A landscape garden feature is designed using two identical sectors of a circle, ORP ORP\,ORP and OQSOQSOQS, and a central rhombus-shaped flower bed OSTROSTROSTR, as shown in the layout. The points PPP, OOO, and Q Q\,Q lie on a straight stone boundary of length 12 metres, such that O O\,O is the midpoint of PQPQPQ. The radii OR OR\,OR and OS OS\,OS form two sides of the rhombus, and the angle ∠ROS \angle ROS\,∠ROS is denoted by θ \theta\,θ radians.
Show that the total area of the garden feature, A A\,A square metres, is given by
A=18(π−θ+2sinθ) A = 18(\pi - \theta + 2\sin\theta) A=18(π−θ+2sinθ)Use calculus to show that the maximum value of A A\,A occurs when θ=π3\displaystyle \theta = \frac{\pi}{3}θ=3π. Fully justify that this value of θ \theta\,θ gives a maximum.
Determine the exact maximum value of AAA.
Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if the total boundary length PQ PQ\,PQ were increased to 24 metres.
Practise AQA A Level Maths 1.8 E: Trigonometry with exam-style questions for A Level Maths. 239 questions covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context, matched to the AQA A Level Maths (7357) 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.
