 3S = 8x² + 10x × 75/(4x²)
  4. 4S = 8x² + 375/(2x)

AnswerS = 8x² + 375/(2x) shown

Part (b) · 5 marks

Using calculus, find x to 3 significant figures giving minimum S, and justify it is a minimum.

  1. 1dS/dx = 16x − 375/(2x²)
  2. 2Set dS/dx = 0: 16x = 375/(2x²), so x³ = 375/32
  3. 3x = 2.2714… ≈ 2.27
  4. 4d²S/dx² = 16 + 375/x³ ≈ 48
  5. 5x > 0 so d²S/dx² > 0, so S is a minimum

Answer(i) x = 2.27 (ii) d²S/dx² = 16 + 375/x³ (≈ 48) > 0, so minimum

Part (c) · 2 marks

Find the minimum value of S to 3 significant figures.

  1. 1S = 8(2.2714)² + 375/(2 × 2.2714)
  2. 2= 41.27 + 82.55 = 123.82…
  3. 3S ≈ 124 cm²

AnswerS = 124 cm²

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