Let the radius of base be r

So the vol. of the cone =  \frac{1}{3}  \pi  r^{2}  x r = \frac{1}{3}  \pi r^{3}

Vol. of hemisphere= \frac{1}{2} X  \frac{4}{3} X  \pi X  r^{3}

Vol.of the cylinder:  [tex]\\Ratio  : \frac{1}{3} \pi r^{3} : \frac{2}{3} X \pi X r^{3} :  \pi r^{2}r=\pi r^{3} \\   3 X\frac{1}{3} \pi r^{3} :   3 X  \frac{2}{3} X \pi X r^{3} : 3  X \pi r^{2}r=\pi r^{3}   = 1:2:3[/tex]


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The formulas for the volumes of the solids are :

   V =  1/3 π R² H:   R = radius of the base  and   H  = the height of the cone
   Base = π R²

    V =  2/3 π  R³        , R = radius
    Base = π R²

Cylinder :
    V = π R² H     , R = radius  and  H is the height
     Base = π R²

All of them have same base =>  Same Radius.   They have same height too.  The height of a hemisphere is same as the radius.  Hence,  R = H.

So ratio of their volumes =  1/3 π R² H : 2/3 π R³ :  π R² H 
                        = H : 2 R : 3 H
                        = 1 : 2 : 3

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