Question

A cone, hemisphere, and cylinder have same base, radii, and equal heights. What is the ratio of their volumes?

Answer 1

`V_("cone"):V_("hemi"):V_("cyl")::1:2:3`

Explanation:

Firstly the formulae we need

`V_("cone")=1/3pir^2h`

`r="base radius ", h="perpendicular height"`

`V_("hemi")=2/3pir^3`

which is half the volume of a sphere with radius`=r`

`V_("cyl")=pir^2h`

`r="base radius ", h="perpendicular height"`

All three solids have the same height. This means that the hemisphere determines the height which will be `r=`radius

so for all relevant solids `h=r`

`V_("cone"):V_("hemi"):V_("cyl")::1/3pir^2r:2/3pir^3:pir^2r`

`V_("cone"):V_("hemi"):V_("cyl")::1/3cancel(pir^3):2/3cancel(pir^3):cancel(pir^3)`

`V_("cone"):V_("hemi"):V_("cyl")::1/3:2/3:1`

`V_("cone"):V_("hemi"):V_("cyl")::1:2:3`