Results 1 to 3 of 3

Math Help - Depleting Volume

  1. #1
    Newbie
    Joined
    Mar 2014
    From
    Jupiter
    Posts
    6

    Depleting Volume

    Hi,

    I am trying to figure out how an answer from my textbook was worked out. The textbook contains MANY ridiculous errors, so I would like to get a second opinion from someone here regarding the accuracy of the solution. It is possible that I just do not understand the logic, but I want to be sure before I go wasting any more time on it. Here is the question:

    Code:
    A plastic juice bottle holds 2 L of liquid. In an experiment, a small hole is drilled in the bottom of the bottle. The volume of liquid, V, remaining after t seconds can be modelled by V(t) = 2 - t/5 + t^2/200, where t >= 0. How long does it take for the 2L of liquid to drain from the bottle?
    The solution provided is as follows:

    Code:
    V(t) = 2 - t/5 + t^2/200
    
    V(t) = 2000 - 200t + 5t^2
    
    V(t) = 5(400 - 40t + t^2)
    
    V(t) = 5(20 - t)(20 + t)
    
    20 - t = 0
    
    20 = t
    
    It takes 20 seconds to drain the bottle.
    I do not understand how they are getting to step #2 in their solution and I would really appreciate some clarification. I have been racking my brain all day on other problems so it is possible I am overlooking something really obvious at this point.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123

    Re: Depleting Volume

    Quote Originally Posted by VinH View Post
    Hi,

    I am trying to figure out how an answer from my textbook was worked out. The textbook contains MANY ridiculous errors, so I would like to get a second opinion from someone here regarding the accuracy of the solution. It is possible that I just do not understand the logic, but I want to be sure before I go wasting any more time on it. Here is the question:

    Code:
    A plastic juice bottle holds 2 L of liquid. In an experiment, a small hole is drilled in the bottom of the bottle. The volume of liquid, V, remaining after t seconds can be modelled by V(t) = 2 - t/5 + t^2/200, where t >= 0. How long does it take for the 2L of liquid to drain from the bottle?
    The solution provided is as follows:

    Code:
    V(t) = 2 - t/5 + t^2/200
    
    V(t) = 2000 - 200t + 5t^2
    
    V(t) = 5(400 - 40t + t^2)
    
    V(t) = 5(20 - t)(20 + t)
    
    20 - t = 0
    
    20 = t
    
    It takes 20 seconds to drain the bottle.
    I do not understand how they are getting to step #2 in their solution and I would really appreciate some clarification. I have been racking my brain all day on other problems so it is possible I am overlooking something really obvious at this point.

    Thank you.
    Hello,

    V(t) = 2 - t/5 + t^2/200 | *1000 to get the volume in cm³

    V(t) = 2000 - 200t + 5t^2

    But much more funny is this transformation:

    V(t) = 5(400 - 40t + t^2)

    V(t) = 5(20 - t)(20 + t) --> this is definitely wrong, it should read

    V(t) = 5(20 - t)^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470

    Re: Depleting Volume

    The first step is just silly! Earboth correctly deduced that they must have converted from liters to cubic centimeters but I would object to using "V(t)" to mean both "volume in liters" and "volume in cubic centimeters". Personally, I would not have made that conversion. Given that V(t)= 2- t/5+ t^2/200.
    When all 2 liters have drained, V(t) will be equal to 0 so we need to solve the equation 2- t/5+ t^2/200= 0.

    Now multiply both side by 200 to get rid of the fractions: 400- 40t+ t^2= (t- 20)^2= 0 (which is what they have after factoring "5" out).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 18th 2011, 01:08 PM
  2. volume flux, mass flux and volume flo
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: June 7th 2011, 06:30 PM
  3. Replies: 1
    Last Post: May 14th 2010, 04:08 PM
  4. divergence = flux / volume is independant of the limiting volume
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: April 26th 2010, 07:31 PM
  5. Volume
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 3rd 2009, 10:17 PM

Search Tags


/mathhelpforum @mathhelpforum