Apps. and Interpretation for IBDP Maths Ebook 1 | Page 205

57
Paper 1 – Optimization Problems
Example
In a shop, the profit of selling x laptops can be modelled by
where 0 �x
� 10 .
(a) Find P� ( x)
.
(b)
(c)
3 2
P( x) 12x 288x 1728x
� � � ,
Hence, find the number of laptops to be sold such that the profit is maximized.
Write down the exact maximum profit.
[2]
[3]
[1]
15
Solution
(a)
2
P�( x) �12(3 x ) � 288(2 x) � 1728(1)
(A1) for correct derivative
2
�( ) � 36 �576 � 1728
A1 N2
P x x x
(b) P�( x) � 0
2
36x
576x
1728 0
� � � (M1) for setting equation
By considering the graph of
y x x
x � or
2
� 36 �576 � 1728 , 4
x � 12 (Rejected). (M1) for valid approach
Thus, the required number of laptops is 4. A1 N3
(c) $3072 A1 N1
Exercise 57
1. In a factory, the average cost of producing a titanium ring of x kilograms can be
2 54
modelled by C( x) � x � 6 � , where 1�x
� 7.
x
(a) Find C� ( x)
.
(b)
[2]
Hence, find the mass of a titanium ring to be produced such that the average cost
is minimized.
[3]
[2]
[3]
[1]
www.seprodstore.com
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