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Question
A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter 4 \[\frac{2}{3}\] cm and height 3 cm. Find the number of cones so formed.
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Solution
The radius of solid metallic sphere, `R= 28/2`= 14 cm
The volume of sphere
`= 4/3 pi R^3`
`= 4/3 xx pi xx (14)^3`
`= 4/3pi xx 14 xx 14 xx 14`
`= (10976 pi)/3 cm^3`
Given, the sphere is recast into smaller cones.
The radius of cone,
`r = 14/(3 xx 2)`
`= 7/3 cm`
The height of cone h = 3 cm
Let n be the no. of smaller cones.
Clearly, the volume of solid sphere = n × volume of one smaller cone
\[\frac{10976}{3}\pi = n \times \frac{1}{3}\pi \times \left( \frac{7}{3} \right)^2 \times 3\]
\[n \times \frac{49}{3} = 10976\]
\[n = \frac{10976 \times 3}{49}\]
\[n = 672\]
Thus, the no. of smaller cones = 672
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