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4PM1/01 Jan 2022, Question 8: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

A cylinder with radius r cm and height h cm has volume 90π cm³ and surface area S cm²; derive an expression for S, then use calculus to minimise it and find the least surface area.

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

Part (a) · 3 marks

Starting from the given volume and surface area of the cylinder, derive the stated formula for S in terms of r only.

  1. 1πr²h = 90π, so h = 90/r²
  2. 2S = 2πr² + 2πrh
  3. 3S = 2πr² + 2πr × 90/r² = 2πr² + 180π/r

AnswerS = 2πr² + 180π/r shown

Part (b) · 5 marks

Differentiate S with respect to r, solve for the stationary r to 3 significant figures, and confirm it is a minimum.

  1. 1dS/dr = 4πr − 180π/r²
  2. 24πr − 180π/r² = 0 gives r³ = 45
  3. 3r = ∛45 = 3.5568… = 3.56
  4. 4d²S/dr² = 4π + 360π/r³ = 37.7 > 0, so S is a minimum

Answerr = 3.56; d²S/dr² > 0 so minimum

Part (c) · 2 marks

Substitute the r found in part b into S and state the least surface area to 3 significant figures.

  1. 1S = 2π(3.5568…)² + 180π/3.5568…
  2. 2= 238.47… = 238 cm²

AnswerS = 238 cm²

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