The population of a small town is decreasing exponentially at a rate of 14.3% each year. The current population is 9,400 people. The town’s

Question

The population of a small town is decreasing exponentially at a rate of 14.3% each year. The current population is 9,400 people. The town’s tax status will change once the population is below 6,000 people. Create an inequality that can be used to determine after how many years, t, the town’s tax status will change, and use it to answer the question below.

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Aubrey 3 days 2021-10-10T18:49:24+00:00 1 Answer 0

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    2021-10-10T18:50:25+00:00

    Answer:

    After 2.9 years the town’s tax status will change

    The towns tax status change within the next 3 years

    Step-by-step explanation:

    The question below is

    Will the towns tax status change within the next 3 years ?

    Let

    y —–> the population of a small town

    t —-> the number of years

    we have a exponential function of the form

    y=a(b)^{t}

    where

    a is the initial value

    b is the base

    In this problem

    a=9,400\ people

    b=100\%-14.3\%=85.7\%=85.7/100=0.857

    substitute

    y=9,400(0.857)^{t}

    Remember that

    The town’s tax status will change once the population is below 6,000 people

    so

    The inequality that represent this situation is

    9,400(0.857)^{t}< 6,000    

    Solve for t

    (0.857)^{t}< 6,000/9,400

    Apply log both sides

    (t)log(0.857)< log(6,000/9,400)

    -0.067t< -0.1950

    Multiply by -1 both sides

    0.067t > 0.1950

    t > 2.9\ years

    so

    After 2.9 years the town’s tax status will change

    therefore

    The answer is

    Yes, the towns tax status change within the next 3 years

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