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Date	November 2021	Marks available	2	Reference code	21N.1.SL.TZ0.8
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	8	Adapted from	N/A

Question

Joey is making a party hat in the form of a cone. The hat is made from a sector, $AOB$ , of a circular piece of paper with a radius of $18 cm$ and $AÔB = θ$ as shown in the diagram.

To make the hat, sides $[OA]$ and $[OB]$ are joined together. The hat has a base radius of $6.5 cm$ .

Write down the perimeter of the base of the hat in terms of $π$ .

[1]

a.i.

Find the value of $θ$ .

[2]

a.ii.

Find the surface area of the outside of the hat.

[2]

Markscheme

$13 π cm$ A1

Note: Answer must be in terms of $π$ .

[1 mark]

a.i.

METHOD 1

$\frac{θ}{360} \times 2 π (18) = 13 π$ OR $\frac{θ}{360} \times 2 π (18) = 40.8407 \dots$ (M1)

Note: Award (M1) for correct substitution into length of an arc formula.

$(θ =) 130 °$ A1

METHOD 2

$\frac{θ}{360} \times π \times 18^{2} = π \times 6.5 \times 18$ (M1)

$(θ =) 130 °$ A1

[2 marks]

a.ii.

EITHER

$\frac{130}{360} \times π {(18)}^{2}$ (M1)

Note: Award (M1) for correct substitution into area of a sector formula.

$π (6.5) (18)$ (M1)

Note: Award (M1) for correct substitution into curved area of a cone formula.

THEN

(Area $=$ ) $368 {cm}^{2} (367.566 \dots, 117 π)$ A1

Note: Allow FT from their part (a)(ii) even if their angle is not obtuse.

[2 marks]

Examiners report

Although most candidates understood what to do in part (a), many of them wrote a decimal approximation instead and did not give their answer in terms of $π$ as required in this part. Many candidates were able to use the length of arc formula in (a)(ii). Some candidates went wrong with this question by confusing between the two values: $6.5 cm$ and $18 cm$ for the radius. In part (b), although candidates were able to find the surface area of the outside of the hat, several added the surface area of the base to their calculation. A few candidates used the cosine rule to find chord $AB$ which was then used as the circumference of the base of the cone.

a.i.

a.ii.

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.1—3d space, volume, angles, midpoints

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Hide related questions

Topic 3—Geometry and trigonometry » SL 3.4—The circle, arc and area of sector, degrees only

Topic 3—Geometry and trigonometry

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Markscheme

Examiners report

Syllabus sections

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