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Question

The surface area of a solid metallic sphere is $$2464\ cm^{2}$$. If is melted and recast into solid right circular cones of radius $$3.5\ cm$$ and height $$7\ cm$$. Calculate the number of cones recasted. (Take $$\pi = \dfrac{22}7)$$


Solution

Let the radius of sphere $$= r\ cm$$
Surface area of sphere $$= 4\pi r^{2} = 2464\ cm^{2}$$ (given)
$$r^{2} = \cfrac {2464}{4\pi}$$
$$r^{2} = \cfrac {2464 \times 7}{4\times 22} = 196$$
$$r = 14\ cm$$.
Volume of sphere $$= \dfrac {4}{3}\pi r^{3} = \dfrac {4}{3} \pi (14)^{3}\ cm^3$$
Volume of cone $$= \dfrac {1}{3}\pi r^{2}h = \dfrac {1}{3}\pi (3.5)^{2}\times 7\ cm^3$$
No. of cones recasted $$= \dfrac {\text {Volume of sphere}}{\text {Volume of cone}}$$
$$= \dfrac {\dfrac {4\pi}{3}(14)^{3}}{\dfrac {1}{3}\pi (3.5)^{2}\times 7} = \dfrac {4\times 14\times 14\times 14}{3.5\times 3.5\times 7} =128$$

The number of cones recasted is $$128$$

Mathematics

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