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Math Help - Hourly Decay Rate for Salt

  1. #1
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    Hourly Decay Rate for Salt

    The question:

    A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

    (a.1) Explain why S is an exponential function.

    My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

    (a.2) Find its hourly decay factor.

    ____

    (b) Give a formula for S.

    S =

    (I thought this would be in a a(b)^x format... but I'm not really sure at all.)
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  2. #2
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    Quote Originally Posted by MathBane View Post
    The question:

    A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

    (a.1) Explain why S is an exponential function.

    My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

    (a.2) Find its hourly decay factor.

    ____

    (b) Give a formula for S.

    S =

    (I thought this would be in a a(b)^x format... but I'm not really sure at all.)
    1. In general the equation

    A(t)=A_0 \cdot e^{k \cdot t}

    determines the amount in an exponential process, where k is a constant, t denotes the time and A_0 is the initial amount.

    2. with your problem:

    S_0 = 65 \ lbs

    After one hour you have S(1)=0.81 \cdot 65 . Use this value to get the constant k:

    0.81 \cdot 65 = 65 \cdot e^{k \cdot 1}~\implies~k = \ln(0.81) \approx - 0.21072...

    3. Combining all results you get:

    S(t) = 65 \cdot e^{\ln(0.81) \cdot t} ~\implies~\boxed{S(t) = 65 \cdot 0.81^t}

    t measured in hours.
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  3. #3
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    Quote Originally Posted by MathBane View Post
    The question:

    A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

    (a.1) Explain why S is an exponential function.

    My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

    (a.2) Find its hourly decay factor.

    ____

    (b) Give a formula for S.

    S =

    (I thought this would be in a a(b)^x format... but I'm not really sure at all.)
    You can also derive S_t from the recurrence relation

    S_{t+1} = 0.81S_t

    S_{t+2} = 0.81S_{t+1} = 0.81(0.81S_t)

    and so on until it gets to the nth stage

    S_{n(t)} = t_0 \cdot 0.81^t

    As t_0=65 we get S_{n(t)} = 65 \cdot 0.81^t
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