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Math Help - Instantaneous rate of change for a volume

  1. #1
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    Instantaneous rate of change for a volume

    pv=c, where p=pressure, v= volume and c=constant.

    When the volume is 6 cm^3, the pressure is 2 kg/cm^2 what is the instantaneous rate of change of the volume if the pressure decreases by 200g/cm^2 every minute?

    So instantaneous rate of change is the same as the derivative, from what I understand. Usually to find this I would multiply d/dt into the equation, find the derivatives,isolate for the variable I am looking for and sub in the given values to solve. In this instance I just don't know where c fits in. Can anyone help me figure it out?
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  2. #2
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    Re: Instantaneous rate of change for a volume

    We have v(t)=\frac{c}{p(t)}, so v'(t)=-\frac{c}{p^2(t)}p'(t)=-\frac{v(t)}{p(t)}p'(t). All three quantities are given.
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  3. #3
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    Re: Instantaneous rate of change for a volume

    I'm not really following the third step--I don't quite understand how
    -__c__ (p' (t)) = -v(t) (p'(t))
    _p^2(t)________p(t)

    I'm probably missing something obvious but I've been trying to wrap my brain around it and can't seem to . . .
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  4. #4
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    Re: Instantaneous rate of change for a volume

    -\frac{c}{p^2(t)}}=-\frac{c}{p(t)}\frac{1}{p(t)}=-v(t)\frac{1}{p(t)} because p(t)v(t)=c.
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  5. #5
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    Re: Instantaneous rate of change for a volume

    okay thanks, that makes sense!

    Now just one more question. . . I know that I sub in 6 v(t) and 2 for p (t) but where does the .2kg come in? I thought that was the value of the average rate of change, not the instantaneous rate of change...

    thanks in advance
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  6. #6
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    Re: Instantaneous rate of change for a volume

    sorry I meant .2kg/cm^2 (the change in pressure per minute)
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  7. #7
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    Re: Instantaneous rate of change for a volume

    I think that "the pressure decreases by 200g/cm^2 every minute" means that the instantaneous rate of change of pressure is 200 when measured in g / (cm^2 * min). E.g., this is the same thing as 3.33 g / (cm^2 * sec). Otherwise, it's impossible to find v'(t) because it depends on p'(t), and the average rate of change of p(t) does not determine p'(t) at some particular moment t. That is, it does not determine unless there are some additional requirements on p(t), such as that p(t) is linear, i.e., p(t) = p_0 + C(t - t_0) for some constants C and t_0. But in this case the instantaneous and the average rates of change of p(t) are the same.
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  8. #8
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    Re: Instantaneous rate of change for a volume

    Okay, thank-you. I would have thought this was not the case.

    Aside form this particular question, would there be a general way of determining whether a value such as that is an instantaneous rate of change? I guess it is a case specific issue . . . .
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  9. #9
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    Re: Instantaneous rate of change for a volume

    Yes, I think it should be said in the problem statement.
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