a. The population of France in 1950 was about 41.7 million, while in 1970, it wa
ID: 2884117 • Letter: A
Question
a. The population of France in 1950 was about 41.7 million, while in 1970, it was about 50.7 million. Let P_n denote the population of France at time n, where P_n is given in millions of people and n is given in decades since 1950. Suppose P_n is growing according to the discrete growth equation P_n + 1 = (1 + r)P_n, with P_0 = 41.7 (i.e., the population in 1950). Use the population in 1970 (P_2) to find the value of r. r = _____. Determine how many years it takes for this population to double. Doubling time = _____ years. b. Estimate the population in 2000 based on this model. Population in 2000 is _____ million. Given that the population in 2000 was 59.4 million, find the percent error between the actual and predicted values. Percent Error is _____ per cent. c. A better model fitting the census data for France is a Logistic growth model given by P_n + 1 = F(P_n) = 1.278P_n - 0.00414P^2_n, where again n is in decades after 1950. If P_0 = 41.7, then use this model to predict the populations in 1960 and 1970. Population in 1960 is _____ million. Population in 1970 is _____ million. d. Find all equilibria for this logistic model for France. (P_1eExplanation / Answer
From the given question,
Pn+1=(1+r)Pn
P0=41.7, P2=50.7
P1=(1+r)P0
P2=(1+r)P1=(1+r)2P0
P2/P0= (1+r)2
(50.7/41.7)=(1+r)2
1.1026=1+r
r=0.1026
2P0=(1+r)nP0
(1+r)n= 2
nlog(1.1026)=log 2
n=7.096 years
doubling time=7.1 years
for 2000, n=6
P=(1+0.1026)6(41.7)
P=74.93 million
predicted value is 74.93 million.
Actual value is 59.4 million.
percenage error= (74.93 -59.4)* 100/74.96
=20.71 %
percentage error is 20.71 %
c) better model
Pn+1= 1.278 Pn- 0.00414Pn2
P0=41.7
P1=1.278 P0- 0.00414P02
=1.278 (41.7)- 0.00414(41.7)2
=46.09
population in 1960 is 46.1 million.
P2=1.278 P1- 0.00414P12
=1.278 (46.09)- 0.00414(46.09)2
=50.11
population in 1970 is 50.1 million.
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