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0606/21 Oct/Nov 2025, Question 11: Tangents, Normals and Stationary Points (Further)

Cambridge IGCSE Additional Mathematics 0606 · Paper 21 (Calculator) · Calculator · 8 marks

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The question, in short

Cylinder with radius r cm, height h cm, volume 330 cm³ and total surface area A cm². Show the given formula for A, then find the r giving a stationary A and prove it is a minimum.

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Part (a) · 3 marks

Using volume 330 cm³, express h in terms of r and substitute into total surface area including both ends to show A = 2πr² + 660/r.

  1. 1Volume: πr²h = 330, so h = 330/(πr²)
  2. 2A = 2πr² + 2πrh
  3. 3A = 2πr² + 2πr × 330/(πr²)
  4. 4A = 2πr² + 660/r

AnswerA = 2πr² + 660/r shown

Part (b) · 5 marks

Differentiate A with respect to r, set to zero to find the stationary r, then use the second derivative to show it is a minimum.

  1. 1dA/dr = 4πr − 660r⁻²
  2. 2Stationary: 4πr = 660/r², so r³ = 660/(4π) = 165/π
  3. 3r = ∛(165/π) = 3.74
  4. 4d²A/dr² = 4π + 1320/r³
  5. 5For r > 0 this is positive (= 12π ≈ 37.7 at r = 3.74), so A is a minimum

Answerr = ∛(660/(4π)) = 3.74; d²A/dr² = 4π + 1320/r³ > 0, so minimum

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