Analysis and Approaches for IBDP Maths Ebook 1 | Seite 177

Front Page
59
Paper 1 Section B – Optimization problems
Example
The following diagram shows a quarter of a circle with centre O and radius 2. Let P be a
point on the circumference and Q be a point on the x -axis such that PQ and OQ are
perpendicular, with POQ ˆ
�
� � radians, where 0 ��
� . Let S be the area of the shaded
2
segment.
14
(a) Find the area of the triangle OPQ in terms of � .
[3]
(b) Show that S �� � sin 2�
.
[2]
(c) Find the value of � when S is at a local minimum, justifying that it is a minimum.
[6]
(d) Find a value of � for which S has its greatest value.
[2]
Solution
(a)
(b)
The area of the triangle OPQ
1
� (OQ)(PQ)
2
(M1) for valid approach
1
� (2cos � )(2sin � )
2
(M1) for substitution
� 2sin�cos�
� sin 2�
A1 N3
S
� 1 (2)
2 sin 2
4 � �
A2
��
� sin 2�
AG N0
[3]
[2]
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