A-LevelMathematicsMechanics9709/51 · May/June 2018 · Q710 Marks

A uniform solid cone has height 1.2 m and base radius 0.5 m. A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).

(i)Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is 0.0352πm³. [The volume of a cone is (1/3)πr²h.][4]
(ii)Find the distance of the centre of mass of the object from its base.[6]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[M1] Height of conical tip = 1.2 × 0.2 / 0.5 = 0.48; Use ratio of corresponding sides, similar figures
[A1] Cylindrical height = 1.2 – 0.48 = 0.72; AG
[M1] Volume removed = π0.2² × 0.48/3 + π0.2² × 0.72
[A1] Volume removed = 0.0352π; AG
[B1] Moment of cone removed about the base = 0.0064π(0.72 + 0.48/4) = 0.0064π × 0.84
[B1] Moment of cylinder removed about the base = 0.0288π × 0.72/2 = 0.0288π × 0.36
[B1] Moment of the original cone about the base = π 0.5² × 1.2/3 × 0.3 = 0.1π × 0.3
[M1] 0.1π × 0.3 = 0.0064π × 0.84 + 0.0288π × 0.36 + 0.0648πx; Attempt to take moments about the base
[A1] Note 0.0648π is the volume of the object
[A1] x = 0.22 m

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