pre calc

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Water is poured into a container in the sape of a right circlar cone with raiud 4t an heigh 16 feet. Express the volue (V) of the water in the cone as a funtio of the heigh (h) of the water.
thanks

  • pre calc -

    I can not read your radius so call it R.
    I assume the point of the one is down.
    the radius of the water surface at height h is r = (16/R)h
    the area of that water surface is pi r^2
    The volume of a horizontal slice dh is
    pi r^2 dh
    which is
    pi (16/R)^2 h^2 dh

    so integrating the volume at height h is
    V(h) = pi (16/R)^2 h^3 /3

    if for example R were 4 feet then
    V(h) = pi (16/3) h^3

  • pre calc -

    or, just using similar cones, at height h, the surface of the water has a radius which can be found by

    R/16 = r/h
    r = hR/16

    v = 1/3 pi r^2 * h
    = pi/3 (hR/16)^2 * h
    = pi/3 * R^2/256 * h

    Hmmm. Better check both our maths - we disagree.

  • pre calc -

    or , assuming you are saying the radius of the cone is 4 ft,

    let the height of water be h
    and the radius of the water level be r

    r/h = 4/16 = 1/4
    r = h/4

    V = (1/3)πr^2 h
    = (1/3)π(h^2/16)h
    = (π/48)h^3

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