calculus

Find the maximum volume of right circular cylinder that can be inscribed in a cone of altitude 12 cm and base radius 4 cm, if the axes of the cylinder and con coincide.

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  1. Try to make a sketch of a cylinder inside a cone
    Draw in the altitude, let the height be h
    let the radius of the cylinder be r
    Look at a cross section of the diagram.
    the altitude from the top of the cylinder to the vertex of the cone is 12-h
    and by similar triangles
    (12-h)/r = 12/4 = 3/1
    3r = 12-h
    h = 12-3r

    V(cylinder) = πr^2 h
    = πr^2 (12-3r)
    = 12πr^2 - 3πr^3
    dV/dr = 24πr - 9πr^2 = 0 for a max of V
    3πr(8 - 3r) = 0
    r = 0 , clearly yielding a minimum Volume
    or
    r = 8/3

    max V = ....

    (you do the button pushing)

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