The population of Waterville increased 12% during 4 years. How many years are required for the population to double its initial value?

Question

The population of Waterville increased 12% during 4 years. How many years are required for the population to double its initial value?

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Remi 3 weeks 2022-01-03T01:36:09+00:00 1 Answer 0 views 0

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    2022-01-03T01:37:37+00:00

    Answer:

    24.5 years will be required for the population to double its initial value.

    Step-by-step explanation:

    The population of Waterville can be modeled by the following equation.

    P(t) = P_{0}(1 + r)^{t}

    In which P_{0} is the initial population and r is the growth rate.

    The population of Waterville increased 12% during 4 years.

    This means that P(4) = 1.12P_{0}

    With this, we can find r

    P(t) = P_{0}(1 + r)^{t}

    1.12P_{0} = P_{0}(1 + r)^{4}

    (1+r)^{4} = 1.12

    Applying the fourth root to both sides

    1 + r = 1.0287

    So

    P(t) = P_{0}(1.0287)^{t}

    How many years are required for the population to double its initial value?

    This is t when P(t) = 2P_{0}

    So

    P(t) = P_{0}(1.0287)^{t}

    2P_{0} = P_{0}(1.0287)^{t}

    (1.0287)^{t} = 2

    How we find t?

    Logarithims

    We have that

    \log{a}^{t} = t*\log{a}

    So we apply log to both sides

    \log{(1.0287)^{t}} = \log{2}

    t\log{1.0287} = \log{2}

    t = \frac{\log{2}}{\log{1.0287}}

    t = 24.5

    24.5 years will be required for the population to double its initial value.

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