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Pure Maths

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A container is in the shape of a right circular cone (inverted) with both height and diameter 2 meters. It is being filled with water at a rate of (pi)m^3 per minute. Fine the rate of change of height h of water when the container is 1/8th full(by volume).
(Volume of a right circular cone of radius r and height h is 1/3(pi)r^3h)

Step by step working please.
Thanks and God bless :)

  • Pure Maths -

    for any height h, volume is 1/3 PI r^2 h
    Notice your formula is wrong.
    but for r for any height h is r=h/2. Think on that.

    so volume=1/3 PI *(h/2)^2*h

    V=1/12 * PI * h^3
    and h= cuberoot (12V/PI) solve for h when V=1/8*1/12*PI*2 (max volume, r=1,h=2)
    Now for the calculus work.
    dV/dh=3/12 PI h^2
    but dV/dh*dh/dt=dV/dt

    or dh/dt=dV/dt / dV/dh
    you area given dV/dt as PI m^3/minute
    and you found dV/dh=3/12 PI h^2

    so figure dh/dt

  • Pure Maths -

    I understand the rest but not the 1/8th of the volume part. Can you give me a further explanation on that?

    Thanks :)

  • Pure Maths -

    Answer is dh/dt= m/min

    The volume when it's 1/8th full is Pi/12
    I don't understand how they got that....

  • Pure Maths -

    make a sketch to see that by ratios,
    r = h/2 , like bobpursely noted

    The sneaky part of the question is that when the cone is 1/8 full , the water is NOT 1/8 of the way up

    Full volume = (1/3) π (1^2)(2) = 2π/3 m^3

    we want r and h when volume = (1/8)(2π/3) or π/12
    (Again, see bobpursely above)

    Volume = V = (1/3)π(r^2)(h)
    = (1/3)π(h^2/4)(h) = (π/12) h^3

    so when cone is 1/8 full,
    (π/12) h^3 = π/12
    h^3 = 1
    h = 1 , and r = 1/2

    Now back to actual Calculus,

    V = (π/12 h^3
    dV/dh = (π/4) h^2 dh/dt

    plug in the given dV/dt = π, and h = 1
    π = (π/4) (1^2) dh/dt
    dh/dt = 1/4

    So when the cone is 1/8 full, the height is changing at
    1/4 m/minute

    We could have done the 1/8 part in our heads by realizing that ..
    The volume of two similar solids is proportional to the cube of their sides, and since
    (1/2)^3 = 1/8 ......
    the height must have been 1/2 of the 2 m of the cone, or 1 m

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