Math

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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

  • Math -

    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.

  • Math correction -

    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.

  • Math -

    THank you so much

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