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 compute the populations of juveniles and adults separately using the given recurrence relation and then add them together.
Let's start with the initial population values of 4.17 million juveniles (J0) and 17.24 million adults (A0) on 1 January 2010.
To find the population of juveniles (J1) on 1 January 2011, we use the recurrence relation:
J1 = 0.9257 * J0 + 0.0208 * A0 + 0.014
Substituting the given values, we get:
J1 = 0.9257 * 4.17 + 0.0208 * 17.24 + 0.014
Calculating this, we find:
J1 ≈ 4.0133
Next, to find the population of adults (A1) on 1 January 2011, we use the recurrence relation:
A1 = 0.0661 * J0 + 0.9767 * A0 + 0.056
Substituting the given values, we get:
A1 = 0.0661 * 4.17 + 0.9767 * 17.24 + 0.056
Calculating this, we find:
A1 ≈ 17.5356
Finally, to find the total predicted population on 1 January 2011, we add the populations of juveniles and adults:
Total population ≈ J1 + A1
≈ 4.0133 + 17.5356
≈ 21.5489
Rounding this to two decimal places, we have:
Total population ≈ 21.55 million
Among the available options, the closest value to 21.55 million is option C: 21.68 million.