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  1. #1
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    maximum volume that can fit in a cone...

    Here is a calculus problem, I've been trying to solve but I haven't been able to, please help me! thanks

    A right circular cone has a height of 10 and a radius of 3. find the dimensions of the cylinder with maximum volume that can fit inside this cone such that the base of the cylinder lies on the base of the cone. Also give maximum volume and verify that the volume is a maximum.
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    Quote Originally Posted by panywff View Post
    Here is a calculus problem, I've been trying to solve but I haven't been able to, please help me! thanks

    A right circular cone has a height of 10 and a radius of 3. find the dimensions of the cylinder with maximum volume that can fit inside this cone such that the base of the cylinder lies on the base of the cone. Also give maximum volume and verify that the volume is a maximum.
    See attachment.

    The volume of a cylinder is calculated by:

    V_{cyl} = \pi \cdot r^2 \cdot h

    According to my sketch you can use similar triangles.

    \dfrac h{R-r} = \dfrac HR ~\implies~ h = \dfrac{H(R-r)}{R}

    Substitute this term into the first equation:

    V(r)=\pi \cdot r^2 \cdot \dfrac{H(R-r)}{R} = -\pi \frac HR \cdot r^3 + \pi H\cdot r^2

    Calculate V'(r) and solve V'(r) = 0 for r.

    I've got r = 0~\vee~r = \frac23 R

    Since V''(r) < 0 the volume must be a maximum.
    Attached Thumbnails Attached Thumbnails maximum volume that can fit in a cone...-zylind_im_kegl.gif  
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