19 The population of the Brazil was about 63 million in 1955

19. The population of the Brazil was about 63 million in 1955 and 84.5 million in 1965. Let 1955 be represented by P0 and assume that its population is growing according to the Malthusian growth law,

Pn+1 =(1+r)Pn, where n is in years.

a. Use the data above to find the annual growth rate r, then write an expression for the population in any year following 1955. (Write the solution Pn in terms of P0 with n being the number of years after 1955.)

b. Predict the population in the year 1995. The actual population was about 162 million. What is the error between the model and the actual census data?
c. According to the model, how long until the Brazil population doubled from its 1955 level?

Solution

Pn+1=(1+r)Pn

In exponential model it can be written as

P_n= P₀e^nt

a. P₁₀=84.5 and P₀=63

84.5= 63 e^10r

84.5/63= e^10r

ln(84.5/63)/10=r

r=.02936

P_n=P₀e^.02936n

b)   P₄₀=63 e^.02936*40 = 203.88 million

Actual was 162 mllion and on using the Malthusian law we are getting 203.88 million,therefore the error is of 203.88-163= 40.88 million

c) In 1955 the population was 63 million. Double of 63 million =126 million

126=63e^.02936n

2=e^.02936n

ln2/.02936=n

n=23.6 yrs