Question

a. Develop the model that represents the population of Chicago

b. predict the population in 2019

c. predict the population in 2024

d. predict when the population will be double its 2011 population

Relates to this question:

Exercise 5-12 deal with data from the U.S. Bureau of the Census on populations of major cities. These data allow us to find how fast the population is growing and when it will reach certain levels. Such calculations are very important, because they indicate the future needs of the population for goods and services and how well the area can support the population. The third largest city in the united states is Chicago. Its population in 2011 was 2,705 (in thousands); in 2012, it was 2715. a. convert this information to two ordered pairs (t,p), where t measures years since 2011 and p measures population in thousands. b. find △t, the change in time c. find △p, change in population d. find △p/△t, the average growth rate e. find (△p/△t)/p, the relative growth rate

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Answer 1

(a)

Given population in 2011 is 2705 and in 2012 is 2715 , since there are only points we can assume the growth to be linear . Assume 2011 as t = 0 and 2012 as t = 1 ; The points are (0,2705) and (1,2715)

slope m = (p2 - p1)(t2-t1) = (2715-2705)/(1-0) = 10/1 = 10

Equation is : (p-p1) = m(t-t1) => (p - 2705) = m(t-0) => p = 10t + 2705

(b)

the year 2019 correspons to t = 2019 - 2011 = 8

p(8) = 10*8+2705 = 80+2705 = 2785

c.

the year 2024 correspons to t = 2024 - 2011 = 13

p(8) = 13*10+2705 = 130 +2705 = 2835

d. Given p(t) = 2*p(0) = 2*2705 = 5410

we have to find t ,

Now p(t) = 10*t+2705 = 5410 => 10t = 5410 - 2705 = 2705 => t = 2705/10 => 270.5 years