Analysis and Approaches for IBDP Maths Ebook 1 | Page 53

Front Page
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Paper 2 Section B – Comparing two models
Example
The number of insects in two colonies, A and B , starts increasing at the same time.
The number of insects in colony A after t months is modeled by the function
0.3t
A( t) � 240e
.
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(a) Find the initial number of insects in colony A .
(b)
(c)
Find the number of insects in colony A after seven months.
How long does it take for the number of insects in colony A to reach
800?
The number of insects in colony B after t months is modeled by the function
B( t) � 360e
kt .
(d) After ten months, there are 1000 insects in colony B . Find the value of k .
(e)
The number of insects in colony A first exceeds the number of insects
in colony B after n months, where n� . Find the value of n .
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Solution
(a)
(b)
Initial number of insects
0.3(0)
� 240e
(A1) for substitution
� 240
A1 N2
Number of insects in colony A after seven months
0.3(7)
� 240e
(A1) for substitution
� 1959.880779
(A1) for correct working
� 1960
A1 N3
(c) At ( ) � 800
(A1) for setting equation
0.3t
240e � 800
0.3t
240e �800 � 0
0.3t
By considering the graph of y�240e
� 800 (M1) for correct working
t � 4.0132427
�It takes 4.01 months. A1 N3
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