What changes will be occured in the height of a right circular cone if its radius of base is decreased by `50%` and its volaume be decreased by `25%` ?

Explanation

Let the radius of base of the cone be r units and height be h units.
Now , if the volume of the cone be V cubic - units, then `V=(1)/(3)pi r^2h`.............(1)
Now , let the volume of the cone be `V_1` cubic - unit and height be `h_1` units , when its radius of base is decreased by `50%` and volume is decreased by `25%` .
`therefore V_1=(1)/(3)pi (r-r xx(50)/(100))^2xxh_1=(1)/(3)pi (r-(r)/(2))^2xxh_1=(1)/(3)pi xx(r^2)/(4)xxh_1`
i.e., `V-Vxx(25)/(100) =(1)/(3) pi xx (r^2)/(4)xxh_1 or , (3V)/(4)=(1)/(3)pi xx (r^2)/(4)h_1 or , 9V=pi r^2h_1`
or, `9xx(1)/(3)pi r^2h=pi r^2 h_1 ` [ by (1)] or, `3h=h_1 therefore h_1=3h`
`therefore` the percentage of change in height `=(3h-h)/(h)xx100%=(2h)/(h)xx100%=200%`
Hence the required increase in height in present of the cone `=200%` .