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Related rates volume pouring in and out question

  1. Sep 29, 2005 #1
    At a rate of 10ft^3/min, water is pouring into a conical cistern that is 16ft deep and 8ft in diameter at the top. But the cistern has developed a small leak. At the same time the water is 12ft deep, the water level is observed to be rising at 1/3 ft/min. How fast is the water leaking out?

    i got .. Vout = 1.46ft^3/min
    is this answer correct? if not, what is the correct answer and why?
     
  2. jcsd
  3. Sep 29, 2005 #2
    help needed fast!!~~~ thanks anybody/.? :P
     
  4. Sep 29, 2005 #3

    lurflurf

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    Homework Helper

    That is wrong, how did you arrive at that?

    vnet=vin-vout
    vi=10
    the height of water at time t can be
    h(t)
    the radius of the water cone can be expressed in terms of height
    r(h(t))
    you know
    r(0)=0
    r(16)=4
    and
    r(t)=a+b*h(t) for real numbers a and b
    (it is a linear function)
    V(t)=pi*r(h(t))^2*h(t)/3
    you know for some time t
    h(t)=12
    h'(t)=1/3
    find
    V'(t)
    use h(t) and h'(t) to write V(t) as a number then
    vout=vin-V'(t)
     
  5. Sep 30, 2005 #4
    oh thanks alot.
    so as for the answer, i got Vout = 10-9*pi = 10-9*3.14 = -18.26
    so conclusion : water is pouring out at a rate of 18.26 ft^3/s when water is 12ft deep.
    is that correct?
     
  6. Sep 30, 2005 #5

    lurflurf

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    Homework Helper

    That should be 10-3*pi
    a negative vout would imply water is coming in
     
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