A-LevelMathematicsCircular measure9709/12 · May/June 2021 · Q1211 Marks
The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are A, B, C, D, E and F. Points P and Q are situated where straight sections of the rope meet the pipe with centre A. [Figure 12.1]
- (a)Show that angle PAQ = π/3 radians.[2]
- (b)Find the length of the rope.[4]
- (c)Find the area of the hexagon ABCDEF, giving your answer in terms of √3.[2]
- (d)Find the area of the complete region enclosed by the rope.[3]
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Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [M1] [By symmetry] [6 × PÂQ = 2π], [PÂQ=] 2π÷6,
- [A1] AG
- [B1] Each straight section of rope has length 40 cm
- [*M1] Each curved section round each pipe has length rθ=20×π/3=
- [DM1] Total length = 6×((their 40)+kπ)
- [A1] 240 + 40π or 366 (AWRT) (cm)
- [B1] [Triangle area =] 1/2×40×40×sin(π/3) or 1/2×40×20√3 or 400√3 or 693(AWRT)
- [B1] [Total area of hexagon = 6 × 400√3 =] 2400√3
- [B1] Each rectangle area = 40 × 20 (= 800)
- [B1] Each sector area = 1/2r²θ = 1/2×20²×π/3=200π/3
- [B1] Total area = 2400√3+4800+400π or 10200 (cm²) (AWRT)
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