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.
