Analysis and Approaches for IBDP Maths Ebook 1 | Page 179

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2. The following diagram shows a circle with centre O , diameter AB and radius 10. Let C
be a point on the circumference with BOC ˆ
�
� � radians, where 0 ��
� . A rectangle is
2
inscribed in the circle. Let P be the total area of the shaded regions.
14
(a) Show that P �100( � � 2sin 2 �)
.
[4]
(b) Find the value of � when P is a local minimum, justifying that it is a minimum.
[6]
(c) Find the minimum value of P in terms of � .
[2]
(d) Write down the values of � for which P has its greatest value.
[2]
3 2
3. The temperature Q of a substance at time t is given by Q( t) � t �12t � 36t
, where
t � 0 .
(a) Find the t -intercepts of Q .
[3]
(b) Find the value of t when Q is a local maximum, justifying that it is a maximum.
[6]
(c) Find the minimum value of Q .
[2]
(d) Let R( t) �Q( t) � 20 . Write down the minimum value of R .
[2]
3 2
4. The price P of a share at time t is given by P( t) � �t �9t � 24t
� 720 , where
0 �t
� 12 .
(a) Show that the t -intercept of P is 12.
[2]
(b) Find the value of t when P is a local minimum, justifying that it is a minimum.
[6]
(c) Given that P(4) � 704 , find the maximum value of P .
[3]
(d) Let Q( t) �P( t � 3) . Write down the maximum value of Q .
[2]
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