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Question
The surface area of a solid metallic sphere is 2464 cm2. It is melted and recast into solid right circular cones of radius 3.5 cm and height 7 cm. Calculate:
- the radius of the sphere.
- the number of cones recast. (Take π = `22/7`)
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Solution
i. Total surface area of the sphere = 4πr2, where r is the radius of the sphere.
Thus,
4πr2 = 2464 cm2
`=> 4 xx 22/7 xx r^2 = 2464`
`=>` r2 = 196
`=>` r = 14 cm
∴ R = 14 cm
ii. Volume of sphere melted = `4/3 piR^3`
= `4/3 xx pi xx 14 xx 14 xx 14`
Radius of each cone recasted = r = 3.5 cm
Height of each cone recasted = h = 7 cm
∴ Volume of each cone recasted = `1/3 pir^2h`
= `1/3 xx pi xx 3.5 xx 3.5 xx 7`
∴ Number of cones recasted
= `"Volume of sphere melted"/"Volume of each cone formed"`
= `(4/3 xx pi xx 14 xx 14 xx 14)/(1/3 xx pi xx 3.5 xx 3.5 xx 7)`
= `(4 xx 14 xx 14 xx 14)/(3.5 xx 3.5 xx 7)`
= 4 × 4 × 4 × 2
= 128
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