Question

A cone,a hemisphere and a cylinder stand on equal bases and have the same height.Show that their volumes are in the ratio 1:2:3.

Answer 1

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]

Answer 2

the formulas for the volumes of the solids are :

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

Hemisphere:
    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