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Question
A hollow sphere of internal and external diameter 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.
Solution
External diameter = 8 cm
Therefore, radius (R) = 4 cm
Internal diameter = 4 cm
Therefore, radius (r) = 2 cm
Volume of metal used in hollow sphere = `4/3pi(R^3 - r^3)`
= `4/3 xx 22/7 xx (4^3 - 2^3)`
= `88/21(64 - 8)`
= `88/21 xx 56 cm^3` ...(i)
Diameter of cone = 8 cm
Therefore, radius = 4 cm
Let height of cone = h
∴ Volume = `1/3pir^2h`
= `1/3 xx 22/7 xx 4 xx 4 xx h`
= `352/21 h` ...(ii)
From (i) and (ii)
`352/21 h = 88/21 xx 56`
`=> h = (88 xx 56 xx 21)/(21 xx 352) = 14 cm`
Height of the cone = 14 cm.
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