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Date	May 2021	Marks available	3	Reference code	21M.1.SL.TZ2.9
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Find	Question number	9	Adapted from	N/A

Question

A garden includes a small lawn. The lawn is enclosed by an arc $AB$ $AB$ of a circle with centre $O$ $O$ and radius $6 m$ $6 m$ , such that $AÔB = 135 °$ $AÔB = 135 °$ . The straight border of the lawn is defined by chord $[AB]$ $[AB]$ .

The lawn is shown as the shaded region in the following diagram.

A footpath is to be laid around the curved side of the lawn. Find the length of the footpath.

[3]

a.

Find the area of the lawn.

[4]

b.

Markscheme

$135 ° \times \frac{12 π}{360 °}$ $135 ° \times \frac{12 π}{360 °}$ (M1)(A1)

$14.1 (m) (14.1371 \dots)$ $14.1 (m) (14.1371 \dots)$ A1

[3 marks]

a.

evidence of splitting region into two areas (M1)

$135 ° \times \frac{π 6^{2}}{360 °} - \frac{6 \times 6 \times sin 135 °}{2}$ $135 ° \times \frac{π 6^{2}}{360 °} - \frac{6 \times 6 \times \sin 135 °}{2}$ (M1)(M1)

Note: Award M1 for correctly substituting into area of sector formula, M1 for evidence of substituting into area of triangle formula.

$42.4115 \dots - 12.7279 \dots$ $42.4115 \dots - 12.7279 \dots$

$29.7 m^{2} (29.6835 \dots)$ $29.7 m^{2} (29.6835 \dots)$ A1

[4 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.2—2d and 3d trig

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21M.2.SL.TZ2.2b:
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21M.2.SL.TZ2.2d:
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19M.2.AHL.TZ2.H_1:
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Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.
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EXN.1.SL.TZ0.11a:
Find the area of the field.
EXN.1.SL.TZ0.11b:
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Find $BD$ . Fully justify any assumptions you make.
21M.1.SL.TZ1.9:
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Calculate the maximum possible area of the field.
21M.1.SL.TZ2.9b:
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21N.1.AHL.TZ0.11:
The following diagram shows a corner of a field bounded by two walls defined by lines $L_{1}$ and $L_{2}$ . The walls meet at a point $A$ , making an angle of $40 °$ .

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He wants the enclosure to take up as little of the current field as possible.

Find the minimum possible area of the triangular enclosure $ABC$ .
21N.2.SL.TZ0.4a:
Find the angle of depression from $M$ to $N$ .
21N.2.SL.TZ0.4b.i:
Find $CV$ .
21N.2.SL.TZ0.4c:
By finding an appropriate value, determine whether Joshua is correct.
21N.2.SL.TZ0.4d:
To avoid water leaking into the ground, the five interior sides of the reservoir have been painted with a watertight material.

Find the area that was painted.
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Topic 3—Geometry and trigonometry » SL 3.4—The circle, arc and area of sector, degrees only

Topic 3—Geometry and trigonometry