The volume of a sphere with radius r cm decreases at a rate of 22 cm /s  .  Find the rate of change of r when r =3 cm?

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  • 2 months ago

    If you cant's do your own DE, then drop the class and get a refund if you still can.  How did you manage to get by pre-calculus?

  • ?
    Lv 7
    2 months ago

    Write V as a function of r.

    Calculate dV/dt as a function of r and dr/dt.

    Fill in the known values for dV/dt and r. Solve for dr/dt.

  • zipper
    Lv 7
    2 months ago

    Excuse me!  BUT ME DOES NOT CARE TO DO YOUR WORK FOR YOU! You will never learn anything that way!

  • ?
    Lv 7
    2 months ago

    Mistake: it should be "22 cm^3/s".

    Solution:

    4pi(r^3)/3=V

    =>

    4pi(r^2)dr/dt=dV/dt

    =>

    4pi(3^2)dr/dt=-22

    =>

    dr/dt=-11/(18pi)~ -0.19 cm/s.

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  • 2 months ago

    The volume changes at 22cm³/s or is that (22cm/s)³ ? Meaningless.

  • Ian H
    Lv 7
    2 months ago

    dV/dt = dV/dr*dr/dt = 4πr^2 dr/dt

    dr/dt = 22/(4π*3^2) = 11/(18π) ~ 0.1945 cm/s

  • Jeremy
    Lv 6
    2 months ago

    Volume of the sphere as a function of the radius r:

    V(r) = 4/3 * π * r³

    Rate of change of the volume = First derivative of V as a function of time (t)

    dV/dt = 4/3 * π * 3r² * dr/dt <=== Note: r = 3 cm; dV/dt = 22 cm³/s

    22 = 4/3 * π * 3(3²) * dr/dt

    22 = 36π * dr/dt

    ===> dr/dt = 22 / (36π) ≃ 0.1945 cm/s (to 4 d.p.)

    The radius decreases by 0.1945 cm/s

  • 2 months ago

    V = (4/3)πr³

    so, dV/dt = 4πr².dr/dt

    Then, with dV/dt = -22 and r = 3 we have:

    -22 = 4π(3)².dr/dt

    i.e. dr/dt = -11/18π => -0.19

    Hence, the radius decreases at a rate of 0.19 cm/s

    Note: the volume is decreasing at 22 cm³/s

    :)>

  • 2 months ago

    dV/dt = d/dt((4/3)πr³) = (4/3)π d/dt(r³) =

    =  (4/3)π d/dt(r³) * (dr/dr)

    =  (4/3)π d/dr(r³) * (dr/dt)

    =  (4/3)π * (3r²) * (dr/dt)

    =  4 r² π * (dr/dt)

    (dr/dt) = (dV/dt) / (4 r² π)

    where dV/dt=-22 cm³/s

    The rate of change of r when r =3 cm is

    (dr/dt) = -22 / (4*3² π) = -11/(18π) = -0.1945... cm/s

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