Results 1 to 3 of 3

Thread: Hourly Decay Rate for Salt

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    24

    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 $\displaystyle a(b)^x$ format... but I'm not really sure at all.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,854
    Thanks
    138
    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 $\displaystyle a(b)^x$ format... but I'm not really sure at all.)
    1. In general the equation

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

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

    2. with your problem:

    $\displaystyle S_0 = 65 \ lbs$

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

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

    3. Combining all results you get:

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

    t measured in hours.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    -1
    e^(i*pi)'s Avatar
    Joined
    Feb 2009
    From
    West Midlands, England
    Posts
    3,053
    Thanks
    1
    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 $\displaystyle a(b)^x$ format... but I'm not really sure at all.)
    You can also derive $\displaystyle S_t$ from the recurrence relation

    $\displaystyle S_{t+1} = 0.81S_t$

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

    and so on until it gets to the nth stage

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

    As $\displaystyle t_0=65$ we get $\displaystyle S_{n(t)} = 65 \cdot 0.81^t$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Decay Rate
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 2nd 2010, 02:19 PM
  2. Decay rate
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Nov 12th 2009, 10:23 AM
  3. decay rate of drug
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Jul 12th 2009, 06:22 PM
  4. Hourly rate.
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Jun 18th 2009, 03:53 PM
  5. Regular Hourly Rate
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Jun 12th 2007, 03:30 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum