Question

According to the U.S Bureau of Census in the year 1850 the population of the US was 23,191,876 in 1900 the population was 62,947,714

A. Assuming that the population grew exponentially during the period compute
the growth constant k.

B. Assuming continued growth at the same rate predict the 1950 population

Answer 1

p =pi e^kt
let t = 0 at 1850
so pi = 23,191,876
p = 23,191,876 e^kt
in 1900 t = 1900-1850 = 50
so
62,947,714 = 23,191,876 e^50k
1.275 = e^50k
take ln of both sides
50 k = .2429
k = .00486

P at t is 100 years = 23,191,876 e^(.486)
= 23,191,876 (1.6258)
=37,705,351
I approximated all the natural logs so you should do the arithmetic more accurately although obviously the census and law are not that accurate anyway.

Answer 2

p =pi e^kt
let t = 0 at 1850
so pi = 23,191,876
p = 23,191,876 e^kt
in 1900 t = 1900-1850 = 50
so
62,947,714 = 23,191,876 e^50k
2.714 = e^50k
take ln of both sides
50 k = .9984
k = .01997

P at t is 100 years = 23,191,876 e^(1.997)
= 23,191,876 (7.367)
=170,852,745
I approximated all the natural logs so you should do the arithmetic more accurately although obviously the census and law are not that accurate anyway.

Answer 3

THank you so much