A solid cylinder of diamter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere and the total height of the toy, if the height conical part is thrice its radius.

Explanation

Radius of the cylinder, R = 6 cm.
Height of the cylinder, H = 15 cm
`"volume of the cyliner "=piR^(2)H`
`=(pixx6xx6xx15)cm^(3)`
`=(540pi)cm^(3)`
Let the radius of each of the conical and hemispherical be r.
Then, height of the conical part, h = 3r.
Volume of each toy
= volume of the hemisphere + volume of the cone
`=2/3pir^(3)+1/3pir^(2)h=(2/3pir^(2)h=+1/3pir^(3))cm^(3)`
`=(2/3pir^(3)+pir^(3))cm^(3)=((5pir^(3))/3)cm^(3)`
Now, total volume of 12 toys = volume of the cylinder
`rArr" "12xx(5pir^(2))/3=540pirArr=(540)/20=27rArrr=""^(3)sqrt(27)=3cm.`
Hence, radius of the hemisphere =n 3 cm.
Total height of the toy = (r+h) = (r+3r) = 4r
`=(4xx3)cm = 12 cm`.
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