Two spheres of same metal weight 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.

Answer

`12 cm.`

Explanation

Since both the spheres are made of same metal, their weights are directly peoportional to their volumes, or the ratio of their volumes is equal to the ratio of their weightls.
Radius of first small sphere, `r_(1)=3 cm`.
Let the radius of second small sphere be `r_(2)` cm.
`:." "("volume of first sphere")/("volume of second small sphere")=1/7`
`rArr" "(4/3pir_(1)^(3))/(4/3pir_(2)^(3))=1/7rArr3^(3)/r_(2)^(3)=1/7rArr27xx7=189.`
Sum of volumes of two small spheres
`=4/3pi(r_(1)^(3)+r_(2)^(3))=[4/3pi(3^(3)+189)]cm^(3)`
`=(4/3pixx216)cm^(3)`
Let the radius of the new sphere be R.
Volume of new sphere = sum of volumes of two small spheres
`rArr" "4/3piR^(3)=4/3pixx216rArrR^(3)=216`
`rArr" "R=""^(3)sqrt(216)=6 cm rArr"diameter"=2R^(3)=12 cm.`