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Math Help - Exponential function

  1. #1
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    Exponential function

    I'm trying to solve the following problem...

    Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college co-educational. The number of students who learned of the new program 't' hours later is given by the function...

    f(t) = 3000 / (1 + Be^{-kt})

    If 600 students on campus had heard about the new program 2 hours after the ceremony, how many students had heard about the policy after 4 hours?

    I've tried to tackle this two ways - by solving for b, and solving for k.

    I'm assuming here that f(0) = 3000 / (1 + Be^{-k[0]}) = 300 and that f(2) = 3000 / (1 + Be^{-2k}) = 600

    If f(0) = 300, then 3000 / 10 = 300. Therefore, '1 + B e^-k(0)' has to equal 10. This would mean that B = 9 (as e^0 = 1).

    If I use this value of b in the equation f(2) = 3000 / (1 + 9e^{-2k}) = 600 then try to solve for k.

    3000 / 600 = 1 + 9e^{-2k}<br />
= 4 = 9e^{-2k}<br />
= ln 4/9 = -2k<br />
= ln 4/9(-2) = k<br />
k = 1.622

    If I use this value of k in f(2) = 3000 / 1 + 9e^{-2*1.622} I get 2,220, nowhere near the 600 I should be getting.

    Instead, I tried to solve for B and k using f(2) = 600.

    f(2) = 3000 / 1 + Be^{-kt} = 600

    So now I'm assuming that '1 + Be^(-kt) is equal to 5. One of the problems I'm having is to figure out how I can find out the value of B and k in that value of 5 (or 4, if you take into account the + 1 at the start)
    Last edited by astuart; August 8th 2012 at 04:41 AM.
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  2. #2
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    Re: Exponential function

    Quote Originally Posted by astuart View Post
    I'm trying to solve the following problem...

    Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college co-educational. The number of students who learned of the new program 't' hours later is given by the function...

    f(t) = 3000 / (1 + Be^{-kt})

    If 600 students on campus had heard about the new program 2 hours after the ceremony, how many students had heard about the policy after 4 hours?

    I've tried to tackle this two ways - by solving for b, and solving for k.

    I'm assuming here that f(0) = 3000 / (1 + Be^{-k[0]}) = 300 and that f(2) = 3000 / (1 + Be^{-2k}) = 600

    If f(0) = 300, then 3000 / 10 = 300. Therefore, '1 + B e^-k(0)' has to equal 10. This would mean that B = 9 (as e^0 = 1).

    However, if I use 9 as B, when i plug that into the original equation and try and solve f(4), I get the wrong answer - about 2848, when I should be getting 1080.

    I then tried to solve f(2) = 600, then solve for B and k.

    f(2) = 3000 / (1 + Be^{-k[2]}) = 600

    I reach an issue where using 9 for B (As in the original equation) doesn't work, and I can't figure out how to isolate one variable without knowing the other..
    How can you get an answer for f(4) when you haven't evaluated k yet?
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  3. #3
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    Re: Exponential function

    Quote Originally Posted by Prove It View Post
    How can you get an answer for f(4) when you haven't evaluated k yet?
    Whoops, that's an error. I'll fix up the original post..
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    Re: Exponential function

    Fixed that up, sorry about that..
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    Re: Exponential function

    Anybody?
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  6. #6
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    Re: Exponential function

    Surely \displaystyle \begin{align*} -2k = \ln{\left(\frac{4}{9}\right)} \end{align*} means that \displaystyle \begin{align*} k = -\frac{1}{2}\ln{\left(\frac{4}{9}\right)} \end{align*}...
    Thanks from astuart
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  7. #7
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    Re: Exponential function

    Quote Originally Posted by Prove It View Post
    Surely \displaystyle \begin{align*} -2k = \ln{\left(\frac{4}{9}\right)} \end{align*} means that \displaystyle \begin{align*} k = -\frac{1}{2}\ln{\left(\frac{4}{9}\right)} \end{align*}...
    Whoops. That was meant to be ln (4/9) / -2 (or * 1/2). Don't know how I didn't pick that up this morning - I can understand how I missed it last night.. Really need to double check calculations better.

    Thanks!
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