1.8 E: Trigonometry

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Question 52

A landscape garden feature is designed using two identical sectors of a circle, $ORP\,$ and $O QS$ , and a central rhombus-shaped flower bed $O S T R$ , as shown in the layout. The points $P$ , $O$ , and $Q\,$ lie on a straight stone boundary of length 12 metres, such that $O\,$ is the midpoint of $P Q$ . The radii $OR\,$ and $OS\,$ form two sides of the rhombus, and the angle $\angle ROS\,$ is denoted by $\theta\,$ radians.

Layout of two identical sectors ORP and OQS beside rhombus OSTR, with P, O and Q collinear, PQ equal to 12 metres, and angle ROS labelled theta.

Show that the total area of the garden feature, $A\,$ square metres, is given by

18(\pi - \theta + 2\sin\theta)

[4]

bi.

Use calculus to show that the maximum value of $A\,$ occurs when $θ=π3\displaystyle \theta = \frac{\pi}{3}$ . Fully justify that this value of $\theta\,$ gives a maximum.

[5]

bii.

Determine the exact maximum value of $A$ .

[2]

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if the total boundary length $PQ\,$ were increased to 24 metres.

[2]

1.8 E: Trigonometry

Question 52

Layout of two identical sectors ORP and OQS beside rhombus OSTR, with P, O and Q collinear, PQ equal to 12 metres, and angle ROS labelled theta.

Show that the total area of the garden feature, $A\,$ square metres, is given by

18(\pi - \theta + 2\sin\theta)

[4]

bi.

Use calculus to show that the maximum value of $A\,$ occurs when $θ=π3\displaystyle \theta = \frac{\pi}{3}$ . Fully justify that this value of $\theta\,$ gives a maximum.

[5]

bii.

Determine the exact maximum value of $A$ .

[2]

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if the total boundary length $PQ\,$ were increased to 24 metres.

[2]

1.8 E: Trigonometry

Question 52

Markscheme

1.8 E: Trigonometry

Question 52

Markscheme

1.8 E: Trigonometry

Question 52

1.8 E: Trigonometry Questions

1.8 E: Trigonometry

Question 52

1.8 E: Trigonometry Questions