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Math Help - Finding change of volume in cylinder - related rate

  1. #1
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    Finding change of volume in cylinder - related rate

    Engine cylinder 15 cm deep, being bored out
    radius increases by .10 mm/min
    How fast does volume change when dia is 9.5 cm?

    pi(r^2)h = V
    pi(4.25)^2(15) = 851.12 cm^3

    dr/dt = .01 cm
    dv/dt = ?

    f'gh + fg'h + fgh'
    f' = pi
    g' = 2r(dr/dt)
    h' = dh/dt

    pi(4.25^2)(15) + (pi)((4.25)(0.01) + (pi)(r^2)(1) = 911.91 cm^3

    which is very wrong. Where did I go wrong? thanks.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Finding change of volume in cylinder - related rate

    Try this:

     \frac {dV}{dt} = (\frac {dV}{dr})(\frac {dr}{dt}) - assuming that h is a constant (is it?)

    V = \pi r^2h, so \frac {dV}{dr} = 2 \pi r h

    \frac {dr}{dt} = 0.1 mm/min = 0.01 cm/min.

    \frac {dV}{dt} = 2 \pi (\frac {9.5 \ cm } 2 )(15 \ cm ) \times 0.01\ cm/min = 4.47 \ cm^3/min.
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    Smile Re: Finding change of volume in cylinder - related rate

    dv/dt =dr/dt * dv/dr

    dv/dr = 2pi(h)(r) = 447.67

    dv/dt = .01 * 447.67

    ans = 4.4767 cm^3 per minute
    Last edited by MathJack; February 11th 2013 at 10:57 AM.
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  4. #4
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    Re: Finding change of volume in cylinder - related rate

    but I thought you were supposed to use a chain rule when 3 things multiply together like that
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    Re: Finding change of volume in cylinder - related rate

    no, look over the chain rule, here we did not multiply three things together, we just used the information they gave us to set up a simple equation
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: Finding change of volume in cylinder - related rate

    Well, you could use the chain rule, but it's a bit of overkill to apply to constants:

     \frac {dV}{dt} = \frac {\pi r^2 h} {dt} = \frac {d \pi} {dt} r^2 h + \pi \frac {dr^2} {dt} h + \pi r^2 \frac {dH}{dt}

    Since  \pi and h are constants their derivatives are both zero, giving you:

     \frac {dV}{dt} = \pi h \frac {dr^2}{dt} = \pi h 2 r \frac {dr}{dt}
    Last edited by ebaines; February 11th 2013 at 11:37 AM.
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  7. #7
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    Re: Finding change of volume in cylinder - related rate

    in a recent thread I was told that pi remains even though the derivative is a constant.
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  8. #8
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    Re: Finding change of volume in cylinder - related rate

    actually you got it there, I see now. Thanks
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