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4PM1/01 Jan 2020, Question 11: Volume of a Prism

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 13 marks

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The question, in short

For a right prism with equilateral triangular cross-section of side x cm and volume 72 cm³, work with its total surface area S, using calculus to minimise S.

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

Part (a) · 6 marks

Show that the total surface area S equals √3x²/2 + 288√3/x.

  1. 1Let h be the length of the prism. Area of triangle = ½x² sin 60° = (√3/4)x²
  2. 2Volume: (√3/4)x²h = 72
  3. 3h = 288/(√3x²) = 96√3/x²
  4. 4S = 2 × (√3/4)x² + 3xh
  5. 5S = (√3/2)x² + 3x × 96√3/x²
  6. 6S = √3x²/2 + 288√3/x

AnswerS = √3x²/2 + 288√3/x shown

Part (b) · 5 marks

Using calculus, find x to 4 significant figures giving minimum S, and justify it is a minimum.

  1. 1dS/dx = √3x − 288√3/x²
  2. 2Set dS/dx = 0: √3x = 288√3/x², so x³ = 288
  3. 3x = ∛288 = 6.604
  4. 4d²S/dx² = √3 + 576√3/x³
  5. 5At x = 6.604, d²S/dx² = 3√3 ≈ 5.20 > 0, so S is a minimum

Answerx = 6.604; d²S/dx² > 0 so this gives a minimum

Part (c) · 2 marks

Find the minimum value of S to 3 significant figures.

  1. 1S = √3(6.604)²/2 + 288√3/6.604
  2. 2= 37.77 + 75.53 = 113.3
  3. 3S = 113

AnswerS = 113

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