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4PM1/01R Jan 2022, Question 7: Volume of a Prism

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

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The question, in short

Right triangular prism with triangular ends 3x, 4x, 5x cm and length y cm; total surface area 144 cm². Show V = 72x − 6x³, then maximise V using calculus and find its maximum.

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

Part (a) · 5 marks

Show that the prism volume V, in terms of x, equals 72x − 6x³ using the given total surface area of 144 cm².

  1. 1Surface area: 2 × ½ × 3x × 4x + 3xy + 4xy + 5xy = 144
  2. 212x² + 12xy = 144, so y = (144 − 12x²)/(12x)
  3. 3V = ½ × 3x × 4x × y = 6x²y
  4. 4V = 6x² × (144 − 12x²)/(12x) = 6x(12 − x²)
  5. 5V = 72x − 6x³

AnswerV = 72x − 6x³ shown

Part (b) · 4 marks

Use calculus to find the x value giving maximum V, and justify that it is a maximum.

  1. 1dV/dx = 72 − 18x²
  2. 272 − 18x² = 0, x² = 4, x = 2 (x > 0)
  3. 3d²V/dx² = −36x
  4. 4At x = 2, d²V/dx² = −72 < 0, so V is a maximum

Answerx = 2; d²V/dx² = −36x = −72 < 0, so maximum

Part (c) · 2 marks

Calculate the maximum value of V.

  1. 1V = 72 × 2 − 6 × 2³
  2. 2= 144 − 48 = 96

Answer96

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