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4MA1/1H Jan 2020, Question 15: Stationary Points and Turning Points

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 8 marks

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The question, in short

Use a cuboid with side lengths x + 1, 3 − x and 2x + 5 to derive its volume formula and locate the maximum volume.

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

Part (a) · 3 marks

Show the cuboid volume expands to 15 + 16x − x² − 2x³.

  1. 1V = (x + 1)(2x + 5)(3 − x)
  2. 2(x + 1)(2x + 5) = 2x² + 7x + 5
  3. 3(2x² + 7x + 5)(3 − x) = 6x² + 21x + 15 − 2x³ − 7x² − 5x
  4. 4Collect terms: V = 15 + 16x − x² − 2x³

Answer15 + 16x − x² − 2x³

Part (b) · 5 marks

Find the x giving maximum volume, to 3 significant figures, showing working.

  1. 1dV/dx = 16 − 2x − 6x²
  2. 2At a maximum dV/dx = 0: 6x² + 2x − 16 = 0
  3. 3x = (−2 ± √(4 + 384)) / 12 = (−2 ± √388) / 12
  4. 4x = 1.474… or x = −1.808…
  5. 5A length cannot be negative, so x = 1.47

Answer1.47

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