A landscape architect is designing a fan-shaped flower bed which is a sector of a circle with radius rrr metres and central angle ϕ\phiϕ radians. The design requires the flower bed to have a total perimeter of 18 m18\text{ m}18 m and an area of 20 m220\text{ m}^220 m2.
Show that 5ϕ2−21ϕ+18=05\phi^2 - 21\phi + 18 = 05ϕ2−21ϕ+18=0 is NOT correct for these dimensions, and instead determine the correct quadratic equation in terms of ϕ\phiϕ only, in the form aϕ2+bϕ+c=0a\phi^2 + b\phi + c = 0aϕ2+bϕ+c=0, where a,b,ca, b, ca,b,c are integers.
Hence find the two possible pairs of values for rrr and ϕ\phiϕ.
Practise Edexcel A Level Maths Radians with exam-style questions for A Level Maths. 52 questions covering 5.1 Radian Measure, 5.2 Arc Length, 5.3 Areas of Sectors and Segments, 5.4 Solving Trigonometric Equations, and 5.5 Small Angle Approximations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.