O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2017 · Q68 Marks

The volume of a closed cylinder of base radius x cm and height h cm is 500 cm³.

(i)Express h in terms of x.[1]
(ii)Show that the total surface area of the cylinder is given by A = 2πx² + 1000/x cm².[2]
(iii)Given that x can vary, find the stationary value of A and show that this value is a minimum.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[B1] πx²h = 500 → h=500/(πx²); Ignore units Condone r for x
[M1] A = 2πx² + 2πxh; Correct expression for A and insert for their h.
[A1] = 2πx² + 2πx × 500/(πx²) = 2πx² + 1000/x; Answer given Condone r for x.
[M1] Differentiate: at least one power reduced by 1
[A1] dA/dx = 4πx - 1000/x²
[A1] dA/dx=0 → x³ = 1000/(4π); isw or (x = 4.3(0))
[A1] A = 2π (4.3)² + 1000/4.3 = 349 cm²; awrt 349
[B1] d²A/dx² = 4π + 2000/x³ (>0) or a positive value (→ min); Correct second differential (need not be evaluated) and conclusion. or Examine correct gradient either side of x = 4.3 and conclusion

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2017 Paper 2 Variant 2, question 6.

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