Calculus

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The population of a region is growing exponentially. There were 40 million people in 1980 (when t=0) and 50 million people in 1990. Find an exponential model for the population (in millions of people) at any time t, in years after 1980.

P(t)=

Predicted population in the year 2000 = _______million people.

What is the doubling time?
Doubling time = _______ years.

  • Calculus -

    Since one of the questions deals with "doubling time", I will use 2 as the base.
    (Of course we could use any base, many mathematicians would automatically use e as the base.)

    P(t) = a(2)^(kt) , where P(t) is the population in millions, a is the beginning population, and t is the time in years.
    clearly , a = 40

    P(t) = 40(2)^(kt)
    when t=10, (1990), N = 50
    50 = 40(2)^(10k)
    1.25 = 2^(10k)
    take the ln of both sides, hope you remember your log rules
    10k = ln 1.25/ln 2
    10k = .32193
    k = .032193

    so P(t) = 40(2)^(.032193t)

    in 2000, t = 20
    P(20) = 40(2)^(.032193(20))
    = 62.5 million

    for the formula
    P(t) = a(2)^(t/d), d = the doubling time
    so changing .032193t to t/d
    = .032193t
    = t/31.06

    so the doubling time is 31.06

    another way would be to set
    80 = 40(2)^(.032193t)
    2 = (2)^(.032193t)
    .032193t = ln 2/ln 2 = 1
    t = 31.06

  • Calculus -

    Curiously, this is very close to the current population doubling time for India.

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