21M.1.SL.TZ2.9

pestleMathematicsAISLPaper 121M· sl-3-2-2d-and-3d-trigsource ↗

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

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]

Find the area of the lawn.

[4]

Markscheme / solution

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

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

[3 marks]

evidence of splitting region into two areas (M1)

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

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

$42.4115 \dots - 12.7279 \dots$

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

[4 marks]

Examiners’ report

[N/A]