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
To find the total population predicted by the model for 1 January 2011, we need to find the values of Jn+1 and An+1 using the given recurrence relation and initial values.
Given:
J0 = 4.17 million (initial size of juveniles)
A0 = 17.24 million (initial size of adults)
To find J1, we substitute n = 0 into the recurrence relation:
J1 = (J0 * 0.9257) + (A0 * 0.0208) + 0.014
Similarly, to find A1:
A1 = (J0 * 0.0661) + (A0 * 0.9767) + 0.056
Now we plug in the values for J0 and A0:
J1 = (4.17 * 0.9257) + (17.24 * 0.0208) + 0.014
= 3.851769 + 0.358592 + 0.014
= 4.224361
A1 = (4.17 * 0.0661) + (17.24 * 0.9767) + 0.056
= 0.275307 + 16.845008 + 0.056
= 17.176371
The total population is the sum of J1 and A1:
Total population = J1 + A1
= 4.224361 + 17.176371
= 21.400732
Rounded to two decimal places, the total population predicted by the model for 1 January 2011 is 21.40 million.
Therefore, the correct option is B) 21.40.