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The future population of a small South American country of 10 million people can be modeled by the exponential equation P(t) = P0(e)0.02t, where P(t) is the future population in t years and P0 is the current population in millions of people.

Which of the following graphs represents this model, what is the expected population of this country (rounded to the nearest tenth of a million) in 40 years, and how long will it take (rounded to the nearest year) for the current population to reach 30 million?

Graph 1, 21.7 million, 50 years
Graph 2, 4.9 million, 30 years
Graph 2, 4.5 million, the population will not reach 30 million because this graph is an exponential decay model
Graph 1, 22.3 million, 55 years

The future population of a small South American country of 10 million people can be modeled by the exponential equation Pt P0e002t where Pt is the future popula class=
The future population of a small South American country of 10 million people can be modeled by the exponential equation Pt P0e002t where Pt is the future popula class=

Respuesta :

Edufirst
p(t) = p0 [e^0.02t]

a) Graph

Notice that p(t) is a growing function

Graph 2 is of a decreasing function and graph 1 is of an increasing function, then the right choice is graph 1.

b) Population in 40 years

Wth the value t =0 you obtain p0 = 10 millions

t = 40, p (40) = 10 * [e ^ 0.02*40] = 10 [e^0.8] = 22.26

c) Time to reach 30 million

Ln = natural logarithm

p(t)/p0 = e^0.02t

p(t) = 30
p0 = 10

0.02t = Ln[30/10]

t = Ln [3]/0.02 = 54.93 years

Answer: graph 1, 22.3 million, 55 years.
 

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