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Date May 2014 Marks available 4 Reference code 14M.1.sl.TZ1.8
Level SL only Paper 1 Time zone TZ1
Command term Calculate Question number 8 Adapted from N/A

Question

A child’s wooden toy consists of a hemisphere, of radius 9 cm , attached to a cone with the same base radius. O is the centre of the base of the cone and V is vertically above O.

Angle OVB is \({27.9^ \circ }\).

Diagram not to scale.


Calculate OV, the height of the cone.

[2]
a.

Calculate the volume of wood used to make the toy.

[4]
b.

Markscheme

\(\tan 27.9^\circ  = \frac{9}{{{\text{OV}}}}\)     (M1)

 

Note: Award (M1) for correct substitution in trig formula.

 

\({\text{OV}} = 17.0\left( {{\text{cm}}} \right)\left( {16.9980 \ldots } \right)\)     (A1)     (C2)

[2 marks]

a.

\(\frac{{\pi {{(9)}^2}(16.9980 \ldots )}}{3} + \frac{1}{2} \times \frac{{4\pi {{(9)}^3}}}{3}\)     (M1)(M1)(M1)

 

Note: Award (M1) for correctly substituted volume of the cone, (M1) for correctly substituted volume of a sphere divided by two (hemisphere), (M1) for adding the correctly substituted volume of the cone to either a correctly substituted sphere or hemisphere.

 

\( = 2970{\text{ c}}{{\text{m}}^3}{\text{ (2968.63}} \ldots {\text{)}}\)     (A1)(ft)     (C4)

 

Note: The answer is \(2970{\text{ c}}{{\text{m}}^3}\), the units are required.

 

[4 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 5 - Geometry and trigonometry » 5.5 » Volume and surface areas of the three-dimensional solids defined in 5.4.
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