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. Then, we can add Jn+1 and An+1 to get the total population.
Let's start by substituting the initial values into the recurrence relation. For 1 January 2010:
Jn = 4.17
An = 17.24
By substituting these values into the equation, we can find Jn+1 and An+1:
Jn+1 = (Jn * 0.9257) + (An * 0.0208) + 0.014
An+1 = (Jn * 0.0661) + (An * 0.9767) + 0.056
Calculating Jn+1:
Jn+1 = (4.17 * 0.9257) + (17.24 * 0.0208) + 0.014
Jn+1 ≈ 3.8589 + 0.3591 + 0.014
Jn+1 ≈ 4.232
Calculating An+1:
An+1 = (4.17 * 0.0661) + (17.24 * 0.9767) + 0.056
An+1 ≈ 0.2759 + 16.8579 + 0.056
An+1 ≈ 17.19
Finally, we add Jn+1 and An+1 to get the total population:
Total Population = Jn+1 + An+1
Total Population ≈ 4.232 + 17.19
Total Population ≈ 21.422
Rounding to two decimal places, the total population predicted by the model for 1 January 2011 is 21.42 million.
Therefore, the correct option is B) 21.40.