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math - algebra

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f(t) is an exponential function with
f(0) = 200 and f(3) = 2000.

a) Use the form f(t) = Pe^rt to find the equation of this exponential function.

b) In the form f(x) = Pa^x, a is the growth factor. find the growth factor.

c)What input value gives an output value of 5,000?

  • math - algebra - ,

    F(t) = Pe^rt

    a. F(0) = P*e^0 = 200,
    P*e^0 = 200,
    P*1 = 200,
    P = 200.

    F(3) = 200*e^3r = 2000,
    200e^3r = 2000,
    Divide both sides by 200:
    e^3r = 10,
    Take Ln of both sides:
    3r*Lne = Ln10,
    3r = Ln10 / Lne = 2.30259 / 1 = 2.3026,
    r = 0.7675.

    Eq: F(t) = 200e^(0.7675t).

    b. F(x) = 200a^x.
    F(3) = 200a^3 = 2000,
    200a^3 = 2000,
    a^3 = 10,
    a = 2.15443.

    Eq: F(x) = 200(2.15443)^x

    c. F(x) = 200a^x = 5000,
    200a^x = 5000,
    a^x = 25,
    Take Log of botn sides:
    x*Loga = Log25,
    X = Log25 / Loga=Log25 / Log(2.15443),

    X = 4.194.

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