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A country's population in 1990 was 37 million. in 1999 it was 40 million. estimate the population in 2016 using the exponential growth formula. round your answer to the nearest million

User Cphill
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2 Answers

3 votes
1. The exponential growth formula that predicts the growth of population as a function of years y is:


P(y)=P_0 e^(ky)

where
P_0 is the initial population, and k is the growth rate.

2. So we predict the population in 2016 to be
P(2016)=P_o e^(2016k)

We need to find P_o, and also the growth factor k.



P(1990)=37million=P_o e^(1990k)


P(1999)=40million=P_o e^(1999k)


P_0= (40million)/(e^(1999k)) =(37million)/(e^(1990k))


(e^(1999k))/(e^(1990k))= (40million)/(37million)= 1.081


e^((1999k-1990k)) = e^(9k) = ( e^(9) )^(k) = 1.081

e is approximately 2.718, multiply it 9 times to get e^9: 8095.53

so
8095.53 ^(k)=1.081

then

k=log_8_0_9_5_._5_31.081 whose value is 0.0087, using a scientific calculator or logarithm calculator online.

Now,

37million=P_o e^(1990k)

so
P_0 = (37 million)/(e^(1990k))

3.
P(2016)=P_o e^(2016k)= (37million)/(e^(1990k)) *e^(2016k)=37million*e^((2016-1990)k)


=37 million* e^(26*0.0087) =37 million* e^(0.2262) =37*1.254million

=46.35 million
User Forefinger
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7.1k points
5 votes
This is the concept of application of exponential growth, suppose in 1990 time,t=0 and in 1990 time,t=9. Using the exponential growth formula given by:
f(t)=ae^(kt)
thus substituting the value we get:
40=37e^(9k)
this can be written as:
(40/37)=e^(9k)
introducing the natural logs we get:
ln(40/37)=9k
hence;
k=1/9ln(40/37)=0.0087
Therefore our formual will be given by:
f(t)=37e^(0.0087t)
N/B: The population is in millions. Thus to get the population in 2016 we shall proceed as follows;
t=26
thus
f(t)=37e^(26*0.0087)
f(t)=46.35 million

User Liam Allan
by
7.6k points