O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2015 · Q79 Marks

A cone, of height 8 cm and base radius 6cm, is placed over a cylinder of radius r cm and height h cm and is in contact with the cylinder along the cylinder's upper rim. The arrangement is symmetrical and the diagram shows a vertical cross-section through the vertex of the cone. [Figure 7.1]

(i)Use similar triangles to express h in terms of r.[2]
(ii)Hence show that the volume, V cm³, of the cylinder is given by V = 8πr² – (4/3)πr³.[1]
(iii)Given that r can vary, find the value of r which gives a stationary value of V. Find this stationary value of V in terms of π and determine its nature.[6]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.

[M1] (i) h / (6-r) = 8/6 Uses correct ratio. Cannot be implied
[A1] (i) h = (4/3)(6-r)
[B1] (ii) V = πr²h = πr²(4/3)(6-r) = 8πr² - (4/3)πr³ AG all steps must be seen Penalise missing brackets at any point in working
[M1] (iii) dV/dr = 16πr – 4πr² Differentiate at least one power reduced by one
[A1] (iii)
[M1] (iii) dV/dr = 0 → r = 4 Attempt to solve – must get r = ...
[A1] (iii) Correct value of r. Ignore r =0
[A1] (iii) V = (128/3)π (= 42.7π) Correct value of V. Condone 134.
[B1] (iii) d²V/dr² = 16π – 8πr <0 when r = 4 → max d²V/dr² must be correct and some indication of a negative value seen plus maximum stated

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