The variation in a population is modelled by the recurrence relation
(Jn+1 0.9257 0.0208) (Jn) + (0.014)
(An+1 0.0661 0.9767) (An) (0.056)
where Jn and An are the sizes, in millions, of the subpopulations of
juveniles and adults, respectively, n years after 1 January 2010. On that
date there were 4.17 million juveniles and 17.24 million adults. Choose the
option that gives the total population, in millions, to two decimal places,
predicted by the model for 1 January 2011.
Options
A 21.33 B 21.40 C 21.68 D 21.83
E 22.14 F 22.21 G 22.25 H 22.36
Could you please explain what is the meaning of the two numbers 0.9257 and 0.0208 in (Jn+1 0.9257 0.0208)?
To find the total population on 1 January 2011 using the given recurrence relation, we need to use the initial population values and apply the relation to calculate the new population values.
Given:
J0 = 4.17 (initial juvenile population on 1 January 2010)
A0 = 17.24 (initial adult population on 1 January 2010)
Using the recurrence relation:
Jn+1 = 0.9257Jn + 0.0208An + 0.014
An+1 = 0.0661Jn + 0.9767An + 0.056
To calculate J1, substitute n=0:
J1 = 0.9257J0 + 0.0208A0 + 0.014
To calculate A1, substitute n=0:
A1 = 0.0661J0 + 0.9767A0 + 0.056
Finally, the total population on 1 January 2011 is obtained by adding J1 and A1:
Total population = J1 + A1
Let's calculate the values:
J1 = 0.9257(4.17) + 0.0208(17.24) + 0.014
J1 ≈ 3.851169 + 0.358432 + 0.014
J1 ≈ 4.223601 (approx)
A1 = 0.0661(4.17) + 0.9767(17.24) + 0.056
A1 ≈ 0.275487 + 16.841528 + 0.056
A1 ≈ 17.173071 (approx)
Total population = J1 + A1
Total population ≈ 4.223601 + 17.173071
Total population ≈ 21.396672 (approx)
Therefore, the predicted total population on 1 January 2011, rounded to two decimal places, is approximately 21.40 million.
The correct option is B: 21.40.