If the curved surface area of a toy of shaped a right circular cone of height 24 cm is 550 sq-cm , then what be the volume of the toy ?

Explanation

Let the radius of base of the toy be r cm and its slant height be l cm.
Then `l^2=r^2+24^2[ because "height"= 24 "cm"] rArr l^2=r^2+576rArr l =sqrt(r^2+576)`.
The curved surface area of the toy `=550` sq-cm.
`therefore (22)/(7)xxr xx l =550 or , r xx sqrt(r^2+576)=(550xx7)/(22)or , rxxsqrt(r^2+576)=25xx7`
` or , r^4+(625-49)r^2-625xx49=0 or r^4 +625r^2-49r^2- 625xx49=0`
`or , r^2(r^2+625)-49(r^2+625)=0 or , (r^2+625)(r^2-49)=0`
`therefore` either `r^2+625=0`, which is impossible , or `r^2-49=0rArr r^2= 7^2rArr r =7`.
So, the radius of the base of the toy = 7cm.
`therefore` the volume of the toy `=(1)/(3)xx(22)/(7)xx7^2xx24` cubic -cm `=(1)/(3)xx(22)/(7)xx49xx24 c c =1232c c `.
Hence the required volume of the toy `=1232c c `.