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Help with a volume question calculus style

  1. Jun 4, 2004 #1
    Help with a volume question....calculus style

    This is a problem solving question, so it is worded. I have cut off the first part which just explains the situation

    We have a dam that opens to let water out to flood the plains. When a volume of 300 megalitres has passed through the gates, they automatically close. Within this volume limit, the rate of flow of water (R)
    (in megalitres/minute) can be modelled by the following equation:

    R = cos 0.001t, where t is the time in minutes.

    If the gates open at midnight, at what time (to the nearest minute) will they automatically close?

    This is causing me some considerable trouble, as i am only in grade 12. Any help with this would be greatly appreciated, thankyou
     
  2. jcsd
  3. Jun 4, 2004 #2

    AKG

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    Seeing as how you're in grade 12, this should be a cinch ;). It's a pretty simple problem, so I think most hints would give it away, but here it goes (let t=0 at midnight):

    [tex]R = \cos (0.001t)[/tex]

    [tex]\frac{dV}{dt} = \cos (0.001t) [/tex]

    [tex]dV = \cos(0.001t) dt[/tex]

    [tex]\int_{V_0} ^{V_{f}} dV = \int_{0} ^{t_{f}} \cos(0.001t) dt[/tex]

    [tex]300 = 1000\sin (0.001t) |_{0} ^{t_{f}} [/tex]

    [tex]0.3 = \sin (0.001t_{f})[/tex]

    [tex]t_f = 1000 \arcsin (0.3)[/tex]

    Based on this you can figure out when it closes.
     
    Last edited: Jun 4, 2004
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