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4PM1/02 Jun 2024, Question 8: Volume of a Prism

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

For a solid right triangular prism with isosceles triangular cross-section, volume 3.6 cm³, derive and use a surface area formula in x, then minimise it.

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

Part (a) · 4 marks

Show the total external surface area S equals x²/2 + 72(√2 + 1)/(5x).

  1. 1x² = DE² + DE², so DE = x/√2
  2. 2Volume: ½ × (x/√2)² × y = 3.6, so y = 72/(5x²)
  3. 3S = 2 × x²/4 + 2 × (x/√2) × y + x × y
  4. 4= x²/2 + √2 × 72/(5x) + 72/(5x)
  5. 5= x²/2 + 72(√2 + 1)/(5x)

AnswerS = x²/2 + 72(√2 + 1)/(5x) shown

Part (b) · 4 marks

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

  1. 1dS/dx = x − 72(√2 + 1)/(5x²) = 0
  2. 2x³ = 72(√2 + 1)/5
  3. 3x = 3.2637… ≈ 3.26
  4. 4d²S/dx² = 1 + 144(√2 + 1)/(5x³) > 0 for x > 0, so minimum

Answerx = 3.26 (S is a minimum since d²S/dx² = 1 + 144(√2 + 1)/(5x³) > 0)

Part (c) · 2 marks

Hence find the minimum value of S to 2 significant figures.

  1. 1S = 3.2637²/2 + 72(√2 + 1)/(5 × 3.2637)
  2. 2= 15.977… ≈ 16

AnswerS = 16

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