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4PM1/02 Jun 2022, Question 8: Volume of a Prism

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 12 marks

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The question, in short

Open trapezium-prism basket with base 2x by h, top 8x wide, slant edges 5x, volume 2250 cm³; work with its surface area S.

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

Part (a) · 5 marks

Show that the five-face surface area S simplifies to 40x² + 1350/x using the given dimensions and volume.

  1. 1Height of trapezium = √((5x)² − (3x)²) = 4x
  2. 2V = (1/2)(2x + 8x) × 4x × h = 20x²h = 2250 ⇒ h = 2250/(20x²)
  3. 3S = 2 × 20x² + 2xh + 2(5xh) = 40x² + 12xh
  4. 4S = 40x² + 12x × 2250/(20x²) = 40x² + 1350/x

AnswerS = 40x² + 1350/x shown

Part (b) · 5 marks

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

  1. 1dS/dx = 80x − 1350/x²
  2. 2dS/dx = 0 ⇒ x³ = 1350/80 = 135/8
  3. 3x = 2.56
  4. 4d²S/dx² = 80 + 2700/x³ > 0 for x > 0, so S is a minimum

Answerx = 2.56; d²S/dx² = 80 + 2700/x³ > 0, so a minimum

Part (c) · 2 marks

Calculate the minimum value of S, to 3 significant figures.

  1. 1S = 40(2.565)² + 1350/2.565
  2. 2= 789 (3 s.f.)

AnswerS = 789

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