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Math Help - max/min on an interval #2

  1. #1
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    max/min on an interval #2

    A swimming pool is treated periodically to control the growth of bacteria. Suppoe that t days after a treatment, the concentration of bacteria per cubic cm is C(t)=30t^9-240t+500. Find the lowest concentration of bacteria during the first week after the treatment.

    Ok.. what I did.. I think i did right, but the back of the book.. just wondering if they are wrong. Just making sure, please confirm with me.

    My work is below:
    c'(t)=60t-240

    critical numbers
    c'(t)=0 when 60t-240=0
    t=4

    domain
    0<=t<=7

    test
    f(4)=0
    f(0)=-240
    f(7)=180

    therefore the lowest concentration of bacteria during the first week after the treatment is 0 cm^3 on the 4th day

    That is my answer. However, the back of the book says.. 20
    Last edited by mr fantastic; August 12th 2009 at 03:54 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    A swimming pool is treated periodically to control the growth of bacteria. Suppoe that t days after a treatment, the concentration of bacteria per cubic cm is C(t)=30t^9-240t+500. Find the lowest concentration of bacteria during the first week after the treatment.

    C(t) = 30t^2 - 240t + 500 instead of 30t^9 ... ?

    Ok.. what I did.. I think i did right, but the back of the book.. just wondering if they are wrong. Just making sure, please confirm with me.

    My work is below:
    c'(t)=60t-240

    critical numbers
    c'(t)=0 when 60t-240=0
    t=4

    domain
    0<=t<=7

    test
    f(4)=0 C(4) = 20
    f(0)=-240 C(0) = 500
    f(7)=180 C(7) = 290

    therefore the lowest concentration of bacteria during the first week after the treatment is 0 cm^3 on the 4th day

    That is my answer. However, the back of the book says.. 20
    correct ... C(4) = 20 bacteria/cm^3
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  3. #3
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    Hello, skeske1234!

    A swimming pool is treated periodically to control the growth of bacteria.
    Suppoe that t days after a treatment,
    the concentration of bacteria per cm³ is: .  C(t)\:=\:30t^2-240t+500
    Find the lowest concentration of bacteria during the first week after the treatment.

    My work: . C'(t)\:=\:60t-240\:=\:0 \quad\Rightarrow\quad t \:=\:4

    Domain: . 0 \leq t \leq 7

    This is correct . . . the rest is wrong.
    You were testing values into the derivative, not the C-function.

    . . \begin{array}{cccc} C(0) &=& 500 \\ C(4) &=& {\color{red}20} & \text{minimum}\\ C(7) &=& 290 \end{array}


    Edit: Once again, skeeter is too fast for me . . .
    .
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  4. #4
    Super Member Matt Westwood's Avatar
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    When you've got t = 4 you need to substitute 4 for t in the original equation, not its derivative, to get the concentration at that point.
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